1,1,44,57,0.0958712,"\int x^2 \sin \left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x^2*Sin[a + b*Log[c*x^n]],x]","-\frac{x^3 \left(b n \cos \left(a+b \log \left(c x^n\right)\right)-3 \sin \left(a+b \log \left(c x^n\right)\right)\right)}{b^2 n^2+9}","\frac{3 x^3 \sin \left(a+b \log \left(c x^n\right)\right)}{b^2 n^2+9}-\frac{b n x^3 \cos \left(a+b \log \left(c x^n\right)\right)}{b^2 n^2+9}",1,"-((x^3*(b*n*Cos[a + b*Log[c*x^n]] - 3*Sin[a + b*Log[c*x^n]]))/(9 + b^2*n^2))","A",1
2,1,44,57,0.0686328,"\int x \sin \left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x*Sin[a + b*Log[c*x^n]],x]","-\frac{x^2 \left(b n \cos \left(a+b \log \left(c x^n\right)\right)-2 \sin \left(a+b \log \left(c x^n\right)\right)\right)}{b^2 n^2+4}","\frac{2 x^2 \sin \left(a+b \log \left(c x^n\right)\right)}{b^2 n^2+4}-\frac{b n x^2 \cos \left(a+b \log \left(c x^n\right)\right)}{b^2 n^2+4}",1,"-((x^2*(b*n*Cos[a + b*Log[c*x^n]] - 2*Sin[a + b*Log[c*x^n]]))/(4 + b^2*n^2))","A",1
3,1,40,52,0.0546183,"\int \sin \left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[Sin[a + b*Log[c*x^n]],x]","\frac{x \left(\sin \left(a+b \log \left(c x^n\right)\right)-b n \cos \left(a+b \log \left(c x^n\right)\right)\right)}{b^2 n^2+1}","\frac{x \sin \left(a+b \log \left(c x^n\right)\right)}{b^2 n^2+1}-\frac{b n x \cos \left(a+b \log \left(c x^n\right)\right)}{b^2 n^2+1}",1,"(x*(-(b*n*Cos[a + b*Log[c*x^n]]) + Sin[a + b*Log[c*x^n]]))/(1 + b^2*n^2)","A",1
4,1,38,19,0.0283023,"\int \frac{\sin \left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Sin[a + b*Log[c*x^n]]/x,x]","\frac{\sin (a) \sin \left(b \log \left(c x^n\right)\right)}{b n}-\frac{\cos (a) \cos \left(b \log \left(c x^n\right)\right)}{b n}","-\frac{\cos \left(a+b \log \left(c x^n\right)\right)}{b n}",1,"-((Cos[a]*Cos[b*Log[c*x^n]])/(b*n)) + (Sin[a]*Sin[b*Log[c*x^n]])/(b*n)","A",1
5,1,40,57,0.0707338,"\int \frac{\sin \left(a+b \log \left(c x^n\right)\right)}{x^2} \, dx","Integrate[Sin[a + b*Log[c*x^n]]/x^2,x]","-\frac{\sin \left(a+b \log \left(c x^n\right)\right)+b n \cos \left(a+b \log \left(c x^n\right)\right)}{b^2 n^2 x+x}","-\frac{\sin \left(a+b \log \left(c x^n\right)\right)}{x \left(b^2 n^2+1\right)}-\frac{b n \cos \left(a+b \log \left(c x^n\right)\right)}{x \left(b^2 n^2+1\right)}",1,"-((b*n*Cos[a + b*Log[c*x^n]] + Sin[a + b*Log[c*x^n]])/(x + b^2*n^2*x))","A",1
6,1,44,57,0.0695672,"\int \frac{\sin \left(a+b \log \left(c x^n\right)\right)}{x^3} \, dx","Integrate[Sin[a + b*Log[c*x^n]]/x^3,x]","-\frac{2 \sin \left(a+b \log \left(c x^n\right)\right)+b n \cos \left(a+b \log \left(c x^n\right)\right)}{x^2 \left(b^2 n^2+4\right)}","-\frac{2 \sin \left(a+b \log \left(c x^n\right)\right)}{x^2 \left(b^2 n^2+4\right)}-\frac{b n \cos \left(a+b \log \left(c x^n\right)\right)}{x^2 \left(b^2 n^2+4\right)}",1,"-((b*n*Cos[a + b*Log[c*x^n]] + 2*Sin[a + b*Log[c*x^n]])/((4 + b^2*n^2)*x^2))","A",1
7,1,61,97,0.1592771,"\int x^2 \sin ^2\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x^2*Sin[a + b*Log[c*x^n]]^2,x]","\frac{x^3 \left(-6 b n \sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)-9 \cos \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+4 b^2 n^2+9\right)}{6 \left(4 b^2 n^2+9\right)}","\frac{3 x^3 \sin ^2\left(a+b \log \left(c x^n\right)\right)}{4 b^2 n^2+9}-\frac{2 b n x^3 \sin \left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{4 b^2 n^2+9}+\frac{2 b^2 n^2 x^3}{3 \left(4 b^2 n^2+9\right)}",1,"(x^3*(9 + 4*b^2*n^2 - 9*Cos[2*(a + b*Log[c*x^n])] - 6*b*n*Sin[2*(a + b*Log[c*x^n])]))/(6*(9 + 4*b^2*n^2))","A",1
8,1,57,98,0.1215044,"\int x \sin ^2\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x*Sin[a + b*Log[c*x^n]]^2,x]","\frac{x^2 \left(-b n \sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)-\cos \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+b^2 n^2+1\right)}{4 b^2 n^2+4}","\frac{x^2 \sin ^2\left(a+b \log \left(c x^n\right)\right)}{2 \left(b^2 n^2+1\right)}-\frac{b n x^2 \sin \left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{2 \left(b^2 n^2+1\right)}+\frac{b^2 n^2 x^2}{4 \left(b^2 n^2+1\right)}",1,"(x^2*(1 + b^2*n^2 - Cos[2*(a + b*Log[c*x^n])] - b*n*Sin[2*(a + b*Log[c*x^n])]))/(4 + 4*b^2*n^2)","A",1
9,1,56,88,0.0899777,"\int \sin ^2\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[Sin[a + b*Log[c*x^n]]^2,x]","\frac{x \left(-2 b n \sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)-\cos \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+4 b^2 n^2+1\right)}{8 b^2 n^2+2}","\frac{x \sin ^2\left(a+b \log \left(c x^n\right)\right)}{4 b^2 n^2+1}-\frac{2 b n x \sin \left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{4 b^2 n^2+1}+\frac{2 b^2 n^2 x}{4 b^2 n^2+1}",1,"(x*(1 + 4*b^2*n^2 - Cos[2*(a + b*Log[c*x^n])] - 2*b*n*Sin[2*(a + b*Log[c*x^n])]))/(2 + 8*b^2*n^2)","A",1
10,1,36,39,0.0648552,"\int \frac{\sin ^2\left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Sin[a + b*Log[c*x^n]]^2/x,x]","-\frac{\sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)-2 \left(a+b \log \left(c x^n\right)\right)}{4 b n}","\frac{\log (x)}{2}-\frac{\sin \left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{2 b n}",1,"-1/4*(-2*(a + b*Log[c*x^n]) + Sin[2*(a + b*Log[c*x^n])])/(b*n)","A",1
11,1,57,95,0.1106563,"\int \frac{\sin ^2\left(a+b \log \left(c x^n\right)\right)}{x^2} \, dx","Integrate[Sin[a + b*Log[c*x^n]]^2/x^2,x]","\frac{-2 b n \sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+\cos \left(2 \left(a+b \log \left(c x^n\right)\right)\right)-4 b^2 n^2-1}{2 \left(4 b^2 n^2 x+x\right)}","-\frac{\sin ^2\left(a+b \log \left(c x^n\right)\right)}{x \left(4 b^2 n^2+1\right)}-\frac{2 b n \sin \left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{x \left(4 b^2 n^2+1\right)}-\frac{2 b^2 n^2}{x \left(4 b^2 n^2+1\right)}",1,"(-1 - 4*b^2*n^2 + Cos[2*(a + b*Log[c*x^n])] - 2*b*n*Sin[2*(a + b*Log[c*x^n])])/(2*(x + 4*b^2*n^2*x))","A",1
12,1,58,98,0.1094782,"\int \frac{\sin ^2\left(a+b \log \left(c x^n\right)\right)}{x^3} \, dx","Integrate[Sin[a + b*Log[c*x^n]]^2/x^3,x]","-\frac{b n \sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)-\cos \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+b^2 n^2+1}{4 x^2 \left(b^2 n^2+1\right)}","-\frac{\sin ^2\left(a+b \log \left(c x^n\right)\right)}{2 x^2 \left(b^2 n^2+1\right)}-\frac{b n \sin \left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{2 x^2 \left(b^2 n^2+1\right)}-\frac{b^2 n^2}{4 x^2 \left(b^2 n^2+1\right)}",1,"-1/4*(1 + b^2*n^2 - Cos[2*(a + b*Log[c*x^n])] + b*n*Sin[2*(a + b*Log[c*x^n])])/((1 + b^2*n^2)*x^2)","A",1
13,1,122,160,0.5246722,"\int x^2 \sin ^3\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x^2*Sin[a + b*Log[c*x^n]]^3,x]","\frac{x^3 \left(-9 b n \left(b^2 n^2+1\right) \cos \left(a+b \log \left(c x^n\right)\right)+b n \left(b^2 n^2+9\right) \cos \left(3 \left(a+b \log \left(c x^n\right)\right)\right)-2 \sin \left(a+b \log \left(c x^n\right)\right) \left(\left(b^2 n^2+9\right) \cos \left(2 \left(a+b \log \left(c x^n\right)\right)\right)-13 b^2 n^2-9\right)\right)}{12 \left(b^4 n^4+10 b^2 n^2+9\right)}","\frac{x^3 \sin ^3\left(a+b \log \left(c x^n\right)\right)}{3 \left(b^2 n^2+1\right)}-\frac{b n x^3 \sin ^2\left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{3 \left(b^2 n^2+1\right)}+\frac{2 b^2 n^2 x^3 \sin \left(a+b \log \left(c x^n\right)\right)}{b^4 n^4+10 b^2 n^2+9}-\frac{2 b^3 n^3 x^3 \cos \left(a+b \log \left(c x^n\right)\right)}{3 \left(b^4 n^4+10 b^2 n^2+9\right)}",1,"(x^3*(-9*b*n*(1 + b^2*n^2)*Cos[a + b*Log[c*x^n]] + b*n*(9 + b^2*n^2)*Cos[3*(a + b*Log[c*x^n])] - 2*(-9 - 13*b^2*n^2 + (9 + b^2*n^2)*Cos[2*(a + b*Log[c*x^n])])*Sin[a + b*Log[c*x^n]]))/(12*(9 + 10*b^2*n^2 + b^4*n^4))","A",1
14,1,125,158,0.4868013,"\int x \sin ^3\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x*Sin[a + b*Log[c*x^n]]^3,x]","\frac{x^2 \left(-3 b n \left(9 b^2 n^2+4\right) \cos \left(a+b \log \left(c x^n\right)\right)+3 b n \left(b^2 n^2+4\right) \cos \left(3 \left(a+b \log \left(c x^n\right)\right)\right)-4 \sin \left(a+b \log \left(c x^n\right)\right) \left(\left(b^2 n^2+4\right) \cos \left(2 \left(a+b \log \left(c x^n\right)\right)\right)-13 b^2 n^2-4\right)\right)}{4 \left(9 b^4 n^4+40 b^2 n^2+16\right)}","\frac{2 x^2 \sin ^3\left(a+b \log \left(c x^n\right)\right)}{9 b^2 n^2+4}-\frac{3 b n x^2 \sin ^2\left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{9 b^2 n^2+4}+\frac{12 b^2 n^2 x^2 \sin \left(a+b \log \left(c x^n\right)\right)}{9 b^4 n^4+40 b^2 n^2+16}-\frac{6 b^3 n^3 x^2 \cos \left(a+b \log \left(c x^n\right)\right)}{9 b^4 n^4+40 b^2 n^2+16}",1,"(x^2*(-3*b*n*(4 + 9*b^2*n^2)*Cos[a + b*Log[c*x^n]] + 3*b*n*(4 + b^2*n^2)*Cos[3*(a + b*Log[c*x^n])] - 4*(-4 - 13*b^2*n^2 + (4 + b^2*n^2)*Cos[2*(a + b*Log[c*x^n])])*Sin[a + b*Log[c*x^n]]))/(4*(16 + 40*b^2*n^2 + 9*b^4*n^4))","A",1
15,1,121,149,0.4732054,"\int \sin ^3\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[Sin[a + b*Log[c*x^n]]^3,x]","-\frac{x \left(-3 \left(b^3 n^3+b n\right) \cos \left(3 \left(a+b \log \left(c x^n\right)\right)\right)+3 b n \left(9 b^2 n^2+1\right) \cos \left(a+b \log \left(c x^n\right)\right)+2 \sin \left(a+b \log \left(c x^n\right)\right) \left(\left(b^2 n^2+1\right) \cos \left(2 \left(a+b \log \left(c x^n\right)\right)\right)-13 b^2 n^2-1\right)\right)}{36 b^4 n^4+40 b^2 n^2+4}","\frac{x \sin ^3\left(a+b \log \left(c x^n\right)\right)}{9 b^2 n^2+1}-\frac{3 b n x \sin ^2\left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{9 b^2 n^2+1}+\frac{6 b^2 n^2 x \sin \left(a+b \log \left(c x^n\right)\right)}{9 b^4 n^4+10 b^2 n^2+1}-\frac{6 b^3 n^3 x \cos \left(a+b \log \left(c x^n\right)\right)}{9 b^4 n^4+10 b^2 n^2+1}",1,"-((x*(3*b*n*(1 + 9*b^2*n^2)*Cos[a + b*Log[c*x^n]] - 3*(b*n + b^3*n^3)*Cos[3*(a + b*Log[c*x^n])] + 2*(-1 - 13*b^2*n^2 + (1 + b^2*n^2)*Cos[2*(a + b*Log[c*x^n])])*Sin[a + b*Log[c*x^n]]))/(4 + 40*b^2*n^2 + 36*b^4*n^4))","A",1
16,1,45,43,0.0588758,"\int \frac{\sin ^3\left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Sin[a + b*Log[c*x^n]]^3/x,x]","\frac{\cos \left(3 \left(a+b \log \left(c x^n\right)\right)\right)}{12 b n}-\frac{3 \cos \left(a+b \log \left(c x^n\right)\right)}{4 b n}","\frac{\cos ^3\left(a+b \log \left(c x^n\right)\right)}{3 b n}-\frac{\cos \left(a+b \log \left(c x^n\right)\right)}{b n}",1,"(-3*Cos[a + b*Log[c*x^n]])/(4*b*n) + Cos[3*(a + b*Log[c*x^n])]/(12*b*n)","A",1
17,1,125,158,0.3323914,"\int \frac{\sin ^3\left(a+b \log \left(c x^n\right)\right)}{x^2} \, dx","Integrate[Sin[a + b*Log[c*x^n]]^3/x^2,x]","\frac{3 \left(b^3 n^3+b n\right) \cos \left(3 \left(a+b \log \left(c x^n\right)\right)\right)-3 b n \left(9 b^2 n^2+1\right) \cos \left(a+b \log \left(c x^n\right)\right)+2 \sin \left(a+b \log \left(c x^n\right)\right) \left(\left(b^2 n^2+1\right) \cos \left(2 \left(a+b \log \left(c x^n\right)\right)\right)-13 b^2 n^2-1\right)}{4 x \left(9 b^4 n^4+10 b^2 n^2+1\right)}","-\frac{\sin ^3\left(a+b \log \left(c x^n\right)\right)}{x \left(9 b^2 n^2+1\right)}-\frac{3 b n \sin ^2\left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{x \left(9 b^2 n^2+1\right)}-\frac{6 b^2 n^2 \sin \left(a+b \log \left(c x^n\right)\right)}{x \left(9 b^4 n^4+10 b^2 n^2+1\right)}-\frac{6 b^3 n^3 \cos \left(a+b \log \left(c x^n\right)\right)}{x \left(9 b^4 n^4+10 b^2 n^2+1\right)}",1,"(-3*b*n*(1 + 9*b^2*n^2)*Cos[a + b*Log[c*x^n]] + 3*(b*n + b^3*n^3)*Cos[3*(a + b*Log[c*x^n])] + 2*(-1 - 13*b^2*n^2 + (1 + b^2*n^2)*Cos[2*(a + b*Log[c*x^n])])*Sin[a + b*Log[c*x^n]])/(4*(1 + 10*b^2*n^2 + 9*b^4*n^4)*x)","A",1
18,1,125,158,0.38859,"\int \frac{\sin ^3\left(a+b \log \left(c x^n\right)\right)}{x^3} \, dx","Integrate[Sin[a + b*Log[c*x^n]]^3/x^3,x]","\frac{-3 b n \left(9 b^2 n^2+4\right) \cos \left(a+b \log \left(c x^n\right)\right)+3 b n \left(b^2 n^2+4\right) \cos \left(3 \left(a+b \log \left(c x^n\right)\right)\right)+4 \sin \left(a+b \log \left(c x^n\right)\right) \left(\left(b^2 n^2+4\right) \cos \left(2 \left(a+b \log \left(c x^n\right)\right)\right)-13 b^2 n^2-4\right)}{4 x^2 \left(9 b^4 n^4+40 b^2 n^2+16\right)}","-\frac{2 \sin ^3\left(a+b \log \left(c x^n\right)\right)}{x^2 \left(9 b^2 n^2+4\right)}-\frac{3 b n \sin ^2\left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{x^2 \left(9 b^2 n^2+4\right)}-\frac{12 b^2 n^2 \sin \left(a+b \log \left(c x^n\right)\right)}{x^2 \left(9 b^4 n^4+40 b^2 n^2+16\right)}-\frac{6 b^3 n^3 \cos \left(a+b \log \left(c x^n\right)\right)}{x^2 \left(9 b^4 n^4+40 b^2 n^2+16\right)}",1,"(-3*b*n*(4 + 9*b^2*n^2)*Cos[a + b*Log[c*x^n]] + 3*b*n*(4 + b^2*n^2)*Cos[3*(a + b*Log[c*x^n])] + 4*(-4 - 13*b^2*n^2 + (4 + b^2*n^2)*Cos[2*(a + b*Log[c*x^n])])*Sin[a + b*Log[c*x^n]])/(4*(16 + 40*b^2*n^2 + 9*b^4*n^4)*x^2)","A",1
19,1,171,202,0.4979636,"\int x^2 \sin ^4\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x^2*Sin[a + b*Log[c*x^n]]^4,x]","\frac{x^3 \left(-128 b^3 n^3 \sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+16 b^3 n^3 \sin \left(4 \left(a+b \log \left(c x^n\right)\right)\right)-12 \left(16 b^2 n^2+9\right) \cos \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+3 \left(4 b^2 n^2+9\right) \cos \left(4 \left(a+b \log \left(c x^n\right)\right)\right)-72 b n \sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+36 b n \sin \left(4 \left(a+b \log \left(c x^n\right)\right)\right)+64 b^4 n^4+180 b^2 n^2+81\right)}{8 \left(64 b^4 n^4+180 b^2 n^2+81\right)}","\frac{3 x^3 \sin ^4\left(a+b \log \left(c x^n\right)\right)}{16 b^2 n^2+9}-\frac{4 b n x^3 \sin ^3\left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{16 b^2 n^2+9}+\frac{36 b^2 n^2 x^3 \sin ^2\left(a+b \log \left(c x^n\right)\right)}{64 b^4 n^4+180 b^2 n^2+81}-\frac{24 b^3 n^3 x^3 \sin \left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{64 b^4 n^4+180 b^2 n^2+81}+\frac{8 b^4 n^4 x^3}{64 b^4 n^4+180 b^2 n^2+81}",1,"(x^3*(81 + 180*b^2*n^2 + 64*b^4*n^4 - 12*(9 + 16*b^2*n^2)*Cos[2*(a + b*Log[c*x^n])] + 3*(9 + 4*b^2*n^2)*Cos[4*(a + b*Log[c*x^n])] - 72*b*n*Sin[2*(a + b*Log[c*x^n])] - 128*b^3*n^3*Sin[2*(a + b*Log[c*x^n])] + 36*b*n*Sin[4*(a + b*Log[c*x^n])] + 16*b^3*n^3*Sin[4*(a + b*Log[c*x^n])]))/(8*(81 + 180*b^2*n^2 + 64*b^4*n^4))","A",1
20,1,169,210,0.4376774,"\int x \sin ^4\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x*Sin[a + b*Log[c*x^n]]^4,x]","\frac{x^2 \left(-16 b^3 n^3 \sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+2 b^3 n^3 \sin \left(4 \left(a+b \log \left(c x^n\right)\right)\right)-4 \left(4 b^2 n^2+1\right) \cos \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+\left(b^2 n^2+1\right) \cos \left(4 \left(a+b \log \left(c x^n\right)\right)\right)-4 b n \sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+2 b n \sin \left(4 \left(a+b \log \left(c x^n\right)\right)\right)+12 b^4 n^4+15 b^2 n^2+3\right)}{16 \left(4 b^4 n^4+5 b^2 n^2+1\right)}","\frac{x^2 \sin ^4\left(a+b \log \left(c x^n\right)\right)}{2 \left(4 b^2 n^2+1\right)}-\frac{b n x^2 \sin ^3\left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{4 b^2 n^2+1}+\frac{3 b^2 n^2 x^2 \sin ^2\left(a+b \log \left(c x^n\right)\right)}{2 \left(4 b^4 n^4+5 b^2 n^2+1\right)}-\frac{3 b^3 n^3 x^2 \sin \left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{2 \left(4 b^4 n^4+5 b^2 n^2+1\right)}+\frac{3 b^4 n^4 x^2}{4 \left(4 b^4 n^4+5 b^2 n^2+1\right)}",1,"(x^2*(3 + 15*b^2*n^2 + 12*b^4*n^4 - 4*(1 + 4*b^2*n^2)*Cos[2*(a + b*Log[c*x^n])] + (1 + b^2*n^2)*Cos[4*(a + b*Log[c*x^n])] - 4*b*n*Sin[2*(a + b*Log[c*x^n])] - 16*b^3*n^3*Sin[2*(a + b*Log[c*x^n])] + 2*b*n*Sin[4*(a + b*Log[c*x^n])] + 2*b^3*n^3*Sin[4*(a + b*Log[c*x^n])]))/(16*(1 + 5*b^2*n^2 + 4*b^4*n^4))","A",1
21,1,168,191,0.3915945,"\int \sin ^4\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[Sin[a + b*Log[c*x^n]]^4,x]","\frac{x \left(-128 b^3 n^3 \sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+16 b^3 n^3 \sin \left(4 \left(a+b \log \left(c x^n\right)\right)\right)-4 \left(16 b^2 n^2+1\right) \cos \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+\left(4 b^2 n^2+1\right) \cos \left(4 \left(a+b \log \left(c x^n\right)\right)\right)-8 b n \sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+4 b n \sin \left(4 \left(a+b \log \left(c x^n\right)\right)\right)+192 b^4 n^4+60 b^2 n^2+3\right)}{8 \left(64 b^4 n^4+20 b^2 n^2+1\right)}","\frac{x \sin ^4\left(a+b \log \left(c x^n\right)\right)}{16 b^2 n^2+1}-\frac{4 b n x \sin ^3\left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{16 b^2 n^2+1}+\frac{12 b^2 n^2 x \sin ^2\left(a+b \log \left(c x^n\right)\right)}{64 b^4 n^4+20 b^2 n^2+1}-\frac{24 b^3 n^3 x \sin \left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{64 b^4 n^4+20 b^2 n^2+1}+\frac{24 b^4 n^4 x}{64 b^4 n^4+20 b^2 n^2+1}",1,"(x*(3 + 60*b^2*n^2 + 192*b^4*n^4 - 4*(1 + 16*b^2*n^2)*Cos[2*(a + b*Log[c*x^n])] + (1 + 4*b^2*n^2)*Cos[4*(a + b*Log[c*x^n])] - 8*b*n*Sin[2*(a + b*Log[c*x^n])] - 128*b^3*n^3*Sin[2*(a + b*Log[c*x^n])] + 4*b*n*Sin[4*(a + b*Log[c*x^n])] + 16*b^3*n^3*Sin[4*(a + b*Log[c*x^n])]))/(8*(1 + 20*b^2*n^2 + 64*b^4*n^4))","A",1
22,1,51,73,0.0853577,"\int \frac{\sin ^4\left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Sin[a + b*Log[c*x^n]]^4/x,x]","\frac{12 \left(a+b \log \left(c x^n\right)\right)-8 \sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+\sin \left(4 \left(a+b \log \left(c x^n\right)\right)\right)}{32 b n}","-\frac{\sin ^3\left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{4 b n}-\frac{3 \sin \left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{8 b n}+\frac{3 \log (x)}{8}",1,"(12*(a + b*Log[c*x^n]) - 8*Sin[2*(a + b*Log[c*x^n])] + Sin[4*(a + b*Log[c*x^n])])/(32*b*n)","A",1
23,1,170,202,0.5093245,"\int \frac{\sin ^4\left(a+b \log \left(c x^n\right)\right)}{x^2} \, dx","Integrate[Sin[a + b*Log[c*x^n]]^4/x^2,x]","-\frac{128 b^3 n^3 \sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)-16 b^3 n^3 \sin \left(4 \left(a+b \log \left(c x^n\right)\right)\right)-4 \left(16 b^2 n^2+1\right) \cos \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+\left(4 b^2 n^2+1\right) \cos \left(4 \left(a+b \log \left(c x^n\right)\right)\right)+8 b n \sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)-4 b n \sin \left(4 \left(a+b \log \left(c x^n\right)\right)\right)+192 b^4 n^4+60 b^2 n^2+3}{8 x \left(64 b^4 n^4+20 b^2 n^2+1\right)}","-\frac{\sin ^4\left(a+b \log \left(c x^n\right)\right)}{x \left(16 b^2 n^2+1\right)}-\frac{4 b n \sin ^3\left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{x \left(16 b^2 n^2+1\right)}-\frac{12 b^2 n^2 \sin ^2\left(a+b \log \left(c x^n\right)\right)}{x \left(64 b^4 n^4+20 b^2 n^2+1\right)}-\frac{24 b^3 n^3 \sin \left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{x \left(64 b^4 n^4+20 b^2 n^2+1\right)}-\frac{24 b^4 n^4}{x \left(64 b^4 n^4+20 b^2 n^2+1\right)}",1,"-1/8*(3 + 60*b^2*n^2 + 192*b^4*n^4 - 4*(1 + 16*b^2*n^2)*Cos[2*(a + b*Log[c*x^n])] + (1 + 4*b^2*n^2)*Cos[4*(a + b*Log[c*x^n])] + 8*b*n*Sin[2*(a + b*Log[c*x^n])] + 128*b^3*n^3*Sin[2*(a + b*Log[c*x^n])] - 4*b*n*Sin[4*(a + b*Log[c*x^n])] - 16*b^3*n^3*Sin[4*(a + b*Log[c*x^n])])/((1 + 20*b^2*n^2 + 64*b^4*n^4)*x)","A",1
24,1,169,210,0.4549436,"\int \frac{\sin ^4\left(a+b \log \left(c x^n\right)\right)}{x^3} \, dx","Integrate[Sin[a + b*Log[c*x^n]]^4/x^3,x]","-\frac{16 b^3 n^3 \sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)-2 b^3 n^3 \sin \left(4 \left(a+b \log \left(c x^n\right)\right)\right)-4 \left(4 b^2 n^2+1\right) \cos \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+\left(b^2 n^2+1\right) \cos \left(4 \left(a+b \log \left(c x^n\right)\right)\right)+4 b n \sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)-2 b n \sin \left(4 \left(a+b \log \left(c x^n\right)\right)\right)+12 b^4 n^4+15 b^2 n^2+3}{16 x^2 \left(4 b^4 n^4+5 b^2 n^2+1\right)}","-\frac{\sin ^4\left(a+b \log \left(c x^n\right)\right)}{2 x^2 \left(4 b^2 n^2+1\right)}-\frac{b n \sin ^3\left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{x^2 \left(4 b^2 n^2+1\right)}-\frac{3 b^2 n^2 \sin ^2\left(a+b \log \left(c x^n\right)\right)}{2 x^2 \left(4 b^4 n^4+5 b^2 n^2+1\right)}-\frac{3 b^3 n^3 \sin \left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{2 x^2 \left(4 b^4 n^4+5 b^2 n^2+1\right)}-\frac{3 b^4 n^4}{4 x^2 \left(4 b^4 n^4+5 b^2 n^2+1\right)}",1,"-1/16*(3 + 15*b^2*n^2 + 12*b^4*n^4 - 4*(1 + 4*b^2*n^2)*Cos[2*(a + b*Log[c*x^n])] + (1 + b^2*n^2)*Cos[4*(a + b*Log[c*x^n])] + 4*b*n*Sin[2*(a + b*Log[c*x^n])] + 16*b^3*n^3*Sin[2*(a + b*Log[c*x^n])] - 2*b*n*Sin[4*(a + b*Log[c*x^n])] - 2*b^3*n^3*Sin[4*(a + b*Log[c*x^n])])/((1 + 5*b^2*n^2 + 4*b^4*n^4)*x^2)","A",1
25,1,29,39,0.0149888,"\int \sin (\log (a+b x)) \, dx","Integrate[Sin[Log[a + b*x]],x]","-\frac{(a+b x) (\cos (\log (a+b x))-\sin (\log (a+b x)))}{2 b}","\frac{(a+b x) \sin (\log (a+b x))}{2 b}-\frac{(a+b x) \cos (\log (a+b x))}{2 b}",1,"-1/2*((a + b*x)*(Cos[Log[a + b*x]] - Sin[Log[a + b*x]]))/b","A",1
26,0,0,133,0.3556324,"\int x^m \sin \left(a+\sqrt{-\frac{(1+m)^2}{n^2}} \log \left(c x^n\right)\right) \, dx","Integrate[x^m*Sin[a + Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n]],x]","\int x^m \sin \left(a+\sqrt{-\frac{(1+m)^2}{n^2}} \log \left(c x^n\right)\right) \, dx","\frac{(m+1) x^{m+1} \log (x) e^{\frac{a n \sqrt{-\frac{(m+1)^2}{n^2}}}{m+1}} \left(c x^n\right)^{-\frac{m+1}{n}}}{2 n \sqrt{-\frac{(m+1)^2}{n^2}}}-\frac{x^{m+1} e^{\frac{a (m+1)}{n \sqrt{-\frac{(m+1)^2}{n^2}}}} \left(c x^n\right)^{\frac{m+1}{n}}}{4 n \sqrt{-\frac{(m+1)^2}{n^2}}}",1,"Integrate[x^m*Sin[a + Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n]], x]","F",-1
27,0,0,88,0.190924,"\int x^2 \sin \left(a+3 \sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right) \, dx","Integrate[x^2*Sin[a + 3*Sqrt[-n^(-2)]*Log[c*x^n]],x]","\int x^2 \sin \left(a+3 \sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right) \, dx","\frac{1}{12} \sqrt{-\frac{1}{n^2}} n x^3 e^{-a \sqrt{-\frac{1}{n^2}} n} \left(c x^n\right)^{3/n}-\frac{1}{2} \sqrt{-\frac{1}{n^2}} n x^3 e^{a \sqrt{-\frac{1}{n^2}} n} \log (x) \left(c x^n\right)^{-3/n}",1,"Integrate[x^2*Sin[a + 3*Sqrt[-n^(-2)]*Log[c*x^n]], x]","F",-1
28,0,0,88,0.1608501,"\int x \sin \left(a+2 \sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right) \, dx","Integrate[x*Sin[a + 2*Sqrt[-n^(-2)]*Log[c*x^n]],x]","\int x \sin \left(a+2 \sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right) \, dx","\frac{1}{8} \sqrt{-\frac{1}{n^2}} n x^2 e^{-a \sqrt{-\frac{1}{n^2}} n} \left(c x^n\right)^{2/n}-\frac{1}{2} \sqrt{-\frac{1}{n^2}} n x^2 e^{a \sqrt{-\frac{1}{n^2}} n} \log (x) \left(c x^n\right)^{-2/n}",1,"Integrate[x*Sin[a + 2*Sqrt[-n^(-2)]*Log[c*x^n]], x]","F",-1
29,0,0,82,0.1132933,"\int \sin \left(a+\sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right) \, dx","Integrate[Sin[a + Sqrt[-n^(-2)]*Log[c*x^n]],x]","\int \sin \left(a+\sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right) \, dx","\frac{1}{4} \sqrt{-\frac{1}{n^2}} n x e^{-a \sqrt{-\frac{1}{n^2}} n} \left(c x^n\right)^{\frac{1}{n}}-\frac{1}{2} \sqrt{-\frac{1}{n^2}} n x e^{a \sqrt{-\frac{1}{n^2}} n} \log (x) \left(c x^n\right)^{-1/n}",1,"Integrate[Sin[a + Sqrt[-n^(-2)]*Log[c*x^n]], x]","F",-1
30,1,5,5,0.0008177,"\int \frac{\sin (a)}{x} \, dx","Integrate[Sin[a]/x,x]","\sin (a) \log (x)","\sin (a) \log (x)",1,"Log[x]*Sin[a]","A",1
31,0,0,86,0.1089076,"\int \frac{\sin \left(a+\sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right)}{x^2} \, dx","Integrate[Sin[a + Sqrt[-n^(-2)]*Log[c*x^n]]/x^2,x]","\int \frac{\sin \left(a+\sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right)}{x^2} \, dx","\frac{\sqrt{-\frac{1}{n^2}} n e^{a \sqrt{-\frac{1}{n^2}} n} \left(c x^n\right)^{-1/n}}{4 x}+\frac{\sqrt{-\frac{1}{n^2}} n e^{-a \sqrt{-\frac{1}{n^2}} n} \log (x) \left(c x^n\right)^{\frac{1}{n}}}{2 x}",1,"Integrate[Sin[a + Sqrt[-n^(-2)]*Log[c*x^n]]/x^2, x]","F",-1
32,0,0,88,0.1229582,"\int \frac{\sin \left(a+2 \sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right)}{x^3} \, dx","Integrate[Sin[a + 2*Sqrt[-n^(-2)]*Log[c*x^n]]/x^3,x]","\int \frac{\sin \left(a+2 \sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right)}{x^3} \, dx","\frac{\sqrt{-\frac{1}{n^2}} n e^{a \sqrt{-\frac{1}{n^2}} n} \left(c x^n\right)^{-2/n}}{8 x^2}+\frac{\sqrt{-\frac{1}{n^2}} n e^{-a \sqrt{-\frac{1}{n^2}} n} \log (x) \left(c x^n\right)^{2/n}}{2 x^2}",1,"Integrate[Sin[a + 2*Sqrt[-n^(-2)]*Log[c*x^n]]/x^3, x]","F",-1
33,0,0,117,0.4774465,"\int x^m \sin ^2\left(a+\frac{1}{2} \sqrt{-\frac{(1+m)^2}{n^2}} \log \left(c x^n\right)\right) \, dx","Integrate[x^m*Sin[a + (Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n])/2]^2,x]","\int x^m \sin ^2\left(a+\frac{1}{2} \sqrt{-\frac{(1+m)^2}{n^2}} \log \left(c x^n\right)\right) \, dx","-\frac{x^{m+1} e^{-\frac{2 a n \sqrt{-\frac{(m+1)^2}{n^2}}}{m+1}} \left(c x^n\right)^{\frac{m+1}{n}}}{8 (m+1)}-\frac{1}{4} x^{m+1} \log (x) e^{\frac{2 a n \sqrt{-\frac{(m+1)^2}{n^2}}}{m+1}} \left(c x^n\right)^{-\frac{m+1}{n}}+\frac{x^{m+1}}{2 (m+1)}",1,"Integrate[x^m*Sin[a + (Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n])/2]^2, x]","F",-1
34,0,0,76,0.2921356,"\int x^2 \sin ^2\left(a+\frac{3}{2} \sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right) \, dx","Integrate[x^2*Sin[a + (3*Sqrt[-n^(-2)]*Log[c*x^n])/2]^2,x]","\int x^2 \sin ^2\left(a+\frac{3}{2} \sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right) \, dx","-\frac{1}{24} x^3 e^{-2 a \sqrt{-\frac{1}{n^2}} n} \left(c x^n\right)^{3/n}-\frac{1}{4} x^3 e^{2 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left(c x^n\right)^{-3/n}+\frac{x^3}{6}",1,"Integrate[x^2*Sin[a + (3*Sqrt[-n^(-2)]*Log[c*x^n])/2]^2, x]","F",-1
35,0,0,76,0.1849409,"\int x \sin ^2\left(a+\sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right) \, dx","Integrate[x*Sin[a + Sqrt[-n^(-2)]*Log[c*x^n]]^2,x]","\int x \sin ^2\left(a+\sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right) \, dx","-\frac{1}{16} x^2 e^{-2 a \sqrt{-\frac{1}{n^2}} n} \left(c x^n\right)^{2/n}-\frac{1}{4} x^2 e^{2 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left(c x^n\right)^{-2/n}+\frac{x^2}{4}",1,"Integrate[x*Sin[a + Sqrt[-n^(-2)]*Log[c*x^n]]^2, x]","F",-1
36,0,0,68,0.1280364,"\int \sin ^2\left(a+\frac{1}{2} \sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right) \, dx","Integrate[Sin[a + (Sqrt[-n^(-2)]*Log[c*x^n])/2]^2,x]","\int \sin ^2\left(a+\frac{1}{2} \sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right) \, dx","-\frac{1}{8} x e^{-2 a \sqrt{-\frac{1}{n^2}} n} \left(c x^n\right)^{\frac{1}{n}}-\frac{1}{4} x e^{2 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left(c x^n\right)^{-1/n}+\frac{x}{2}",1,"Integrate[Sin[a + (Sqrt[-n^(-2)]*Log[c*x^n])/2]^2, x]","F",-1
37,1,7,7,0.0010933,"\int \frac{\sin ^2(a)}{x} \, dx","Integrate[Sin[a]^2/x,x]","\sin ^2(a) \log (x)","\sin ^2(a) \log (x)",1,"Log[x]*Sin[a]^2","A",1
38,0,0,74,0.2123954,"\int \frac{\sin ^2\left(a+\frac{1}{2} \sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right)}{x^2} \, dx","Integrate[Sin[a + (Sqrt[-n^(-2)]*Log[c*x^n])/2]^2/x^2,x]","\int \frac{\sin ^2\left(a+\frac{1}{2} \sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right)}{x^2} \, dx","\frac{e^{2 a \sqrt{-\frac{1}{n^2}} n} \left(c x^n\right)^{-1/n}}{8 x}-\frac{e^{-2 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left(c x^n\right)^{\frac{1}{n}}}{4 x}-\frac{1}{2 x}",1,"Integrate[Sin[a + (Sqrt[-n^(-2)]*Log[c*x^n])/2]^2/x^2, x]","F",-1
39,0,0,76,0.172942,"\int \frac{\sin ^2\left(a+\sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right)}{x^3} \, dx","Integrate[Sin[a + Sqrt[-n^(-2)]*Log[c*x^n]]^2/x^3,x]","\int \frac{\sin ^2\left(a+\sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right)}{x^3} \, dx","\frac{e^{2 a \sqrt{-\frac{1}{n^2}} n} \left(c x^n\right)^{-2/n}}{16 x^2}-\frac{e^{-2 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left(c x^n\right)^{2/n}}{4 x^2}-\frac{1}{4 x^2}",1,"Integrate[Sin[a + Sqrt[-n^(-2)]*Log[c*x^n]]^2/x^3, x]","F",-1
40,1,169,226,1.5025899,"\int x^m \sin ^3\left(a+\frac{1}{2} \sqrt{-\frac{(1+m)^2}{n^2}} \log \left(c x^n\right)\right) \, dx","Integrate[x^m*Sin[a + (Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n])/2]^3,x]","\frac{x^{m+1} \left(2 (m+1) \left(5 \sin \left(a+\frac{1}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left(c x^n\right)\right)+\sin \left(3 a+\frac{3}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left(c x^n\right)\right)\right)-5 n \sqrt{-\frac{(m+1)^2}{n^2}} \cos \left(a+\frac{1}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left(c x^n\right)\right)-3 n \sqrt{-\frac{(m+1)^2}{n^2}} \cos \left(3 a+\frac{3}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left(c x^n\right)\right)\right)}{10 (m+1)^2}","-\frac{4 x^{m+1} \sin ^3\left(a+\frac{1}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left(c x^n\right)\right)}{5 (m+1)}+\frac{8 x^{m+1} \sin \left(a+\frac{1}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left(c x^n\right)\right)}{5 (m+1)}-\frac{4 n \sqrt{-\frac{(m+1)^2}{n^2}} x^{m+1} \cos \left(a+\frac{1}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left(c x^n\right)\right)}{5 (m+1)^2}+\frac{6 n \sqrt{-\frac{(m+1)^2}{n^2}} x^{m+1} \sin ^2\left(a+\frac{1}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left(c x^n\right)\right) \cos \left(a+\frac{1}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left(c x^n\right)\right)}{5 (m+1)^2}",1,"(x^(1 + m)*(-5*Sqrt[-((1 + m)^2/n^2)]*n*Cos[a + (Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n])/2] - 3*Sqrt[-((1 + m)^2/n^2)]*n*Cos[3*a + (3*Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n])/2] + 2*(1 + m)*(5*Sin[a + (Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n])/2] + Sin[3*a + (3*Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n])/2])))/(10*(1 + m)^2)","A",1
41,0,0,172,0.3104004,"\int x^2 \sin ^3\left(a+\sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right) \, dx","Integrate[x^2*Sin[a + Sqrt[-n^(-2)]*Log[c*x^n]]^3,x]","\int x^2 \sin ^3\left(a+\sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right) \, dx","-\frac{3}{16} \sqrt{-\frac{1}{n^2}} n x^3 e^{a \sqrt{-\frac{1}{n^2}} n} \left(c x^n\right)^{-1/n}-\frac{1}{48} \sqrt{-\frac{1}{n^2}} n x^3 e^{-3 a \sqrt{-\frac{1}{n^2}} n} \left(c x^n\right)^{3/n}+\frac{3}{32} \sqrt{-\frac{1}{n^2}} n x^3 e^{-a \sqrt{-\frac{1}{n^2}} n} \left(c x^n\right)^{\frac{1}{n}}+\frac{1}{8} \sqrt{-\frac{1}{n^2}} n x^3 e^{3 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left(c x^n\right)^{-3/n}",1,"Integrate[x^2*Sin[a + Sqrt[-n^(-2)]*Log[c*x^n]]^3, x]","F",-1
42,0,0,178,0.3513604,"\int x \sin ^3\left(a+\frac{2}{3} \sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right) \, dx","Integrate[x*Sin[a + (2*Sqrt[-n^(-2)]*Log[c*x^n])/3]^3,x]","\int x \sin ^3\left(a+\frac{2}{3} \sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right) \, dx","-\frac{9}{32} \sqrt{-\frac{1}{n^2}} n x^2 e^{a \sqrt{-\frac{1}{n^2}} n} \left(c x^n\right)^{\left.-\frac{2}{3}\right/n}+\frac{9}{64} \sqrt{-\frac{1}{n^2}} n x^2 e^{-a \sqrt{-\frac{1}{n^2}} n} \left(c x^n\right)^{\left.\frac{2}{3}\right/n}-\frac{1}{32} \sqrt{-\frac{1}{n^2}} n x^2 e^{-3 a \sqrt{-\frac{1}{n^2}} n} \left(c x^n\right)^{2/n}+\frac{1}{8} \sqrt{-\frac{1}{n^2}} n x^2 e^{3 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left(c x^n\right)^{-2/n}",1,"Integrate[x*Sin[a + (2*Sqrt[-n^(-2)]*Log[c*x^n])/3]^3, x]","F",-1
43,0,0,168,0.210609,"\int \sin ^3\left(a+\frac{1}{3} \sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right) \, dx","Integrate[Sin[a + (Sqrt[-n^(-2)]*Log[c*x^n])/3]^3,x]","\int \sin ^3\left(a+\frac{1}{3} \sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right) \, dx","-\frac{9}{16} \sqrt{-\frac{1}{n^2}} n x e^{a \sqrt{-\frac{1}{n^2}} n} \left(c x^n\right)^{\left.-\frac{1}{3}\right/n}+\frac{9}{32} \sqrt{-\frac{1}{n^2}} n x e^{-a \sqrt{-\frac{1}{n^2}} n} \left(c x^n\right)^{\left.\frac{1}{3}\right/n}-\frac{1}{16} \sqrt{-\frac{1}{n^2}} n x e^{-3 a \sqrt{-\frac{1}{n^2}} n} \left(c x^n\right)^{\frac{1}{n}}+\frac{1}{8} \sqrt{-\frac{1}{n^2}} n x e^{3 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left(c x^n\right)^{-1/n}",1,"Integrate[Sin[a + (Sqrt[-n^(-2)]*Log[c*x^n])/3]^3, x]","F",-1
44,1,7,7,0.0008437,"\int \frac{\sin ^3(a)}{x} \, dx","Integrate[Sin[a]^3/x,x]","\sin ^3(a) \log (x)","\sin ^3(a) \log (x)",1,"Log[x]*Sin[a]^3","A",1
45,0,0,176,0.2371722,"\int \frac{\sin ^3\left(a+\frac{1}{3} \sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right)}{x^2} \, dx","Integrate[Sin[a + (Sqrt[-n^(-2)]*Log[c*x^n])/3]^3/x^2,x]","\int \frac{\sin ^3\left(a+\frac{1}{3} \sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right)}{x^2} \, dx","-\frac{\sqrt{-\frac{1}{n^2}} n e^{3 a \sqrt{-\frac{1}{n^2}} n} \left(c x^n\right)^{-1/n}}{16 x}+\frac{9 \sqrt{-\frac{1}{n^2}} n e^{a \sqrt{-\frac{1}{n^2}} n} \left(c x^n\right)^{\left.-\frac{1}{3}\right/n}}{32 x}-\frac{9 \sqrt{-\frac{1}{n^2}} n e^{-a \sqrt{-\frac{1}{n^2}} n} \left(c x^n\right)^{\left.\frac{1}{3}\right/n}}{16 x}-\frac{\sqrt{-\frac{1}{n^2}} n e^{-3 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left(c x^n\right)^{\frac{1}{n}}}{8 x}",1,"Integrate[Sin[a + (Sqrt[-n^(-2)]*Log[c*x^n])/3]^3/x^2, x]","F",-1
46,0,0,178,0.2582653,"\int \frac{\sin ^3\left(a+\frac{2}{3} \sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right)}{x^3} \, dx","Integrate[Sin[a + (2*Sqrt[-n^(-2)]*Log[c*x^n])/3]^3/x^3,x]","\int \frac{\sin ^3\left(a+\frac{2}{3} \sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right)}{x^3} \, dx","-\frac{\sqrt{-\frac{1}{n^2}} n e^{3 a \sqrt{-\frac{1}{n^2}} n} \left(c x^n\right)^{-2/n}}{32 x^2}+\frac{9 \sqrt{-\frac{1}{n^2}} n e^{a \sqrt{-\frac{1}{n^2}} n} \left(c x^n\right)^{\left.-\frac{2}{3}\right/n}}{64 x^2}-\frac{9 \sqrt{-\frac{1}{n^2}} n e^{-a \sqrt{-\frac{1}{n^2}} n} \left(c x^n\right)^{\left.\frac{2}{3}\right/n}}{32 x^2}-\frac{\sqrt{-\frac{1}{n^2}} n e^{-3 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left(c x^n\right)^{2/n}}{8 x^2}",1,"Integrate[Sin[a + (2*Sqrt[-n^(-2)]*Log[c*x^n])/3]^3/x^3, x]","F",-1
47,0,0,112,0.2751971,"\int x^m \sin \left(a+\frac{1}{2} \sqrt{-(1+m)^2} \log \left(c x^2\right)\right) \, dx","Integrate[x^m*Sin[a + (Sqrt[-(1 + m)^2]*Log[c*x^2])/2],x]","\int x^m \sin \left(a+\frac{1}{2} \sqrt{-(1+m)^2} \log \left(c x^2\right)\right) \, dx","\frac{(m+1) e^{\frac{a \sqrt{-(m+1)^2}}{m+1}} x^{m+1} \log (x) \left(c x^2\right)^{\frac{1}{2} (-m-1)}}{2 \sqrt{-(m+1)^2}}-\frac{e^{\frac{a (m+1)}{\sqrt{-(m+1)^2}}} x^{m+1} \left(c x^2\right)^{\frac{m+1}{2}}}{4 \sqrt{-(m+1)^2}}",1,"Integrate[x^m*Sin[a + (Sqrt[-(1 + m)^2]*Log[c*x^2])/2], x]","F",-1
48,1,44,52,0.0636918,"\int \sin \left(a+\frac{1}{2} i \log \left(c x^2\right)\right) \, dx","Integrate[Sin[a + (I/2)*Log[c*x^2]],x]","\frac{x \left(\sin (a) \left(c x^2+2 \log (x)\right)+i \cos (a) \left(c x^2-2 \log (x)\right)\right)}{4 \sqrt{c x^2}}","\frac{i e^{-i a} c x^3}{4 \sqrt{c x^2}}-\frac{i e^{i a} x \log (x)}{2 \sqrt{c x^2}}",1,"(x*(I*Cos[a]*(c*x^2 - 2*Log[x]) + (c*x^2 + 2*Log[x])*Sin[a]))/(4*Sqrt[c*x^2])","A",1
49,0,0,106,0.3546584,"\int x^m \sin ^2\left(a+\frac{1}{4} \sqrt{-(1+m)^2} \log \left(c x^2\right)\right) \, dx","Integrate[x^m*Sin[a + (Sqrt[-(1 + m)^2]*Log[c*x^2])/4]^2,x]","\int x^m \sin ^2\left(a+\frac{1}{4} \sqrt{-(1+m)^2} \log \left(c x^2\right)\right) \, dx","-\frac{e^{\frac{2 a (m+1)}{\sqrt{-(m+1)^2}}} x^{m+1} \left(c x^2\right)^{\frac{m+1}{2}}}{8 (m+1)}-\frac{1}{4} e^{-\frac{2 a (m+1)}{\sqrt{-(m+1)^2}}} x^{m+1} \log (x) \left(c x^2\right)^{\frac{1}{2} (-m-1)}+\frac{x^{m+1}}{2 (m+1)}",1,"Integrate[x^m*Sin[a + (Sqrt[-(1 + m)^2]*Log[c*x^2])/4]^2, x]","F",-1
50,1,60,53,0.1039897,"\int \sin ^2\left(a+\frac{1}{4} i \log \left(c x^2\right)\right) \, dx","Integrate[Sin[a + (I/4)*Log[c*x^2]]^2,x]","\frac{x \left(i \sin (2 a) \left(c x^2-2 \log (x)\right)-\cos (2 a) \left(c x^2+2 \log (x)\right)+4 \sqrt{c x^2}\right)}{8 \sqrt{c x^2}}","-\frac{e^{2 i a} x \log (x)}{4 \sqrt{c x^2}}-\frac{e^{-2 i a} c x^3}{8 \sqrt{c x^2}}+\frac{x}{2}",1,"(x*(4*Sqrt[c*x^2] - Cos[2*a]*(c*x^2 + 2*Log[x]) + I*(c*x^2 - 2*Log[x])*Sin[2*a]))/(8*Sqrt[c*x^2])","A",1
51,0,0,218,0.515436,"\int x^m \sin ^3\left(a+\frac{1}{6} \sqrt{-(1+m)^2} \log \left(c x^2\right)\right) \, dx","Integrate[x^m*Sin[a + (Sqrt[-(1 + m)^2]*Log[c*x^2])/6]^3,x]","\int x^m \sin ^3\left(a+\frac{1}{6} \sqrt{-(1+m)^2} \log \left(c x^2\right)\right) \, dx","\frac{9 e^{\frac{a \sqrt{-(m+1)^2}}{m+1}} x^{m+1} \left(c x^2\right)^{\frac{1}{6} (-m-1)}}{16 \sqrt{-(m+1)^2}}-\frac{9 e^{\frac{a (m+1)}{\sqrt{-(m+1)^2}}} x^{m+1} \left(c x^2\right)^{\frac{m+1}{6}}}{32 \sqrt{-(m+1)^2}}+\frac{e^{\frac{3 a (m+1)}{\sqrt{-(m+1)^2}}} x^{m+1} \left(c x^2\right)^{\frac{m+1}{2}}}{16 \sqrt{-(m+1)^2}}-\frac{(m+1) e^{-\frac{3 a (m+1)}{\sqrt{-(m+1)^2}}} x^{m+1} \log (x) \left(c x^2\right)^{\frac{1}{2} (-m-1)}}{8 \sqrt{-(m+1)^2}}",1,"Integrate[x^m*Sin[a + (Sqrt[-(1 + m)^2]*Log[c*x^2])/6]^3, x]","F",-1
52,1,103,98,0.130945,"\int \sin ^3\left(a+\frac{1}{6} i \log \left(c x^2\right)\right) \, dx","Integrate[Sin[a + (I/6)*Log[c*x^2]]^3,x]","\frac{x \left(-2 c x^2 \sin (3 a)+9 \sin (a) \left(c x^2\right)^{2/3}+18 \sin (a) \sqrt[3]{c x^2}+9 i \cos (a) \sqrt[3]{c x^2} \left(\sqrt[3]{c x^2}-2\right)-2 i \cos (3 a) \left(c x^2-2 \log (x)\right)-4 \sin (3 a) \log (x)\right)}{32 \sqrt{c x^2}}","\frac{9}{32} i e^{-i a} x \sqrt[6]{c x^2}-\frac{9 i e^{i a} x}{16 \sqrt[6]{c x^2}}+\frac{i e^{3 i a} x \log (x)}{8 \sqrt{c x^2}}-\frac{i e^{-3 i a} c x^3}{16 \sqrt{c x^2}}",1,"(x*((9*I)*(c*x^2)^(1/3)*(-2 + (c*x^2)^(1/3))*Cos[a] - (2*I)*Cos[3*a]*(c*x^2 - 2*Log[x]) + 18*(c*x^2)^(1/3)*Sin[a] + 9*(c*x^2)^(2/3)*Sin[a] - 2*c*x^2*Sin[3*a] - 4*Log[x]*Sin[3*a]))/(32*Sqrt[c*x^2])","A",1
53,1,94,111,1.3857317,"\int x \sqrt{\sin \left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[x*Sqrt[Sin[a + b*Log[c*x^n]]],x]","\frac{2 x^2 \left(-1+e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \, _2F_1\left(1,\frac{5}{4}-\frac{i}{b n};\frac{3}{4}-\frac{i}{b n};e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \sqrt{\sin \left(a+b \log \left(c x^n\right)\right)}}{-4+i b n}","\frac{2 x^2 \, _2F_1\left(-\frac{1}{2},\frac{1}{4} \left(-1-\frac{4 i}{b n}\right);\frac{1}{4} \left(3-\frac{4 i}{b n}\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sqrt{\sin \left(a+b \log \left(c x^n\right)\right)}}{(4-i b n) \sqrt{1-e^{2 i a} \left(c x^n\right)^{2 i b}}}",1,"(2*(-1 + E^((2*I)*(a + b*Log[c*x^n])))*x^2*Hypergeometric2F1[1, 5/4 - I/(b*n), 3/4 - I/(b*n), E^((2*I)*(a + b*Log[c*x^n]))]*Sqrt[Sin[a + b*Log[c*x^n]]])/(-4 + I*b*n)","A",0
54,1,96,110,1.3595418,"\int \sqrt{\sin \left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[Sqrt[Sin[a + b*Log[c*x^n]]],x]","\frac{2 x \left(-1+e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \, _2F_1\left(1,\frac{5}{4}-\frac{i}{2 b n};\frac{3}{4}-\frac{i}{2 b n};e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \sqrt{\sin \left(a+b \log \left(c x^n\right)\right)}}{-2+i b n}","\frac{2 x \, _2F_1\left(-\frac{1}{2},-\frac{b n+2 i}{4 b n};\frac{1}{4} \left(3-\frac{2 i}{b n}\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sqrt{\sin \left(a+b \log \left(c x^n\right)\right)}}{(2-i b n) \sqrt{1-e^{2 i a} \left(c x^n\right)^{2 i b}}}",1,"(2*(-1 + E^((2*I)*(a + b*Log[c*x^n])))*x*Hypergeometric2F1[1, 5/4 - (I/2)/(b*n), 3/4 - (I/2)/(b*n), E^((2*I)*(a + b*Log[c*x^n]))]*Sqrt[Sin[a + b*Log[c*x^n]]])/(-2 + I*b*n)","A",0
55,1,32,29,0.081429,"\int \frac{\sqrt{\sin \left(a+b \log \left(c x^n\right)\right)}}{x} \, dx","Integrate[Sqrt[Sin[a + b*Log[c*x^n]]]/x,x]","-\frac{2 E\left(\left.\frac{1}{2} \left(-a-b \log \left(c x^n\right)+\frac{\pi }{2}\right)\right|2\right)}{b n}","\frac{2 E\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)-\frac{\pi }{2}\right)\right|2\right)}{b n}",1,"(-2*EllipticE[(-a + Pi/2 - b*Log[c*x^n])/2, 2])/(b*n)","A",1
56,1,99,111,1.4519903,"\int \frac{\sqrt{\sin \left(a+b \log \left(c x^n\right)\right)}}{x^2} \, dx","Integrate[Sqrt[Sin[a + b*Log[c*x^n]]]/x^2,x]","-\frac{2 i \left(-1+e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \, _2F_1\left(1,\frac{5}{4}+\frac{i}{2 b n};\frac{3}{4}+\frac{i}{2 b n};e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \sqrt{\sin \left(a+b \log \left(c x^n\right)\right)}}{x (b n-2 i)}","-\frac{2 \, _2F_1\left(-\frac{1}{2},\frac{1}{4} \left(\frac{2 i}{b n}-1\right);\frac{1}{4} \left(3+\frac{2 i}{b n}\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sqrt{\sin \left(a+b \log \left(c x^n\right)\right)}}{x (2+i b n) \sqrt{1-e^{2 i a} \left(c x^n\right)^{2 i b}}}",1,"((-2*I)*(-1 + E^((2*I)*(a + b*Log[c*x^n])))*Hypergeometric2F1[1, 5/4 + (I/2)/(b*n), 3/4 + (I/2)/(b*n), E^((2*I)*(a + b*Log[c*x^n]))]*Sqrt[Sin[a + b*Log[c*x^n]]])/((-2*I + b*n)*x)","A",0
57,1,95,111,1.4348622,"\int \frac{\sqrt{\sin \left(a+b \log \left(c x^n\right)\right)}}{x^3} \, dx","Integrate[Sqrt[Sin[a + b*Log[c*x^n]]]/x^3,x]","-\frac{2 i \left(-1+e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \, _2F_1\left(1,\frac{5}{4}+\frac{i}{b n};\frac{3}{4}+\frac{i}{b n};e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \sqrt{\sin \left(a+b \log \left(c x^n\right)\right)}}{x^2 (b n-4 i)}","-\frac{2 \, _2F_1\left(-\frac{1}{2},\frac{1}{4} \left(\frac{4 i}{b n}-1\right);\frac{1}{4} \left(3+\frac{4 i}{b n}\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sqrt{\sin \left(a+b \log \left(c x^n\right)\right)}}{x^2 (4+i b n) \sqrt{1-e^{2 i a} \left(c x^n\right)^{2 i b}}}",1,"((-2*I)*(-1 + E^((2*I)*(a + b*Log[c*x^n])))*Hypergeometric2F1[1, 5/4 + I/(b*n), 3/4 + I/(b*n), E^((2*I)*(a + b*Log[c*x^n]))]*Sqrt[Sin[a + b*Log[c*x^n]]])/((-4*I + b*n)*x^2)","A",0
58,1,159,111,1.8283678,"\int x \sin ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x*Sin[a + b*Log[c*x^n]]^(3/2),x]","\frac{x^2 \left(6 b^2 n^2 \left(-1+e^{2 i a} \left(c x^n\right)^{2 i b}\right) \, _2F_1\left(1,\frac{3}{4}-\frac{i}{b n};\frac{5}{4}-\frac{i}{b n};e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)+(4+i b n) \left(3 b n \sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)-8 \sin ^2\left(a+b \log \left(c x^n\right)\right)\right)\right)}{(-4-i b n) \left(9 b^2 n^2+16\right) \sqrt{\sin \left(a+b \log \left(c x^n\right)\right)}}","\frac{2 x^2 \, _2F_1\left(-\frac{3}{2},\frac{1}{4} \left(-3-\frac{4 i}{b n}\right);\frac{1}{4} \left(1-\frac{4 i}{b n}\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sin ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}{(4-3 i b n) \left(1-e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{3/2}}",1,"(x^2*(6*b^2*n^2*(-1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))*Hypergeometric2F1[1, 3/4 - I/(b*n), 5/4 - I/(b*n), E^((2*I)*(a + b*Log[c*x^n]))] + (4 + I*b*n)*(-8*Sin[a + b*Log[c*x^n]]^2 + 3*b*n*Sin[2*(a + b*Log[c*x^n])])))/((-4 - I*b*n)*(16 + 9*b^2*n^2)*Sqrt[Sin[a + b*Log[c*x^n]]])","A",0
59,1,161,109,1.8938516,"\int \sin ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[Sin[a + b*Log[c*x^n]]^(3/2),x]","\frac{x \left(6 i b^2 n^2 \left(-1+e^{2 i a} \left(c x^n\right)^{2 i b}\right) \, _2F_1\left(1,\frac{3}{4}-\frac{i}{2 b n};\frac{5}{4}-\frac{i}{2 b n};e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)+(b n-2 i) \left(4 \sin ^2\left(a+b \log \left(c x^n\right)\right)-3 b n \sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)\right)\right)}{(b n-2 i) \left(9 b^2 n^2+4\right) \sqrt{\sin \left(a+b \log \left(c x^n\right)\right)}}","\frac{2 x \, _2F_1\left(-\frac{3}{2},\frac{1}{4} \left(-3-\frac{2 i}{b n}\right);\frac{1}{4} \left(1-\frac{2 i}{b n}\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sin ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}{(2-3 i b n) \left(1-e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{3/2}}",1,"(x*((6*I)*b^2*n^2*(-1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))*Hypergeometric2F1[1, 3/4 - (I/2)/(b*n), 5/4 - (I/2)/(b*n), E^((2*I)*(a + b*Log[c*x^n]))] + (-2*I + b*n)*(4*Sin[a + b*Log[c*x^n]]^2 - 3*b*n*Sin[2*(a + b*Log[c*x^n])])))/((-2*I + b*n)*(4 + 9*b^2*n^2)*Sqrt[Sin[a + b*Log[c*x^n]]])","A",0
60,1,58,68,0.1363577,"\int \frac{\sin ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Sin[a + b*Log[c*x^n]]^(3/2)/x,x]","-\frac{2 \left(F\left(\left.\frac{1}{4} \left(-2 a-2 b \log \left(c x^n\right)+\pi \right)\right|2\right)+\sqrt{\sin \left(a+b \log \left(c x^n\right)\right)} \cos \left(a+b \log \left(c x^n\right)\right)\right)}{3 b n}","\frac{2 F\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)-\frac{\pi }{2}\right)\right|2\right)}{3 b n}-\frac{2 \sqrt{\sin \left(a+b \log \left(c x^n\right)\right)} \cos \left(a+b \log \left(c x^n\right)\right)}{3 b n}",1,"(-2*(EllipticF[(-2*a + Pi - 2*b*Log[c*x^n])/4, 2] + Cos[a + b*Log[c*x^n]]*Sqrt[Sin[a + b*Log[c*x^n]]]))/(3*b*n)","A",1
61,1,172,111,1.176437,"\int \frac{\sin ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}{x^2} \, dx","Integrate[Sin[a + b*Log[c*x^n]]^(3/2)/x^2,x]","\frac{6 i b^2 n^2 \left(-1+e^{2 i a} \left(c x^n\right)^{2 i b}\right) \, _2F_1\left(1,\frac{3}{4}+\frac{i}{2 b n};\frac{5}{4}+\frac{i}{2 b n};e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)-(b n+2 i) \left(4 \sin ^2\left(a+b \log \left(c x^n\right)\right)+3 b n \sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)\right)}{x (b n+2 i) (3 b n-2 i) (3 b n+2 i) \sqrt{\sin \left(a+b \log \left(c x^n\right)\right)}}","-\frac{2 \, _2F_1\left(-\frac{3}{2},\frac{1}{4} \left(\frac{2 i}{b n}-3\right);\frac{1}{4} \left(1+\frac{2 i}{b n}\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sin ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}{x (2+3 i b n) \left(1-e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{3/2}}",1,"((6*I)*b^2*n^2*(-1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))*Hypergeometric2F1[1, 3/4 + (I/2)/(b*n), 5/4 + (I/2)/(b*n), E^((2*I)*(a + b*Log[c*x^n]))] - (2*I + b*n)*(4*Sin[a + b*Log[c*x^n]]^2 + 3*b*n*Sin[2*(a + b*Log[c*x^n])]))/((2*I + b*n)*(-2*I + 3*b*n)*(2*I + 3*b*n)*x*Sqrt[Sin[a + b*Log[c*x^n]]])","A",0
62,1,168,111,1.2017653,"\int \frac{\sin ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}{x^3} \, dx","Integrate[Sin[a + b*Log[c*x^n]]^(3/2)/x^3,x]","\frac{6 i b^2 n^2 \left(-1+e^{2 i a} \left(c x^n\right)^{2 i b}\right) \, _2F_1\left(1,\frac{3}{4}+\frac{i}{b n};\frac{5}{4}+\frac{i}{b n};e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)-(b n+4 i) \left(8 \sin ^2\left(a+b \log \left(c x^n\right)\right)+3 b n \sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)\right)}{x^2 (b n+4 i) (3 b n-4 i) (3 b n+4 i) \sqrt{\sin \left(a+b \log \left(c x^n\right)\right)}}","-\frac{2 \, _2F_1\left(-\frac{3}{2},\frac{1}{4} \left(\frac{4 i}{b n}-3\right);\frac{1}{4} \left(1+\frac{4 i}{b n}\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sin ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}{x^2 (4+3 i b n) \left(1-e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{3/2}}",1,"((6*I)*b^2*n^2*(-1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))*Hypergeometric2F1[1, 3/4 + I/(b*n), 5/4 + I/(b*n), E^((2*I)*(a + b*Log[c*x^n]))] - (4*I + b*n)*(8*Sin[a + b*Log[c*x^n]]^2 + 3*b*n*Sin[2*(a + b*Log[c*x^n])]))/((4*I + b*n)*(-4*I + 3*b*n)*(4*I + 3*b*n)*x^2*Sqrt[Sin[a + b*Log[c*x^n]]])","A",0
63,1,96,109,0.3792908,"\int \frac{1}{\sqrt{\sin \left(a+b \log \left(c x^n\right)\right)}} \, dx","Integrate[1/Sqrt[Sin[a + b*Log[c*x^n]]],x]","\frac{2 x \left(-1+e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \, _2F_1\left(1,\frac{3}{4}-\frac{i}{2 b n};\frac{5}{4}-\frac{i}{2 b n};e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)}{(-2-i b n) \sqrt{\sin \left(a+b \log \left(c x^n\right)\right)}}","\frac{2 x \sqrt{1-e^{2 i a} \left(c x^n\right)^{2 i b}} \, _2F_1\left(\frac{1}{2},\frac{1}{4} \left(1-\frac{2 i}{b n}\right);\frac{1}{4} \left(5-\frac{2 i}{b n}\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{(2+i b n) \sqrt{\sin \left(a+b \log \left(c x^n\right)\right)}}",1,"(2*(-1 + E^((2*I)*(a + b*Log[c*x^n])))*x*Hypergeometric2F1[1, 3/4 - (I/2)/(b*n), 5/4 - (I/2)/(b*n), E^((2*I)*(a + b*Log[c*x^n]))])/((-2 - I*b*n)*Sqrt[Sin[a + b*Log[c*x^n]]])","A",0
64,1,32,29,0.0921651,"\int \frac{1}{x \sqrt{\sin \left(a+b \log \left(c x^n\right)\right)}} \, dx","Integrate[1/(x*Sqrt[Sin[a + b*Log[c*x^n]]]),x]","-\frac{2 F\left(\left.\frac{1}{2} \left(-a-b \log \left(c x^n\right)+\frac{\pi }{2}\right)\right|2\right)}{b n}","\frac{2 F\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)-\frac{\pi }{2}\right)\right|2\right)}{b n}",1,"(-2*EllipticF[(-a + Pi/2 - b*Log[c*x^n])/2, 2])/(b*n)","A",1
65,1,96,109,0.9185854,"\int \frac{1}{\sin ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[Sin[a + b*Log[c*x^n]]^(-3/2),x]","\frac{2 x \left(-1+e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \, _2F_1\left(1,\frac{1}{4}-\frac{i}{2 b n};\frac{7}{4}-\frac{i}{2 b n};e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)}{(-2-3 i b n) \sin ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}","\frac{2 x \left(1-e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{3/2} \, _2F_1\left(\frac{3}{2},\frac{1}{4} \left(3-\frac{2 i}{b n}\right);\frac{1}{4} \left(7-\frac{2 i}{b n}\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{(2+3 i b n) \sin ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}",1,"(2*(-1 + E^((2*I)*(a + b*Log[c*x^n])))*x*Hypergeometric2F1[1, 1/4 - (I/2)/(b*n), 7/4 - (I/2)/(b*n), E^((2*I)*(a + b*Log[c*x^n]))])/((-2 - (3*I)*b*n)*Sin[a + b*Log[c*x^n]]^(3/2))","A",0
66,1,57,64,0.1781951,"\int \frac{1}{x \sin ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[1/(x*Sin[a + b*Log[c*x^n]]^(3/2)),x]","\frac{2 \left(E\left(\left.\frac{1}{4} \left(-2 a-2 b \log \left(c x^n\right)+\pi \right)\right|2\right)-\frac{\cos \left(a+b \log \left(c x^n\right)\right)}{\sqrt{\sin \left(a+b \log \left(c x^n\right)\right)}}\right)}{b n}","-\frac{2 E\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)-\frac{\pi }{2}\right)\right|2\right)}{b n}-\frac{2 \cos \left(a+b \log \left(c x^n\right)\right)}{b n \sqrt{\sin \left(a+b \log \left(c x^n\right)\right)}}",1,"(2*(EllipticE[(-2*a + Pi - 2*b*Log[c*x^n])/4, 2] - Cos[a + b*Log[c*x^n]]/Sqrt[Sin[a + b*Log[c*x^n]]]))/(b*n)","A",1
67,1,125,109,1.5063483,"\int \frac{1}{\sin ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[Sin[a + b*Log[c*x^n]]^(-5/2),x]","\frac{2 x \left(i (b n+2 i) \left(-1+e^{2 i a} \left(c x^n\right)^{2 i b}\right) \, _2F_1\left(1,\frac{3}{4}-\frac{i}{2 b n};\frac{5}{4}-\frac{i}{2 b n};e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)-b n \cot \left(a+b \log \left(c x^n\right)\right)-2\right)}{3 b^2 n^2 \sqrt{\sin \left(a+b \log \left(c x^n\right)\right)}}","\frac{2 x \left(1-e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{5/2} \, _2F_1\left(\frac{5}{2},\frac{1}{4} \left(5-\frac{2 i}{b n}\right);\frac{1}{4} \left(9-\frac{2 i}{b n}\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{(2+5 i b n) \sin ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right)}",1,"(2*x*(-2 - b*n*Cot[a + b*Log[c*x^n]] + I*(2*I + b*n)*(-1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))*Hypergeometric2F1[1, 3/4 - (I/2)/(b*n), 5/4 - (I/2)/(b*n), E^((2*I)*(a + b*Log[c*x^n]))]))/(3*b^2*n^2*Sqrt[Sin[a + b*Log[c*x^n]]])","A",0
68,1,61,68,0.1973622,"\int \frac{1}{x \sin ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[1/(x*Sin[a + b*Log[c*x^n]]^(5/2)),x]","\frac{2 \left(F\left(\left.\frac{1}{4} \left(2 a+2 b \log \left(c x^n\right)-\pi \right)\right|2\right)-\frac{\cos \left(a+b \log \left(c x^n\right)\right)}{\sin ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}\right)}{3 b n}","\frac{2 F\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)-\frac{\pi }{2}\right)\right|2\right)}{3 b n}-\frac{2 \cos \left(a+b \log \left(c x^n\right)\right)}{3 b n \sin ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}",1,"(2*(EllipticF[(2*a - Pi + 2*b*Log[c*x^n])/4, 2] - Cos[a + b*Log[c*x^n]]/Sin[a + b*Log[c*x^n]]^(3/2)))/(3*b*n)","A",1
69,1,81,49,0.1343365,"\int \frac{1}{\sin ^{\frac{3}{2}}(a-2 i \log (c x))} \, dx","Integrate[Sin[a - (2*I)*Log[c*x]]^(-3/2),x]","\frac{x (\cos (a)-i \sin (a)) \sqrt{\frac{2 \sin (a) \left(c^4 x^4+1\right)-2 i \cos (a) \left(c^4 x^4-1\right)}{c^2 x^2}}}{\cos (a) \left(c^4 x^4-1\right)+i \sin (a) \left(c^4 x^4+1\right)}","\frac{e^{-2 i a} \left(1-e^{2 i a} c^4 x^4\right)}{2 c^4 x^3 \sin ^{\frac{3}{2}}(a-2 i \log (c x))}",1,"(x*(Cos[a] - I*Sin[a])*Sqrt[((-2*I)*(-1 + c^4*x^4)*Cos[a] + 2*(1 + c^4*x^4)*Sin[a])/(c^2*x^2)])/((-1 + c^4*x^4)*Cos[a] + I*(1 + c^4*x^4)*Sin[a])","A",1
70,1,341,337,1.9981461,"\int (e x)^m \sin ^4\left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[(e*x)^m*Sin[d*(a + b*Log[c*x^n])]^4,x]","\frac{1}{8} x (e x)^m \left(\frac{4 \sin (2 b d n \log (x)) \left((m+1) \sin \left(2 d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)-2 b d n \cos \left(2 d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)\right)}{4 b^2 d^2 n^2+m^2+2 m+1}-\frac{4 \cos (2 b d n \log (x)) \left((m+1) \cos \left(2 d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)+2 b d n \sin \left(2 d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)\right)}{4 b^2 d^2 n^2+m^2+2 m+1}-\frac{\sin (4 b d n \log (x)) \left((m+1) \sin \left(4 d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)-4 b d n \cos \left(4 d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)\right)}{16 b^2 d^2 n^2+m^2+2 m+1}+\frac{\cos (4 b d n \log (x)) \left((m+1) \cos \left(4 d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)+4 b d n \sin \left(4 d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)\right)}{16 b^2 d^2 n^2+m^2+2 m+1}+\frac{3}{m+1}\right)","\frac{(m+1) (e x)^{m+1} \sin ^4\left(d \left(a+b \log \left(c x^n\right)\right)\right)}{e \left(16 b^2 d^2 n^2+(m+1)^2\right)}+\frac{12 b^2 d^2 (m+1) n^2 (e x)^{m+1} \sin ^2\left(d \left(a+b \log \left(c x^n\right)\right)\right)}{e \left(4 b^2 d^2 n^2+(m+1)^2\right) \left(16 b^2 d^2 n^2+(m+1)^2\right)}-\frac{4 b d n (e x)^{m+1} \sin ^3\left(d \left(a+b \log \left(c x^n\right)\right)\right) \cos \left(d \left(a+b \log \left(c x^n\right)\right)\right)}{e \left(16 b^2 d^2 n^2+(m+1)^2\right)}-\frac{24 b^3 d^3 n^3 (e x)^{m+1} \sin \left(d \left(a+b \log \left(c x^n\right)\right)\right) \cos \left(d \left(a+b \log \left(c x^n\right)\right)\right)}{e \left(4 b^2 d^2 n^2+(m+1)^2\right) \left(16 b^2 d^2 n^2+(m+1)^2\right)}+\frac{24 b^4 d^4 n^4 (e x)^{m+1}}{e (m+1) \left(4 b^2 d^2 n^2+(m+1)^2\right) \left(16 b^2 d^2 n^2+(m+1)^2\right)}",1,"(x*(e*x)^m*(3/(1 + m) + (4*Sin[2*b*d*n*Log[x]]*(-2*b*d*n*Cos[2*d*(a - b*n*Log[x] + b*Log[c*x^n])] + (1 + m)*Sin[2*d*(a - b*n*Log[x] + b*Log[c*x^n])]))/(1 + 2*m + m^2 + 4*b^2*d^2*n^2) - (4*Cos[2*b*d*n*Log[x]]*((1 + m)*Cos[2*d*(a - b*n*Log[x] + b*Log[c*x^n])] + 2*b*d*n*Sin[2*d*(a - b*n*Log[x] + b*Log[c*x^n])]))/(1 + 2*m + m^2 + 4*b^2*d^2*n^2) - (Sin[4*b*d*n*Log[x]]*(-4*b*d*n*Cos[4*d*(a - b*n*Log[x] + b*Log[c*x^n])] + (1 + m)*Sin[4*d*(a - b*n*Log[x] + b*Log[c*x^n])]))/(1 + 2*m + m^2 + 16*b^2*d^2*n^2) + (Cos[4*b*d*n*Log[x]]*((1 + m)*Cos[4*d*(a - b*n*Log[x] + b*Log[c*x^n])] + 4*b*d*n*Sin[4*d*(a - b*n*Log[x] + b*Log[c*x^n])]))/(1 + 2*m + m^2 + 16*b^2*d^2*n^2)))/8","A",1
71,1,326,256,1.3096372,"\int (e x)^m \sin ^3\left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[(e*x)^m*Sin[d*(a + b*Log[c*x^n])]^3,x]","\frac{1}{4} x (e x)^m \left(\frac{3 \cos (b d n \log (x)) \left((m+1) \sin \left(d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)-b d n \cos \left(d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)\right)}{b^2 d^2 n^2+m^2+2 m+1}+\frac{3 \sin (b d n \log (x)) \left((m+1) \cos \left(d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)+b d n \sin \left(d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)\right)}{b^2 d^2 n^2+m^2+2 m+1}-\frac{\cos (3 b d n \log (x)) \left((m+1) \sin \left(3 d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)-3 b d n \cos \left(3 d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)\right)}{9 b^2 d^2 n^2+m^2+2 m+1}-\frac{\sin (3 b d n \log (x)) \left((m+1) \cos \left(3 d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)+3 b d n \sin \left(3 d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)\right)}{9 b^2 d^2 n^2+m^2+2 m+1}\right)","\frac{(m+1) (e x)^{m+1} \sin ^3\left(d \left(a+b \log \left(c x^n\right)\right)\right)}{e \left(9 b^2 d^2 n^2+(m+1)^2\right)}+\frac{6 b^2 d^2 (m+1) n^2 (e x)^{m+1} \sin \left(d \left(a+b \log \left(c x^n\right)\right)\right)}{e \left(b^2 d^2 n^2+(m+1)^2\right) \left(9 b^2 d^2 n^2+(m+1)^2\right)}-\frac{3 b d n (e x)^{m+1} \sin ^2\left(d \left(a+b \log \left(c x^n\right)\right)\right) \cos \left(d \left(a+b \log \left(c x^n\right)\right)\right)}{e \left(9 b^2 d^2 n^2+(m+1)^2\right)}-\frac{6 b^3 d^3 n^3 (e x)^{m+1} \cos \left(d \left(a+b \log \left(c x^n\right)\right)\right)}{e \left(b^2 d^2 n^2+(m+1)^2\right) \left(9 b^2 d^2 n^2+(m+1)^2\right)}",1,"(x*(e*x)^m*((3*Cos[b*d*n*Log[x]]*(-(b*d*n*Cos[d*(a - b*n*Log[x] + b*Log[c*x^n])]) + (1 + m)*Sin[d*(a - b*n*Log[x] + b*Log[c*x^n])]))/(1 + 2*m + m^2 + b^2*d^2*n^2) + (3*Sin[b*d*n*Log[x]]*((1 + m)*Cos[d*(a - b*n*Log[x] + b*Log[c*x^n])] + b*d*n*Sin[d*(a - b*n*Log[x] + b*Log[c*x^n])]))/(1 + 2*m + m^2 + b^2*d^2*n^2) - (Cos[3*b*d*n*Log[x]]*(-3*b*d*n*Cos[3*d*(a - b*n*Log[x] + b*Log[c*x^n])] + (1 + m)*Sin[3*d*(a - b*n*Log[x] + b*Log[c*x^n])]))/(1 + 2*m + m^2 + 9*b^2*d^2*n^2) - (Sin[3*b*d*n*Log[x]]*((1 + m)*Cos[3*d*(a - b*n*Log[x] + b*Log[c*x^n])] + 3*b*d*n*Sin[3*d*(a - b*n*Log[x] + b*Log[c*x^n])]))/(1 + 2*m + m^2 + 9*b^2*d^2*n^2)))/4","A",1
72,1,102,154,0.3028082,"\int (e x)^m \sin ^2\left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[(e*x)^m*Sin[d*(a + b*Log[c*x^n])]^2,x]","-\frac{x (e x)^m \left(2 b d (m+1) n \sin \left(2 d \left(a+b \log \left(c x^n\right)\right)\right)+(m+1)^2 \cos \left(2 d \left(a+b \log \left(c x^n\right)\right)\right)-4 b^2 d^2 n^2-m^2-2 m-1\right)}{2 (m+1) (-2 i b d n+m+1) (2 i b d n+m+1)}","\frac{(m+1) (e x)^{m+1} \sin ^2\left(d \left(a+b \log \left(c x^n\right)\right)\right)}{e \left(4 b^2 d^2 n^2+(m+1)^2\right)}-\frac{2 b d n (e x)^{m+1} \sin \left(d \left(a+b \log \left(c x^n\right)\right)\right) \cos \left(d \left(a+b \log \left(c x^n\right)\right)\right)}{e \left(4 b^2 d^2 n^2+(m+1)^2\right)}+\frac{2 b^2 d^2 n^2 (e x)^{m+1}}{e (m+1) \left(4 b^2 d^2 n^2+(m+1)^2\right)}",1,"-1/2*(x*(e*x)^m*(-1 - 2*m - m^2 - 4*b^2*d^2*n^2 + (1 + m)^2*Cos[2*d*(a + b*Log[c*x^n])] + 2*b*d*(1 + m)*n*Sin[2*d*(a + b*Log[c*x^n])]))/((1 + m)*(1 + m - (2*I)*b*d*n)*(1 + m + (2*I)*b*d*n))","C",1
73,1,63,92,0.1661968,"\int (e x)^m \sin \left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[(e*x)^m*Sin[d*(a + b*Log[c*x^n])],x]","\frac{x (e x)^m \left((m+1) \sin \left(d \left(a+b \log \left(c x^n\right)\right)\right)-b d n \cos \left(d \left(a+b \log \left(c x^n\right)\right)\right)\right)}{b^2 d^2 n^2+m^2+2 m+1}","\frac{(m+1) (e x)^{m+1} \sin \left(d \left(a+b \log \left(c x^n\right)\right)\right)}{e \left(b^2 d^2 n^2+(m+1)^2\right)}-\frac{b d n (e x)^{m+1} \cos \left(d \left(a+b \log \left(c x^n\right)\right)\right)}{e \left(b^2 d^2 n^2+(m+1)^2\right)}",1,"(x*(e*x)^m*(-(b*d*n*Cos[d*(a + b*Log[c*x^n])]) + (1 + m)*Sin[d*(a + b*Log[c*x^n])]))/(1 + 2*m + m^2 + b^2*d^2*n^2)","A",1
74,1,235,150,2.0437409,"\int (e x)^m \sin ^{\frac{3}{2}}\left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[(e*x)^m*Sin[d*(a + b*Log[c*x^n])]^(3/2),x]","\frac{2 (e x)^m \left(x (i b d n+2 m+2) \sin \left(d \left(a+b \log \left(c x^n\right)\right)\right) \left(2 (m+1) \sin \left(d \left(a+b \log \left(c x^n\right)\right)\right)-3 b d n \cos \left(d \left(a+b \log \left(c x^n\right)\right)\right)\right)-3 b^2 d^2 n^2 x \left(-1+e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right) \, _2F_1\left(1,-\frac{2 i m-3 b d n+2 i}{4 b d n};-\frac{2 i m-5 b d n+2 i}{4 b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)\right)}{(i b d n+2 m+2) (-3 i b d n+2 m+2) (3 i b d n+2 m+2) \sqrt{\sin \left(d \left(a+b \log \left(c x^n\right)\right)\right)}}","\frac{2 (e x)^{m+1} \, _2F_1\left(-\frac{3}{2},-\frac{2 i m+3 b d n+2 i}{4 b d n};-\frac{2 i m-b d n+2 i}{4 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d}\right) \sin ^{\frac{3}{2}}\left(d \left(a+b \log \left(c x^n\right)\right)\right)}{e (-3 i b d n+2 m+2) \left(1-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)^{3/2}}",1,"(2*(e*x)^m*(-3*b^2*d^2*(-1 + E^((2*I)*d*(a + b*Log[c*x^n])))*n^2*x*Hypergeometric2F1[1, -1/4*(2*I + (2*I)*m - 3*b*d*n)/(b*d*n), -1/4*(2*I + (2*I)*m - 5*b*d*n)/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] + (2 + 2*m + I*b*d*n)*x*Sin[d*(a + b*Log[c*x^n])]*(-3*b*d*n*Cos[d*(a + b*Log[c*x^n])] + 2*(1 + m)*Sin[d*(a + b*Log[c*x^n])])))/((2 + 2*m + I*b*d*n)*(2 + 2*m - (3*I)*b*d*n)*(2 + 2*m + (3*I)*b*d*n)*Sqrt[Sin[d*(a + b*Log[c*x^n])]])","A",0
75,1,488,149,5.6089666,"\int (e x)^m \sqrt{\sin \left(d \left(a+b \log \left(c x^n\right)\right)\right)} \, dx","Integrate[(e*x)^m*Sqrt[Sin[d*(a + b*Log[c*x^n])]],x]","2 x (e x)^m \left(\frac{\sqrt{\sin \left(d \left(a+b \log \left(c x^n\right)\right)\right)} \sin \left(d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)}{2 (m+1) \sin \left(d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)+b d n \cos \left(d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)}-\frac{b d n x^{-i b d n} \sqrt{2-2 e^{2 i a d} \left(c x^n\right)^{2 i b d}} e^{i d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)} \left((3 b d n-2 i m-2 i) \, _2F_1\left(\frac{1}{2},-\frac{2 i m+b d n+2 i}{4 b d n};-\frac{2 i m-3 b d n+2 i}{4 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)+(b d n+2 i m+2 i) x^{2 i b d n} \, _2F_1\left(\frac{1}{2},-\frac{i \left(m+\frac{3}{2} i b d n+1\right)}{2 b d n};-\frac{2 i m-7 b d n+2 i}{4 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)\right)}{(-i b d n+2 m+2) (3 i b d n+2 m+2) \sqrt{-i e^{-i a d} \left(c x^n\right)^{-i b d} \left(-1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)} \left((b d n-2 i m-2 i) e^{2 i d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)}+b d n+2 i m+2 i\right)}\right)","\frac{2 (e x)^{m+1} \, _2F_1\left(-\frac{1}{2},-\frac{2 i m+b d n+2 i}{4 b d n};-\frac{2 i m-3 b d n+2 i}{4 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d}\right) \sqrt{\sin \left(d \left(a+b \log \left(c x^n\right)\right)\right)}}{e (-i b d n+2 m+2) \sqrt{1-e^{2 i a d} \left(c x^n\right)^{2 i b d}}}",1,"2*x*(e*x)^m*(-((b*d*E^(I*d*(a - b*n*Log[x] + b*Log[c*x^n]))*n*Sqrt[2 - 2*E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)]*((2*I + (2*I)*m + b*d*n)*x^((2*I)*b*d*n)*Hypergeometric2F1[1/2, ((-1/2*I)*(1 + m + ((3*I)/2)*b*d*n))/(b*d*n), -1/4*(2*I + (2*I)*m - 7*b*d*n)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)] + (-2*I - (2*I)*m + 3*b*d*n)*Hypergeometric2F1[1/2, -1/4*(2*I + (2*I)*m + b*d*n)/(b*d*n), -1/4*(2*I + (2*I)*m - 3*b*d*n)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)]))/((2 + 2*m - I*b*d*n)*(2 + 2*m + (3*I)*b*d*n)*(2*I + (2*I)*m + b*d*n + E^((2*I)*d*(a - b*n*Log[x] + b*Log[c*x^n]))*(-2*I - (2*I)*m + b*d*n))*x^(I*b*d*n)*Sqrt[((-I)*(-1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)))/(E^(I*a*d)*(c*x^n)^(I*b*d))])) + (Sqrt[Sin[d*(a + b*Log[c*x^n])]]*Sin[d*(a - b*n*Log[x] + b*Log[c*x^n])])/(b*d*n*Cos[d*(a - b*n*Log[x] + b*Log[c*x^n])] + 2*(1 + m)*Sin[d*(a - b*n*Log[x] + b*Log[c*x^n])]))","B",0
76,1,131,150,0.5281749,"\int \frac{(e x)^m}{\sqrt{\sin \left(d \left(a+b \log \left(c x^n\right)\right)\right)}} \, dx","Integrate[(e*x)^m/Sqrt[Sin[d*(a + b*Log[c*x^n])]],x]","-\frac{2 x (e x)^m \left(-1+e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right) \, _2F_1\left(1,-\frac{2 i m-3 b d n+2 i}{4 b d n};-\frac{2 i m-5 b d n+2 i}{4 b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)}{(i b d n+2 m+2) \sqrt{\sin \left(d \left(a+b \log \left(c x^n\right)\right)\right)}}","\frac{2 (e x)^{m+1} \sqrt{1-e^{2 i a d} \left(c x^n\right)^{2 i b d}} \, _2F_1\left(\frac{1}{2},-\frac{2 i m-b d n+2 i}{4 b d n};-\frac{2 i m-5 b d n+2 i}{4 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{e (i b d n+2 m+2) \sqrt{\sin \left(d \left(a+b \log \left(c x^n\right)\right)\right)}}",1,"(-2*(-1 + E^((2*I)*d*(a + b*Log[c*x^n])))*x*(e*x)^m*Hypergeometric2F1[1, -1/4*(2*I + (2*I)*m - 3*b*d*n)/(b*d*n), -1/4*(2*I + (2*I)*m - 5*b*d*n)/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))])/((2 + 2*m + I*b*d*n)*Sqrt[Sin[d*(a + b*Log[c*x^n])]])","A",0
77,1,544,150,5.15711,"\int \frac{(e x)^m}{\sin ^{\frac{3}{2}}\left(d \left(a+b \log \left(c x^n\right)\right)\right)} \, dx","Integrate[(e*x)^m/Sin[d*(a + b*Log[c*x^n])]^(3/2),x]","\frac{(e x)^m \left(b^2 d^2 n^2+4 m^2+8 m+4\right) x^{1+i b d n} \sqrt{2-2 e^{2 i a d} \left(c x^n\right)^{2 i b d}} \, _2F_1\left(\frac{1}{2},-\frac{i \left(m+\frac{3}{2} i b d n+1\right)}{2 b d n};-\frac{2 i m-7 b d n+2 i}{4 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)+\frac{(e x)^m (3 b d n-2 i m-2 i) x^{1-i b d n} \left((b d n-2 i m-2 i) \sqrt{2-2 e^{2 i a d} \left(c x^n\right)^{2 i b d}} \, _2F_1\left(\frac{1}{2},-\frac{2 i m+b d n+2 i}{4 b d n};-\frac{2 i m-3 b d n+2 i}{4 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d}\right) \sqrt{\sin \left(d \left(a+b \log \left(c x^n\right)\right)\right)}-2 x^{i b d n} \sqrt{-i e^{-i a d} \left(c x^n\right)^{-i b d} \left(-1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)} (b d n \cos (b d n \log (x))-2 (m+1) \sin (b d n \log (x)))\right)}{\sqrt{\sin \left(d \left(a+b \log \left(c x^n\right)\right)\right)}}}{b d n (3 b d n-2 i m-2 i) \sqrt{-i e^{-i a d} \left(c x^n\right)^{-i b d} \left(-1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)} \left(2 (m+1) \sin \left(d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)+b d n \cos \left(d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)\right)}","\frac{2 (e x)^{m+1} \left(1-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)^{3/2} \, _2F_1\left(\frac{3}{2},-\frac{2 i m-3 b d n+2 i}{4 b d n};-\frac{2 i m-7 b d n+2 i}{4 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{e (3 i b d n+2 m+2) \sin ^{\frac{3}{2}}\left(d \left(a+b \log \left(c x^n\right)\right)\right)}",1,"((4 + 8*m + 4*m^2 + b^2*d^2*n^2)*x^(1 + I*b*d*n)*(e*x)^m*Sqrt[2 - 2*E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)]*Hypergeometric2F1[1/2, ((-1/2*I)*(1 + m + ((3*I)/2)*b*d*n))/(b*d*n), -1/4*(2*I + (2*I)*m - 7*b*d*n)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)] + ((-2*I - (2*I)*m + 3*b*d*n)*x^(1 - I*b*d*n)*(e*x)^m*(-2*x^(I*b*d*n)*Sqrt[((-I)*(-1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)))/(E^(I*a*d)*(c*x^n)^(I*b*d))]*(b*d*n*Cos[b*d*n*Log[x]] - 2*(1 + m)*Sin[b*d*n*Log[x]]) + (-2*I - (2*I)*m + b*d*n)*Sqrt[2 - 2*E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)]*Hypergeometric2F1[1/2, -1/4*(2*I + (2*I)*m + b*d*n)/(b*d*n), -1/4*(2*I + (2*I)*m - 3*b*d*n)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)]*Sqrt[Sin[d*(a + b*Log[c*x^n])]]))/Sqrt[Sin[d*(a + b*Log[c*x^n])]])/(b*d*n*(-2*I - (2*I)*m + 3*b*d*n)*Sqrt[((-I)*(-1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)))/(E^(I*a*d)*(c*x^n)^(I*b*d))]*(b*d*n*Cos[d*(a - b*n*Log[x] + b*Log[c*x^n])] + 2*(1 + m)*Sin[d*(a - b*n*Log[x] + b*Log[c*x^n])]))","B",0
78,1,214,150,2.4418112,"\int \frac{(e x)^m}{\sin ^{\frac{5}{2}}\left(d \left(a+b \log \left(c x^n\right)\right)\right)} \, dx","Integrate[(e*x)^m/Sin[d*(a + b*Log[c*x^n])]^(5/2),x]","\frac{2 x (e x)^m \left(-(-i b d n+2 m+2) \left(-1+e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right) \, _2F_1\left(1,-\frac{2 i m-3 b d n+2 i}{4 b d n};-\frac{2 i m-5 b d n+2 i}{4 b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)-b d n \cot \left(d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)+b d n \sin (b d n \log (x)) \csc \left(d \left(a+b \log \left(c x^n\right)\right)\right) \csc \left(d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)-2 m-2\right)}{3 b^2 d^2 n^2 \sqrt{\sin \left(d \left(a+b \log \left(c x^n\right)\right)\right)}}","\frac{2 (e x)^{m+1} \left(1-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)^{5/2} \, _2F_1\left(\frac{5}{2},-\frac{2 i m-5 b d n+2 i}{4 b d n};-\frac{2 i m-9 b d n+2 i}{4 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{e (5 i b d n+2 m+2) \sin ^{\frac{5}{2}}\left(d \left(a+b \log \left(c x^n\right)\right)\right)}",1,"(2*x*(e*x)^m*(-2 - 2*m - b*d*n*Cot[d*(a - b*n*Log[x] + b*Log[c*x^n])] - (-1 + E^((2*I)*d*(a + b*Log[c*x^n])))*(2 + 2*m - I*b*d*n)*Hypergeometric2F1[1, -1/4*(2*I + (2*I)*m - 3*b*d*n)/(b*d*n), -1/4*(2*I + (2*I)*m - 5*b*d*n)/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] + b*d*n*Csc[d*(a + b*Log[c*x^n])]*Csc[d*(a - b*n*Log[x] + b*Log[c*x^n])]*Sin[b*d*n*Log[x]]))/(3*b^2*d^2*n^2*Sqrt[Sin[d*(a + b*Log[c*x^n])]])","A",0
79,1,122,144,0.9988564,"\int (e x)^m \sin ^p\left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[(e*x)^m*Sin[d*(a + b*Log[c*x^n])]^p,x]","-\frac{x (e x)^m \left(-1+e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right) \sin ^p\left(d \left(a+b \log \left(c x^n\right)\right)\right) \, _2F_1\left(1,\frac{1}{2} \left(-\frac{i (m+1)}{b d n}+p+2\right);-\frac{i (m+1)}{2 b d n}-\frac{p}{2}+1;e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)}{-i b d n p+m+1}","\frac{(e x)^{m+1} \left(1-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)^{-p} \, _2F_1\left(-p,-\frac{i m+b d n p+i}{2 b d n};\frac{1}{2} \left(-\frac{i (m+1)}{b d n}-p+2\right);e^{2 i a d} \left(c x^n\right)^{2 i b d}\right) \sin ^p\left(d \left(a+b \log \left(c x^n\right)\right)\right)}{e (-i b d n p+m+1)}",1,"-(((-1 + E^((2*I)*d*(a + b*Log[c*x^n])))*x*(e*x)^m*Hypergeometric2F1[1, (2 - (I*(1 + m))/(b*d*n) + p)/2, 1 - ((I/2)*(1 + m))/(b*d*n) - p/2, E^((2*I)*d*(a + b*Log[c*x^n]))]*Sin[d*(a + b*Log[c*x^n])]^p)/(1 + m - I*b*d*n*p))","A",0
80,1,100,114,0.6792735,"\int x^2 \sin ^p\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x^2*Sin[a + b*Log[c*x^n]]^p,x]","\frac{x^3 \left(-1+e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \, _2F_1\left(1,\frac{1}{2} \left(p-\frac{3 i}{b n}+2\right);-\frac{p}{2}-\frac{3 i}{2 b n}+1;e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \sin ^p\left(a+b \log \left(c x^n\right)\right)}{-3+i b n p}","\frac{x^3 \left(1-e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{-p} \, _2F_1\left(-p,-\frac{b n p+3 i}{2 b n};\frac{1}{2} \left(-p-\frac{3 i}{b n}+2\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sin ^p\left(a+b \log \left(c x^n\right)\right)}{3-i b n p}",1,"((-1 + E^((2*I)*(a + b*Log[c*x^n])))*x^3*Hypergeometric2F1[1, (2 - (3*I)/(b*n) + p)/2, 1 - ((3*I)/2)/(b*n) - p/2, E^((2*I)*(a + b*Log[c*x^n]))]*Sin[a + b*Log[c*x^n]]^p)/(-3 + I*b*n*p)","A",0
81,1,98,114,0.6134798,"\int x \sin ^p\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x*Sin[a + b*Log[c*x^n]]^p,x]","\frac{x^2 \left(-1+e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \, _2F_1\left(1,\frac{p}{2}-\frac{i}{b n}+1;-\frac{p}{2}-\frac{i}{b n}+1;e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \sin ^p\left(a+b \log \left(c x^n\right)\right)}{-2+i b n p}","\frac{x^2 \left(1-e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{-p} \, _2F_1\left(\frac{1}{2} \left(-p-\frac{2 i}{b n}\right),-p;\frac{1}{2} \left(-p-\frac{2 i}{b n}+2\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sin ^p\left(a+b \log \left(c x^n\right)\right)}{2-i b n p}",1,"((-1 + E^((2*I)*(a + b*Log[c*x^n])))*x^2*Hypergeometric2F1[1, 1 - I/(b*n) + p/2, 1 - I/(b*n) - p/2, E^((2*I)*(a + b*Log[c*x^n]))]*Sin[a + b*Log[c*x^n]]^p)/(-2 + I*b*n*p)","A",0
82,1,98,112,0.5589542,"\int \sin ^p\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[Sin[a + b*Log[c*x^n]]^p,x]","\frac{x \left(-1+e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \, _2F_1\left(1,\frac{1}{2} \left(p-\frac{i}{b n}+2\right);-\frac{p}{2}-\frac{i}{2 b n}+1;e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \sin ^p\left(a+b \log \left(c x^n\right)\right)}{-1+i b n p}","\frac{x \left(1-e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{-p} \, _2F_1\left(-p,-\frac{b n p+i}{2 b n};\frac{1}{2} \left(-p-\frac{i}{b n}+2\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sin ^p\left(a+b \log \left(c x^n\right)\right)}{1-i b n p}",1,"((-1 + E^((2*I)*(a + b*Log[c*x^n])))*x*Hypergeometric2F1[1, (2 - I/(b*n) + p)/2, 1 - (I/2)/(b*n) - p/2, E^((2*I)*(a + b*Log[c*x^n]))]*Sin[a + b*Log[c*x^n]]^p)/(-1 + I*b*n*p)","A",0
83,1,86,86,0.1478302,"\int \frac{\sin ^p\left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Sin[a + b*Log[c*x^n]]^p/x,x]","\frac{\sec \left(a+b \log \left(c x^n\right)\right) \sqrt{\cos ^2\left(a+b \log \left(c x^n\right)\right)} \sin ^{p+1}\left(a+b \log \left(c x^n\right)\right) \, _2F_1\left(\frac{1}{2},\frac{p+1}{2};\frac{p+3}{2};\sin ^2\left(a+b \log \left(c x^n\right)\right)\right)}{b n (p+1)}","\frac{\cos \left(a+b \log \left(c x^n\right)\right) \sin ^{p+1}\left(a+b \log \left(c x^n\right)\right) \, _2F_1\left(\frac{1}{2},\frac{p+1}{2};\frac{p+3}{2};\sin ^2\left(a+b \log \left(c x^n\right)\right)\right)}{b n (p+1) \sqrt{\cos ^2\left(a+b \log \left(c x^n\right)\right)}}",1,"(Sqrt[Cos[a + b*Log[c*x^n]]^2]*Hypergeometric2F1[1/2, (1 + p)/2, (3 + p)/2, Sin[a + b*Log[c*x^n]]^2]*Sec[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^(1 + p))/(b*n*(1 + p))","A",1
84,1,102,115,0.6353529,"\int \frac{\sin ^p\left(a+b \log \left(c x^n\right)\right)}{x^2} \, dx","Integrate[Sin[a + b*Log[c*x^n]]^p/x^2,x]","-\frac{i \left(-1+e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \, _2F_1\left(1,\frac{1}{2} \left(p+\frac{i}{b n}+2\right);-\frac{p}{2}+\frac{i}{2 b n}+1;e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \sin ^p\left(a+b \log \left(c x^n\right)\right)}{x (b n p-i)}","-\frac{\left(1-e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{-p} \, _2F_1\left(\frac{1}{2} \left(\frac{i}{b n}-p\right),-p;\frac{1}{2} \left(-p+\frac{i}{b n}+2\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sin ^p\left(a+b \log \left(c x^n\right)\right)}{x (1+i b n p)}",1,"((-I)*(-1 + E^((2*I)*(a + b*Log[c*x^n])))*Hypergeometric2F1[1, (2 + I/(b*n) + p)/2, 1 + (I/2)/(b*n) - p/2, E^((2*I)*(a + b*Log[c*x^n]))]*Sin[a + b*Log[c*x^n]]^p)/((-I + b*n*p)*x)","A",0
85,1,100,115,0.6370039,"\int \frac{\sin ^p\left(a+b \log \left(c x^n\right)\right)}{x^3} \, dx","Integrate[Sin[a + b*Log[c*x^n]]^p/x^3,x]","-\frac{i \left(-1+e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \, _2F_1\left(1,\frac{p}{2}+\frac{i}{b n}+1;-\frac{p}{2}+\frac{i}{b n}+1;e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \sin ^p\left(a+b \log \left(c x^n\right)\right)}{x^2 (b n p-2 i)}","-\frac{\left(1-e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{-p} \, _2F_1\left(\frac{1}{2} \left(\frac{2 i}{b n}-p\right),-p;\frac{1}{2} \left(-p+\frac{2 i}{b n}+2\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sin ^p\left(a+b \log \left(c x^n\right)\right)}{x^2 (2+i b n p)}",1,"((-I)*(-1 + E^((2*I)*(a + b*Log[c*x^n])))*Hypergeometric2F1[1, 1 + I/(b*n) + p/2, 1 + I/(b*n) - p/2, E^((2*I)*(a + b*Log[c*x^n]))]*Sin[a + b*Log[c*x^n]]^p)/((-2*I + b*n*p)*x^2)","A",0
86,1,43,56,0.0909285,"\int x^2 \cos \left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x^2*Cos[a + b*Log[c*x^n]],x]","\frac{x^3 \left(b n \sin \left(a+b \log \left(c x^n\right)\right)+3 \cos \left(a+b \log \left(c x^n\right)\right)\right)}{b^2 n^2+9}","\frac{b n x^3 \sin \left(a+b \log \left(c x^n\right)\right)}{b^2 n^2+9}+\frac{3 x^3 \cos \left(a+b \log \left(c x^n\right)\right)}{b^2 n^2+9}",1,"(x^3*(3*Cos[a + b*Log[c*x^n]] + b*n*Sin[a + b*Log[c*x^n]]))/(9 + b^2*n^2)","A",1
87,1,43,56,0.0783598,"\int x \cos \left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x*Cos[a + b*Log[c*x^n]],x]","\frac{x^2 \left(b n \sin \left(a+b \log \left(c x^n\right)\right)+2 \cos \left(a+b \log \left(c x^n\right)\right)\right)}{b^2 n^2+4}","\frac{b n x^2 \sin \left(a+b \log \left(c x^n\right)\right)}{b^2 n^2+4}+\frac{2 x^2 \cos \left(a+b \log \left(c x^n\right)\right)}{b^2 n^2+4}",1,"(x^2*(2*Cos[a + b*Log[c*x^n]] + b*n*Sin[a + b*Log[c*x^n]]))/(4 + b^2*n^2)","A",1
88,1,39,51,0.0540765,"\int \cos \left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[Cos[a + b*Log[c*x^n]],x]","\frac{x \left(b n \sin \left(a+b \log \left(c x^n\right)\right)+\cos \left(a+b \log \left(c x^n\right)\right)\right)}{b^2 n^2+1}","\frac{b n x \sin \left(a+b \log \left(c x^n\right)\right)}{b^2 n^2+1}+\frac{x \cos \left(a+b \log \left(c x^n\right)\right)}{b^2 n^2+1}",1,"(x*(Cos[a + b*Log[c*x^n]] + b*n*Sin[a + b*Log[c*x^n]]))/(1 + b^2*n^2)","A",1
89,1,37,18,0.0298729,"\int \frac{\cos \left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Cos[a + b*Log[c*x^n]]/x,x]","\frac{\sin (a) \cos \left(b \log \left(c x^n\right)\right)}{b n}+\frac{\cos (a) \sin \left(b \log \left(c x^n\right)\right)}{b n}","\frac{\sin \left(a+b \log \left(c x^n\right)\right)}{b n}",1,"(Cos[b*Log[c*x^n]]*Sin[a])/(b*n) + (Cos[a]*Sin[b*Log[c*x^n]])/(b*n)","B",1
90,1,41,56,0.0739636,"\int \frac{\cos \left(a+b \log \left(c x^n\right)\right)}{x^2} \, dx","Integrate[Cos[a + b*Log[c*x^n]]/x^2,x]","\frac{b n \sin \left(a+b \log \left(c x^n\right)\right)-\cos \left(a+b \log \left(c x^n\right)\right)}{b^2 n^2 x+x}","\frac{b n \sin \left(a+b \log \left(c x^n\right)\right)}{x \left(b^2 n^2+1\right)}-\frac{\cos \left(a+b \log \left(c x^n\right)\right)}{x \left(b^2 n^2+1\right)}",1,"(-Cos[a + b*Log[c*x^n]] + b*n*Sin[a + b*Log[c*x^n]])/(x + b^2*n^2*x)","A",1
91,1,61,97,0.1722657,"\int x^2 \cos ^2\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x^2*Cos[a + b*Log[c*x^n]]^2,x]","\frac{x^3 \left(6 b n \sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+9 \cos \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+4 b^2 n^2+9\right)}{6 \left(4 b^2 n^2+9\right)}","\frac{3 x^3 \cos ^2\left(a+b \log \left(c x^n\right)\right)}{4 b^2 n^2+9}+\frac{2 b n x^3 \sin \left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{4 b^2 n^2+9}+\frac{2 b^2 n^2 x^3}{3 \left(4 b^2 n^2+9\right)}",1,"(x^3*(9 + 4*b^2*n^2 + 9*Cos[2*(a + b*Log[c*x^n])] + 6*b*n*Sin[2*(a + b*Log[c*x^n])]))/(6*(9 + 4*b^2*n^2))","A",1
92,1,54,98,0.1063184,"\int x \cos ^2\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x*Cos[a + b*Log[c*x^n]]^2,x]","\frac{x^2 \left(b n \sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+\cos \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+b^2 n^2+1\right)}{4 b^2 n^2+4}","\frac{x^2 \cos ^2\left(a+b \log \left(c x^n\right)\right)}{2 \left(b^2 n^2+1\right)}+\frac{b n x^2 \sin \left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{2 \left(b^2 n^2+1\right)}+\frac{b^2 n^2 x^2}{4 \left(b^2 n^2+1\right)}",1,"(x^2*(1 + b^2*n^2 + Cos[2*(a + b*Log[c*x^n])] + b*n*Sin[2*(a + b*Log[c*x^n])]))/(4 + 4*b^2*n^2)","A",1
93,1,54,88,0.084465,"\int \cos ^2\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[Cos[a + b*Log[c*x^n]]^2,x]","\frac{x \left(2 b n \sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+\cos \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+4 b^2 n^2+1\right)}{8 b^2 n^2+2}","\frac{x \cos ^2\left(a+b \log \left(c x^n\right)\right)}{4 b^2 n^2+1}+\frac{2 b n x \sin \left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{4 b^2 n^2+1}+\frac{2 b^2 n^2 x}{4 b^2 n^2+1}",1,"(x*(1 + 4*b^2*n^2 + Cos[2*(a + b*Log[c*x^n])] + 2*b*n*Sin[2*(a + b*Log[c*x^n])]))/(2 + 8*b^2*n^2)","A",1
94,1,36,39,0.0664645,"\int \frac{\cos ^2\left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Cos[a + b*Log[c*x^n]]^2/x,x]","\frac{2 \left(a+b \log \left(c x^n\right)\right)+\sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)}{4 b n}","\frac{\sin \left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{2 b n}+\frac{\log (x)}{2}",1,"(2*(a + b*Log[c*x^n]) + Sin[2*(a + b*Log[c*x^n])])/(4*b*n)","A",1
95,1,57,95,0.1352417,"\int \frac{\cos ^2\left(a+b \log \left(c x^n\right)\right)}{x^2} \, dx","Integrate[Cos[a + b*Log[c*x^n]]^2/x^2,x]","-\frac{-2 b n \sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+\cos \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+4 b^2 n^2+1}{2 \left(4 b^2 n^2 x+x\right)}","-\frac{\cos ^2\left(a+b \log \left(c x^n\right)\right)}{x \left(4 b^2 n^2+1\right)}+\frac{2 b n \sin \left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{x \left(4 b^2 n^2+1\right)}-\frac{2 b^2 n^2}{x \left(4 b^2 n^2+1\right)}",1,"-1/2*(1 + 4*b^2*n^2 + Cos[2*(a + b*Log[c*x^n])] - 2*b*n*Sin[2*(a + b*Log[c*x^n])])/(x + 4*b^2*n^2*x)","A",1
96,1,120,160,0.5598291,"\int x^2 \cos ^3\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x^2*Cos[a + b*Log[c*x^n]]^3,x]","\frac{x^3 \left(27 \left(b^2 n^2+1\right) \cos \left(a+b \log \left(c x^n\right)\right)+\left(b^2 n^2+9\right) \cos \left(3 \left(a+b \log \left(c x^n\right)\right)\right)+2 b n \sin \left(a+b \log \left(c x^n\right)\right) \left(\left(b^2 n^2+9\right) \cos \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+5 b^2 n^2+9\right)\right)}{12 \left(b^4 n^4+10 b^2 n^2+9\right)}","\frac{x^3 \cos ^3\left(a+b \log \left(c x^n\right)\right)}{3 \left(b^2 n^2+1\right)}+\frac{b n x^3 \sin \left(a+b \log \left(c x^n\right)\right) \cos ^2\left(a+b \log \left(c x^n\right)\right)}{3 \left(b^2 n^2+1\right)}+\frac{2 b^2 n^2 x^3 \cos \left(a+b \log \left(c x^n\right)\right)}{b^4 n^4+10 b^2 n^2+9}+\frac{2 b^3 n^3 x^3 \sin \left(a+b \log \left(c x^n\right)\right)}{3 \left(b^4 n^4+10 b^2 n^2+9\right)}",1,"(x^3*(27*(1 + b^2*n^2)*Cos[a + b*Log[c*x^n]] + (9 + b^2*n^2)*Cos[3*(a + b*Log[c*x^n])] + 2*b*n*(9 + 5*b^2*n^2 + (9 + b^2*n^2)*Cos[2*(a + b*Log[c*x^n])])*Sin[a + b*Log[c*x^n]]))/(12*(9 + 10*b^2*n^2 + b^4*n^4))","A",1
97,1,123,158,0.5044697,"\int x \cos ^3\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x*Cos[a + b*Log[c*x^n]]^3,x]","\frac{x^2 \left(6 \left(9 b^2 n^2+4\right) \cos \left(a+b \log \left(c x^n\right)\right)+2 \left(b^2 n^2+4\right) \cos \left(3 \left(a+b \log \left(c x^n\right)\right)\right)+6 b n \sin \left(a+b \log \left(c x^n\right)\right) \left(\left(b^2 n^2+4\right) \cos \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+5 b^2 n^2+4\right)\right)}{4 \left(9 b^4 n^4+40 b^2 n^2+16\right)}","\frac{2 x^2 \cos ^3\left(a+b \log \left(c x^n\right)\right)}{9 b^2 n^2+4}+\frac{3 b n x^2 \sin \left(a+b \log \left(c x^n\right)\right) \cos ^2\left(a+b \log \left(c x^n\right)\right)}{9 b^2 n^2+4}+\frac{12 b^2 n^2 x^2 \cos \left(a+b \log \left(c x^n\right)\right)}{9 b^4 n^4+40 b^2 n^2+16}+\frac{6 b^3 n^3 x^2 \sin \left(a+b \log \left(c x^n\right)\right)}{9 b^4 n^4+40 b^2 n^2+16}",1,"(x^2*(6*(4 + 9*b^2*n^2)*Cos[a + b*Log[c*x^n]] + 2*(4 + b^2*n^2)*Cos[3*(a + b*Log[c*x^n])] + 6*b*n*(4 + 5*b^2*n^2 + (4 + b^2*n^2)*Cos[2*(a + b*Log[c*x^n])])*Sin[a + b*Log[c*x^n]]))/(4*(16 + 40*b^2*n^2 + 9*b^4*n^4))","A",1
98,1,117,149,0.4167196,"\int \cos ^3\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[Cos[a + b*Log[c*x^n]]^3,x]","\frac{x \left(3 \left(9 b^2 n^2+1\right) \cos \left(a+b \log \left(c x^n\right)\right)+\left(b^2 n^2+1\right) \cos \left(3 \left(a+b \log \left(c x^n\right)\right)\right)+6 b n \sin \left(a+b \log \left(c x^n\right)\right) \left(\left(b^2 n^2+1\right) \cos \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+5 b^2 n^2+1\right)\right)}{36 b^4 n^4+40 b^2 n^2+4}","\frac{x \cos ^3\left(a+b \log \left(c x^n\right)\right)}{9 b^2 n^2+1}+\frac{3 b n x \sin \left(a+b \log \left(c x^n\right)\right) \cos ^2\left(a+b \log \left(c x^n\right)\right)}{9 b^2 n^2+1}+\frac{6 b^2 n^2 x \cos \left(a+b \log \left(c x^n\right)\right)}{9 b^4 n^4+10 b^2 n^2+1}+\frac{6 b^3 n^3 x \sin \left(a+b \log \left(c x^n\right)\right)}{9 b^4 n^4+10 b^2 n^2+1}",1,"(x*(3*(1 + 9*b^2*n^2)*Cos[a + b*Log[c*x^n]] + (1 + b^2*n^2)*Cos[3*(a + b*Log[c*x^n])] + 6*b*n*(1 + 5*b^2*n^2 + (1 + b^2*n^2)*Cos[2*(a + b*Log[c*x^n])])*Sin[a + b*Log[c*x^n]]))/(4 + 40*b^2*n^2 + 36*b^4*n^4)","A",1
99,1,42,42,0.0556707,"\int \frac{\cos ^3\left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Cos[a + b*Log[c*x^n]]^3/x,x]","\frac{\sin \left(a+b \log \left(c x^n\right)\right)}{b n}-\frac{\sin ^3\left(a+b \log \left(c x^n\right)\right)}{3 b n}","\frac{\sin \left(a+b \log \left(c x^n\right)\right)}{b n}-\frac{\sin ^3\left(a+b \log \left(c x^n\right)\right)}{3 b n}",1,"Sin[a + b*Log[c*x^n]]/(b*n) - Sin[a + b*Log[c*x^n]]^3/(3*b*n)","A",1
100,1,122,158,0.4803542,"\int \frac{\cos ^3\left(a+b \log \left(c x^n\right)\right)}{x^2} \, dx","Integrate[Cos[a + b*Log[c*x^n]]^3/x^2,x]","-\frac{3 \left(9 b^2 n^2+1\right) \cos \left(a+b \log \left(c x^n\right)\right)+\left(b^2 n^2+1\right) \cos \left(3 \left(a+b \log \left(c x^n\right)\right)\right)-6 b n \sin \left(a+b \log \left(c x^n\right)\right) \left(\left(b^2 n^2+1\right) \cos \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+5 b^2 n^2+1\right)}{4 x \left(9 b^4 n^4+10 b^2 n^2+1\right)}","-\frac{\cos ^3\left(a+b \log \left(c x^n\right)\right)}{x \left(9 b^2 n^2+1\right)}+\frac{3 b n \sin \left(a+b \log \left(c x^n\right)\right) \cos ^2\left(a+b \log \left(c x^n\right)\right)}{x \left(9 b^2 n^2+1\right)}-\frac{6 b^2 n^2 \cos \left(a+b \log \left(c x^n\right)\right)}{x \left(9 b^4 n^4+10 b^2 n^2+1\right)}+\frac{6 b^3 n^3 \sin \left(a+b \log \left(c x^n\right)\right)}{x \left(9 b^4 n^4+10 b^2 n^2+1\right)}",1,"-1/4*(3*(1 + 9*b^2*n^2)*Cos[a + b*Log[c*x^n]] + (1 + b^2*n^2)*Cos[3*(a + b*Log[c*x^n])] - 6*b*n*(1 + 5*b^2*n^2 + (1 + b^2*n^2)*Cos[2*(a + b*Log[c*x^n])])*Sin[a + b*Log[c*x^n]])/((1 + 10*b^2*n^2 + 9*b^4*n^4)*x)","A",1
101,1,167,191,0.4431609,"\int \cos ^4\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[Cos[a + b*Log[c*x^n]]^4,x]","\frac{x \left(128 b^3 n^3 \sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+16 b^3 n^3 \sin \left(4 \left(a+b \log \left(c x^n\right)\right)\right)+\left(64 b^2 n^2+4\right) \cos \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+\left(4 b^2 n^2+1\right) \cos \left(4 \left(a+b \log \left(c x^n\right)\right)\right)+8 b n \sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+4 b n \sin \left(4 \left(a+b \log \left(c x^n\right)\right)\right)+192 b^4 n^4+60 b^2 n^2+3\right)}{8 \left(64 b^4 n^4+20 b^2 n^2+1\right)}","\frac{x \cos ^4\left(a+b \log \left(c x^n\right)\right)}{16 b^2 n^2+1}+\frac{4 b n x \sin \left(a+b \log \left(c x^n\right)\right) \cos ^3\left(a+b \log \left(c x^n\right)\right)}{16 b^2 n^2+1}+\frac{12 b^2 n^2 x \cos ^2\left(a+b \log \left(c x^n\right)\right)}{64 b^4 n^4+20 b^2 n^2+1}+\frac{24 b^3 n^3 x \sin \left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{64 b^4 n^4+20 b^2 n^2+1}+\frac{24 b^4 n^4 x}{64 b^4 n^4+20 b^2 n^2+1}",1,"(x*(3 + 60*b^2*n^2 + 192*b^4*n^4 + (4 + 64*b^2*n^2)*Cos[2*(a + b*Log[c*x^n])] + (1 + 4*b^2*n^2)*Cos[4*(a + b*Log[c*x^n])] + 8*b*n*Sin[2*(a + b*Log[c*x^n])] + 128*b^3*n^3*Sin[2*(a + b*Log[c*x^n])] + 4*b*n*Sin[4*(a + b*Log[c*x^n])] + 16*b^3*n^3*Sin[4*(a + b*Log[c*x^n])]))/(8*(1 + 20*b^2*n^2 + 64*b^4*n^4))","A",1
102,1,51,73,0.1010275,"\int \frac{\cos ^4\left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Cos[a + b*Log[c*x^n]]^4/x,x]","\frac{12 \left(a+b \log \left(c x^n\right)\right)+8 \sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+\sin \left(4 \left(a+b \log \left(c x^n\right)\right)\right)}{32 b n}","\frac{\sin \left(a+b \log \left(c x^n\right)\right) \cos ^3\left(a+b \log \left(c x^n\right)\right)}{4 b n}+\frac{3 \sin \left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{8 b n}+\frac{3 \log (x)}{8}",1,"(12*(a + b*Log[c*x^n]) + 8*Sin[2*(a + b*Log[c*x^n])] + Sin[4*(a + b*Log[c*x^n])])/(32*b*n)","A",1
103,1,22,29,0.0122404,"\int \cos (\log (6+3 x)) \, dx","Integrate[Cos[Log[6 + 3*x]],x]","\frac{1}{2} (x+2) (\sin (\log (3 (x+2)))+\cos (\log (3 (x+2))))","\frac{1}{2} (x+2) \sin (\log (3 (x+2)))+\frac{1}{2} (x+2) \cos (\log (3 (x+2)))",1,"((2 + x)*(Cos[Log[3*(2 + x)]] + Sin[Log[3*(2 + x)]]))/2","A",1
104,0,0,101,0.3659893,"\int x^m \cos \left(a+\sqrt{-\frac{(1+m)^2}{n^2}} \log \left(c x^n\right)\right) \, dx","Integrate[x^m*Cos[a + Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n]],x]","\int x^m \cos \left(a+\sqrt{-\frac{(1+m)^2}{n^2}} \log \left(c x^n\right)\right) \, dx","\frac{x^{m+1} e^{\frac{a (m+1)}{n \sqrt{-\frac{(m+1)^2}{n^2}}}} \left(c x^n\right)^{\frac{m+1}{n}}}{4 (m+1)}+\frac{1}{2} x^{m+1} \log (x) e^{\frac{a n \sqrt{-\frac{(m+1)^2}{n^2}}}{m+1}} \left(c x^n\right)^{-\frac{m+1}{n}}",1,"Integrate[x^m*Cos[a + Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n]], x]","F",-1
105,0,0,62,0.112635,"\int \cos \left(a+\sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right) \, dx","Integrate[Cos[a + Sqrt[-n^(-2)]*Log[c*x^n]],x]","\int \cos \left(a+\sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right) \, dx","\frac{1}{4} x e^{-a \sqrt{-\frac{1}{n^2}} n} \left(c x^n\right)^{\frac{1}{n}}+\frac{1}{2} x e^{a \sqrt{-\frac{1}{n^2}} n} \log (x) \left(c x^n\right)^{-1/n}",1,"Integrate[Cos[a + Sqrt[-n^(-2)]*Log[c*x^n]], x]","F",-1
106,0,0,117,0.4503867,"\int x^m \cos ^2\left(a+\frac{1}{2} \sqrt{-\frac{(1+m)^2}{n^2}} \log \left(c x^n\right)\right) \, dx","Integrate[x^m*Cos[a + (Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n])/2]^2,x]","\int x^m \cos ^2\left(a+\frac{1}{2} \sqrt{-\frac{(1+m)^2}{n^2}} \log \left(c x^n\right)\right) \, dx","\frac{x^{m+1} e^{-\frac{2 a n \sqrt{-\frac{(m+1)^2}{n^2}}}{m+1}} \left(c x^n\right)^{\frac{m+1}{n}}}{8 (m+1)}+\frac{1}{4} x^{m+1} \log (x) e^{\frac{2 a n \sqrt{-\frac{(m+1)^2}{n^2}}}{m+1}} \left(c x^n\right)^{-\frac{m+1}{n}}+\frac{x^{m+1}}{2 (m+1)}",1,"Integrate[x^m*Cos[a + (Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n])/2]^2, x]","F",-1
107,0,0,68,0.1218347,"\int \cos ^2\left(a+\frac{1}{2} \sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right) \, dx","Integrate[Cos[a + (Sqrt[-n^(-2)]*Log[c*x^n])/2]^2,x]","\int \cos ^2\left(a+\frac{1}{2} \sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right) \, dx","\frac{1}{8} x e^{-2 a \sqrt{-\frac{1}{n^2}} n} \left(c x^n\right)^{\frac{1}{n}}+\frac{1}{4} x e^{2 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left(c x^n\right)^{-1/n}+\frac{x}{2}",1,"Integrate[Cos[a + (Sqrt[-n^(-2)]*Log[c*x^n])/2]^2, x]","F",-1
108,1,158,226,1.6861773,"\int x^m \cos ^3\left(a+\frac{1}{2} \sqrt{-\frac{(1+m)^2}{n^2}} \log \left(c x^n\right)\right) \, dx","Integrate[x^m*Cos[a + (Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n])/2]^3,x]","\frac{x^{m+1} \left(n \sqrt{-\frac{(m+1)^2}{n^2}} \left(5 \sin \left(a+\frac{1}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left(c x^n\right)\right)-3 \sin \left(3 a+\frac{3}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left(c x^n\right)\right)\right)+10 (m+1) \cos \left(a+\frac{1}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left(c x^n\right)\right)-2 (m+1) \cos \left(3 a+\frac{3}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left(c x^n\right)\right)\right)}{10 (m+1)^2}","\frac{4 n \sqrt{-\frac{(m+1)^2}{n^2}} x^{m+1} \sin \left(a+\frac{1}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left(c x^n\right)\right)}{5 (m+1)^2}-\frac{4 x^{m+1} \cos ^3\left(a+\frac{1}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left(c x^n\right)\right)}{5 (m+1)}+\frac{8 x^{m+1} \cos \left(a+\frac{1}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left(c x^n\right)\right)}{5 (m+1)}-\frac{6 n \sqrt{-\frac{(m+1)^2}{n^2}} x^{m+1} \sin \left(a+\frac{1}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left(c x^n\right)\right) \cos ^2\left(a+\frac{1}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left(c x^n\right)\right)}{5 (m+1)^2}",1,"(x^(1 + m)*(10*(1 + m)*Cos[a + (Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n])/2] - 2*(1 + m)*Cos[3*a + (3*Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n])/2] + Sqrt[-((1 + m)^2/n^2)]*n*(5*Sin[a + (Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n])/2] - 3*Sin[3*a + (3*Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n])/2])))/(10*(1 + m)^2)","A",1
109,0,0,128,0.2164837,"\int \cos ^3\left(a+\frac{1}{3} \sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right) \, dx","Integrate[Cos[a + (Sqrt[-n^(-2)]*Log[c*x^n])/3]^3,x]","\int \cos ^3\left(a+\frac{1}{3} \sqrt{-\frac{1}{n^2}} \log \left(c x^n\right)\right) \, dx","\frac{9}{16} x e^{a \sqrt{-\frac{1}{n^2}} n} \left(c x^n\right)^{\left.-\frac{1}{3}\right/n}+\frac{9}{32} x e^{-a \sqrt{-\frac{1}{n^2}} n} \left(c x^n\right)^{\left.\frac{1}{3}\right/n}+\frac{1}{16} x e^{-3 a \sqrt{-\frac{1}{n^2}} n} \left(c x^n\right)^{\frac{1}{n}}+\frac{1}{8} x e^{3 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left(c x^n\right)^{-1/n}",1,"Integrate[Cos[a + (Sqrt[-n^(-2)]*Log[c*x^n])/3]^3, x]","F",-1
110,1,377,110,3.5571621,"\int \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[Sqrt[Cos[a + b*Log[c*x^n]]],x]","-\frac{2 x \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)} \cos \left(a+b \log \left(c x^n\right)-b n \log (x)\right)}{b n \sin \left(a+b \log \left(c x^n\right)-b n \log (x)\right)-2 \cos \left(a+b \log \left(c x^n\right)-b n \log (x)\right)}+\frac{2 e^{i a} b n x \left(c x^n\right)^{i b} \sqrt{2+2 e^{2 i a} \left(c x^n\right)^{2 i b}} \left((3 b n-2 i) \, _2F_1\left(\frac{1}{2},-\frac{b n+2 i}{4 b n};\frac{3}{4}-\frac{i}{2 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right)+(b n+2 i) x^{2 i b n} \, _2F_1\left(\frac{1}{2},\frac{3}{4}-\frac{i}{2 b n};\frac{7}{4}-\frac{i}{2 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right)\right)}{(b n+2 i) (3 b n-2 i) \sqrt{e^{-i a} \left(c x^n\right)^{-i b}+e^{i a} \left(c x^n\right)^{i b}} \left((-2+i b n) x^{2 i b n}-i e^{2 i a} (b n-2 i) \left(c x^n\right)^{2 i b}\right)}","\frac{2 x \, _2F_1\left(-\frac{1}{2},-\frac{b n+2 i}{4 b n};\frac{1}{4} \left(3-\frac{2 i}{b n}\right);-e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)}}{(2-i b n) \sqrt{1+e^{2 i a} \left(c x^n\right)^{2 i b}}}",1,"(2*b*E^(I*a)*n*x*(c*x^n)^(I*b)*Sqrt[2 + 2*E^((2*I)*a)*(c*x^n)^((2*I)*b)]*((2*I + b*n)*x^((2*I)*b*n)*Hypergeometric2F1[1/2, 3/4 - (I/2)/(b*n), 7/4 - (I/2)/(b*n), -(E^((2*I)*a)*(c*x^n)^((2*I)*b))] + (-2*I + 3*b*n)*Hypergeometric2F1[1/2, -1/4*(2*I + b*n)/(b*n), 3/4 - (I/2)/(b*n), -(E^((2*I)*a)*(c*x^n)^((2*I)*b))]))/((2*I + b*n)*(-2*I + 3*b*n)*Sqrt[1/(E^(I*a)*(c*x^n)^(I*b)) + E^(I*a)*(c*x^n)^(I*b)]*((-2 + I*b*n)*x^((2*I)*b*n) - I*E^((2*I)*a)*(-2*I + b*n)*(c*x^n)^((2*I)*b))) - (2*x*Sqrt[Cos[a + b*Log[c*x^n]]]*Cos[a - b*n*Log[x] + b*Log[c*x^n]])/(-2*Cos[a - b*n*Log[x] + b*Log[c*x^n]] + b*n*Sin[a - b*n*Log[x] + b*Log[c*x^n]])","B",0
111,1,24,24,0.0878709,"\int \frac{\sqrt{\cos \left(a+b \log \left(c x^n\right)\right)}}{x} \, dx","Integrate[Sqrt[Cos[a + b*Log[c*x^n]]]/x,x]","\frac{2 E\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)\right)\right|2\right)}{b n}","\frac{2 E\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)\right)\right|2\right)}{b n}",1,"(2*EllipticE[(a + b*Log[c*x^n])/2, 2])/(b*n)","A",1
112,1,163,109,1.6949049,"\int \cos ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[Cos[a + b*Log[c*x^n]]^(3/2),x]","\frac{x \left((b n-2 i) \left(3 b n \sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+4 \cos ^2\left(a+b \log \left(c x^n\right)\right)\right)-6 i b^2 n^2 \left(1+e^{2 i a} \left(c x^n\right)^{2 i b}\right) \, _2F_1\left(1,\frac{3}{4}-\frac{i}{2 b n};\frac{5}{4}-\frac{i}{2 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)\right)}{(b n-2 i) \left(9 b^2 n^2+4\right) \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)}}","\frac{2 x \, _2F_1\left(-\frac{3}{2},\frac{1}{4} \left(-3-\frac{2 i}{b n}\right);\frac{1}{4} \left(1-\frac{2 i}{b n}\right);-e^{2 i a} \left(c x^n\right)^{2 i b}\right) \cos ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}{(2-3 i b n) \left(1+e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{3/2}}",1,"(x*((-6*I)*b^2*n^2*(1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))*Hypergeometric2F1[1, 3/4 - (I/2)/(b*n), 5/4 - (I/2)/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))] + (-2*I + b*n)*(4*Cos[a + b*Log[c*x^n]]^2 + 3*b*n*Sin[2*(a + b*Log[c*x^n])])))/((-2*I + b*n)*(4 + 9*b^2*n^2)*Sqrt[Cos[a + b*Log[c*x^n]]])","A",0
113,1,54,63,0.1155235,"\int \frac{\cos ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Cos[a + b*Log[c*x^n]]^(3/2)/x,x]","\frac{2 \left(F\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)\right)\right|2\right)+\sin \left(a+b \log \left(c x^n\right)\right) \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)}\right)}{3 b n}","\frac{2 F\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)\right)\right|2\right)}{3 b n}+\frac{2 \sin \left(a+b \log \left(c x^n\right)\right) \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)}}{3 b n}",1,"(2*(EllipticF[(a + b*Log[c*x^n])/2, 2] + Sqrt[Cos[a + b*Log[c*x^n]]]*Sin[a + b*Log[c*x^n]]))/(3*b*n)","A",1
114,1,696,110,7.2131984,"\int \cos ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[Cos[a + b*Log[c*x^n]]^(5/2),x]","\frac{30 b^3 n^3 x^{1-i b n} e^{i \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)} \sqrt{2+2 x^{2 i b n} e^{2 i \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)}} \left((b n+2 i) x^{2 i b n} \, _2F_1\left(\frac{1}{2},\frac{3}{4}-\frac{i}{2 b n};\frac{7}{4}-\frac{i}{2 b n};-e^{2 i \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)} x^{2 i b n}\right)+(3 b n-2 i) \, _2F_1\left(\frac{1}{2},-\frac{b n+2 i}{4 b n};\frac{3}{4}-\frac{i}{2 b n};-e^{2 i \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)} x^{2 i b n}\right)\right)}{(2-5 i b n) (b n+2 i) (3 b n-2 i) (5 b n-2 i) \left((b n-2 i) e^{2 i \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)}-b n-2 i\right) \sqrt{x^{-i b n} e^{-i \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)} \left(1+x^{2 i b n} e^{2 i \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)}\right)}}+\sqrt{\cos \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)+b n \log (x)\right)} \left(-\frac{2 x \left(15 b^2 n^2 \cos \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)-b n \sin \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)+2 \cos \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right)}{(5 b n-2 i) (5 b n+2 i) \left(b n \sin \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)-2 \cos \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right)}+\frac{x \sin (2 b n \log (x)) \left(5 b n \cos \left(2 \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right)-2 \sin \left(2 \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right)\right)}{(5 b n-2 i) (5 b n+2 i)}+\frac{x \cos (2 b n \log (x)) \left(5 b n \sin \left(2 \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right)+2 \cos \left(2 \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right)\right)}{(5 b n-2 i) (5 b n+2 i)}\right)","\frac{2 x \, _2F_1\left(-\frac{5}{2},\frac{1}{4} \left(-5-\frac{2 i}{b n}\right);-\frac{b n+2 i}{4 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right) \cos ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right)}{(2-5 i b n) \left(1+e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{5/2}}",1,"(30*b^3*E^(I*(a + b*(-(n*Log[x]) + Log[c*x^n])))*n^3*x^(1 - I*b*n)*Sqrt[2 + 2*E^((2*I)*(a + b*(-(n*Log[x]) + Log[c*x^n])))*x^((2*I)*b*n)]*((2*I + b*n)*x^((2*I)*b*n)*Hypergeometric2F1[1/2, 3/4 - (I/2)/(b*n), 7/4 - (I/2)/(b*n), -(E^((2*I)*(a + b*(-(n*Log[x]) + Log[c*x^n])))*x^((2*I)*b*n))] + (-2*I + 3*b*n)*Hypergeometric2F1[1/2, -1/4*(2*I + b*n)/(b*n), 3/4 - (I/2)/(b*n), -(E^((2*I)*(a + b*(-(n*Log[x]) + Log[c*x^n])))*x^((2*I)*b*n))]))/((2 - (5*I)*b*n)*(2*I + b*n)*(-2*I + 3*b*n)*(-2*I + 5*b*n)*(-2*I - b*n + E^((2*I)*(a + b*(-(n*Log[x]) + Log[c*x^n])))*(-2*I + b*n))*Sqrt[(1 + E^((2*I)*(a + b*(-(n*Log[x]) + Log[c*x^n])))*x^((2*I)*b*n))/(E^(I*(a + b*(-(n*Log[x]) + Log[c*x^n])))*x^(I*b*n))]) + Sqrt[Cos[a + b*n*Log[x] + b*(-(n*Log[x]) + Log[c*x^n])]]*((-2*x*(2*Cos[a + b*(-(n*Log[x]) + Log[c*x^n])] + 15*b^2*n^2*Cos[a + b*(-(n*Log[x]) + Log[c*x^n])] - b*n*Sin[a + b*(-(n*Log[x]) + Log[c*x^n])]))/((-2*I + 5*b*n)*(2*I + 5*b*n)*(-2*Cos[a + b*(-(n*Log[x]) + Log[c*x^n])] + b*n*Sin[a + b*(-(n*Log[x]) + Log[c*x^n])])) + (x*Sin[2*b*n*Log[x]]*(5*b*n*Cos[2*(a + b*(-(n*Log[x]) + Log[c*x^n]))] - 2*Sin[2*(a + b*(-(n*Log[x]) + Log[c*x^n]))]))/((-2*I + 5*b*n)*(2*I + 5*b*n)) + (x*Cos[2*b*n*Log[x]]*(2*Cos[2*(a + b*(-(n*Log[x]) + Log[c*x^n]))] + 5*b*n*Sin[2*(a + b*(-(n*Log[x]) + Log[c*x^n]))]))/((-2*I + 5*b*n)*(2*I + 5*b*n)))","B",0
115,1,58,63,0.1346518,"\int \frac{\cos ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Cos[a + b*Log[c*x^n]]^(5/2)/x,x]","\frac{6 E\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)\right)\right|2\right)+\sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right) \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)}}{5 b n}","\frac{6 E\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)\right)\right|2\right)}{5 b n}+\frac{2 \sin \left(a+b \log \left(c x^n\right)\right) \cos ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}{5 b n}",1,"(6*EllipticE[(a + b*Log[c*x^n])/2, 2] + Sqrt[Cos[a + b*Log[c*x^n]]]*Sin[2*(a + b*Log[c*x^n])])/(5*b*n)","A",1
116,1,99,109,0.3753684,"\int \frac{1}{\sqrt{\cos \left(a+b \log \left(c x^n\right)\right)}} \, dx","Integrate[1/Sqrt[Cos[a + b*Log[c*x^n]]],x]","-\frac{2 i x \left(1+e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \, _2F_1\left(1,\frac{3}{4}-\frac{i}{2 b n};\frac{5}{4}-\frac{i}{2 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)}{(b n-2 i) \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)}}","\frac{2 x \sqrt{1+e^{2 i a} \left(c x^n\right)^{2 i b}} \, _2F_1\left(\frac{1}{2},\frac{1}{4} \left(1-\frac{2 i}{b n}\right);\frac{1}{4} \left(5-\frac{2 i}{b n}\right);-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{(2+i b n) \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)}}",1,"((-2*I)*(1 + E^((2*I)*(a + b*Log[c*x^n])))*x*Hypergeometric2F1[1, 3/4 - (I/2)/(b*n), 5/4 - (I/2)/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))])/((-2*I + b*n)*Sqrt[Cos[a + b*Log[c*x^n]]])","A",0
117,1,24,24,0.0796318,"\int \frac{1}{x \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)}} \, dx","Integrate[1/(x*Sqrt[Cos[a + b*Log[c*x^n]]]),x]","\frac{2 F\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)\right)\right|2\right)}{b n}","\frac{2 F\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)\right)\right|2\right)}{b n}",1,"(2*EllipticF[(a + b*Log[c*x^n])/2, 2])/(b*n)","A",1
118,1,431,109,3.7087652,"\int \frac{1}{\cos ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[Cos[a + b*Log[c*x^n]]^(-3/2),x]","\frac{x \left((3 b n-2 i) x^{-i b n} \left(2 x^{i b n} \sqrt{e^{-i a} \left(c x^n\right)^{-i b}+e^{i a} \left(c x^n\right)^{i b}} (b n \cos (b n \log (x))-2 \sin (b n \log (x)))-(b n-2 i) \sqrt{2+2 e^{2 i a} \left(c x^n\right)^{2 i b}} \, _2F_1\left(\frac{1}{2},-\frac{b n+2 i}{4 b n};\frac{3}{4}-\frac{i}{2 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)}\right)-\left(b^2 n^2+4\right) x^{i b n} \sqrt{2+2 e^{2 i a} \left(c x^n\right)^{2 i b}} \, _2F_1\left(\frac{1}{2},\frac{3}{4}-\frac{i}{2 b n};\frac{7}{4}-\frac{i}{2 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)}\right)}{b n (3 b n-2 i) \sqrt{e^{-i a} \left(c x^n\right)^{-i b}+e^{i a} \left(c x^n\right)^{i b}} \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)} \left(b n \sin \left(a+b \log \left(c x^n\right)-b n \log (x)\right)-2 \cos \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)}","\frac{2 x \left(1+e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{3/2} \, _2F_1\left(\frac{3}{2},\frac{1}{4} \left(3-\frac{2 i}{b n}\right);\frac{1}{4} \left(7-\frac{2 i}{b n}\right);-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{(2+3 i b n) \cos ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}",1,"(x*(-((4 + b^2*n^2)*x^(I*b*n)*Sqrt[2 + 2*E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Sqrt[Cos[a + b*Log[c*x^n]]]*Hypergeometric2F1[1/2, 3/4 - (I/2)/(b*n), 7/4 - (I/2)/(b*n), -(E^((2*I)*a)*(c*x^n)^((2*I)*b))]) + ((-2*I + 3*b*n)*(-((-2*I + b*n)*Sqrt[2 + 2*E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Sqrt[Cos[a + b*Log[c*x^n]]]*Hypergeometric2F1[1/2, -1/4*(2*I + b*n)/(b*n), 3/4 - (I/2)/(b*n), -(E^((2*I)*a)*(c*x^n)^((2*I)*b))]) + 2*x^(I*b*n)*Sqrt[1/(E^(I*a)*(c*x^n)^(I*b)) + E^(I*a)*(c*x^n)^(I*b)]*(b*n*Cos[b*n*Log[x]] - 2*Sin[b*n*Log[x]])))/x^(I*b*n)))/(b*n*(-2*I + 3*b*n)*Sqrt[1/(E^(I*a)*(c*x^n)^(I*b)) + E^(I*a)*(c*x^n)^(I*b)]*Sqrt[Cos[a + b*Log[c*x^n]]]*(-2*Cos[a - b*n*Log[x] + b*Log[c*x^n]] + b*n*Sin[a - b*n*Log[x] + b*Log[c*x^n]]))","B",0
119,1,54,59,0.1488312,"\int \frac{1}{x \cos ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[1/(x*Cos[a + b*Log[c*x^n]]^(3/2)),x]","\frac{2 \left(\frac{\sin \left(a+b \log \left(c x^n\right)\right)}{\sqrt{\cos \left(a+b \log \left(c x^n\right)\right)}}-E\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)\right)\right|2\right)\right)}{b n}","\frac{2 \sin \left(a+b \log \left(c x^n\right)\right)}{b n \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)}}-\frac{2 E\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)\right)\right|2\right)}{b n}",1,"(2*(-EllipticE[(a + b*Log[c*x^n])/2, 2] + Sin[a + b*Log[c*x^n]]/Sqrt[Cos[a + b*Log[c*x^n]]]))/(b*n)","A",1
120,1,147,109,1.1285697,"\int \frac{1}{\cos ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[Cos[a + b*Log[c*x^n]]^(-5/2),x]","\frac{2 x \left((2-i b n) \left(1+e^{2 i a} \left(c x^n\right)^{2 i b}\right) \, _2F_1\left(1,\frac{3}{4}-\frac{i}{2 b n};\frac{5}{4}-\frac{i}{2 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \cos \left(a+b \log \left(c x^n\right)\right)+b n \sin \left(a+b \log \left(c x^n\right)\right)-2 \cos \left(a+b \log \left(c x^n\right)\right)\right)}{3 b^2 n^2 \cos ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}","\frac{2 x \left(1+e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{5/2} \, _2F_1\left(\frac{5}{2},\frac{1}{4} \left(5-\frac{2 i}{b n}\right);\frac{1}{4} \left(9-\frac{2 i}{b n}\right);-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{(2+5 i b n) \cos ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right)}",1,"(2*x*(-2*Cos[a + b*Log[c*x^n]] + (2 - I*b*n)*(1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))*Cos[a + b*Log[c*x^n]]*Hypergeometric2F1[1, 3/4 - (I/2)/(b*n), 5/4 - (I/2)/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))] + b*n*Sin[a + b*Log[c*x^n]]))/(3*b^2*n^2*Cos[a + b*Log[c*x^n]]^(3/2))","A",0
121,1,54,63,0.1448507,"\int \frac{1}{x \cos ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[1/(x*Cos[a + b*Log[c*x^n]]^(5/2)),x]","\frac{2 \left(F\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)\right)\right|2\right)+\frac{\sin \left(a+b \log \left(c x^n\right)\right)}{\cos ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}\right)}{3 b n}","\frac{2 F\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)\right)\right|2\right)}{3 b n}+\frac{2 \sin \left(a+b \log \left(c x^n\right)\right)}{3 b n \cos ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}",1,"(2*(EllipticF[(a + b*Log[c*x^n])/2, 2] + Sin[a + b*Log[c*x^n]]/Cos[a + b*Log[c*x^n]]^(3/2)))/(3*b*n)","A",1
122,1,82,48,0.1180061,"\int \frac{1}{\cos ^{\frac{3}{2}}(a-2 i \log (c x))} \, dx","Integrate[Cos[a - (2*I)*Log[c*x]]^(-3/2),x]","-\frac{x (\cos (a)-i \sin (a)) \sqrt{\frac{2 \cos (a) \left(c^4 x^4+1\right)+2 i \sin (a) \left(c^4 x^4-1\right)}{c^2 x^2}}}{\cos (a) \left(c^4 x^4+1\right)+i \sin (a) \left(c^4 x^4-1\right)}","-\frac{e^{-2 i a} \left(1+e^{2 i a} c^4 x^4\right)}{2 c^4 x^3 \cos ^{\frac{3}{2}}(a-2 i \log (c x))}",1,"-((x*(Cos[a] - I*Sin[a])*Sqrt[(2*(1 + c^4*x^4)*Cos[a] + (2*I)*(-1 + c^4*x^4)*Sin[a])/(c^2*x^2)])/((1 + c^4*x^4)*Cos[a] + I*(-1 + c^4*x^4)*Sin[a]))","A",1
123,1,312,266,4.0250742,"\int x^m \cos ^4\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x^m*Cos[a + b*Log[c*x^n]]^4,x]","\frac{1}{8} x^{m+1} \left(-\frac{4 \sin (2 b n \log (x)) \left((m+1) \sin \left(2 \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)-2 b n \cos \left(2 \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)\right)}{4 b^2 n^2+m^2+2 m+1}+\frac{4 \cos (2 b n \log (x)) \left((m+1) \cos \left(2 \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)+2 b n \sin \left(2 \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)\right)}{4 b^2 n^2+m^2+2 m+1}-\frac{\sin (4 b n \log (x)) \left((m+1) \sin \left(4 \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)-4 b n \cos \left(4 \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)\right)}{16 b^2 n^2+m^2+2 m+1}+\frac{\cos (4 b n \log (x)) \left((m+1) \cos \left(4 \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)+4 b n \sin \left(4 \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)\right)}{16 b^2 n^2+m^2+2 m+1}+\frac{3}{m+1}\right)","\frac{(m+1) x^{m+1} \cos ^4\left(a+b \log \left(c x^n\right)\right)}{16 b^2 n^2+(m+1)^2}+\frac{12 b^2 (m+1) n^2 x^{m+1} \cos ^2\left(a+b \log \left(c x^n\right)\right)}{\left(4 b^2 n^2+(m+1)^2\right) \left(16 b^2 n^2+(m+1)^2\right)}+\frac{4 b n x^{m+1} \sin \left(a+b \log \left(c x^n\right)\right) \cos ^3\left(a+b \log \left(c x^n\right)\right)}{16 b^2 n^2+(m+1)^2}+\frac{24 b^3 n^3 x^{m+1} \sin \left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{\left(4 b^2 n^2+(m+1)^2\right) \left(16 b^2 n^2+(m+1)^2\right)}+\frac{24 b^4 n^4 x^{m+1}}{(m+1) \left(4 b^2 n^2+(m+1)^2\right) \left(16 b^2 n^2+(m+1)^2\right)}",1,"(x^(1 + m)*(3/(1 + m) - (4*Sin[2*b*n*Log[x]]*(-2*b*n*Cos[2*(a - b*n*Log[x] + b*Log[c*x^n])] + (1 + m)*Sin[2*(a - b*n*Log[x] + b*Log[c*x^n])]))/(1 + 2*m + m^2 + 4*b^2*n^2) + (4*Cos[2*b*n*Log[x]]*((1 + m)*Cos[2*(a - b*n*Log[x] + b*Log[c*x^n])] + 2*b*n*Sin[2*(a - b*n*Log[x] + b*Log[c*x^n])]))/(1 + 2*m + m^2 + 4*b^2*n^2) - (Sin[4*b*n*Log[x]]*(-4*b*n*Cos[4*(a - b*n*Log[x] + b*Log[c*x^n])] + (1 + m)*Sin[4*(a - b*n*Log[x] + b*Log[c*x^n])]))/(1 + 2*m + m^2 + 16*b^2*n^2) + (Cos[4*b*n*Log[x]]*((1 + m)*Cos[4*(a - b*n*Log[x] + b*Log[c*x^n])] + 4*b*n*Sin[4*(a - b*n*Log[x] + b*Log[c*x^n])]))/(1 + 2*m + m^2 + 16*b^2*n^2)))/8","A",1
124,1,292,201,1.9385808,"\int x^m \cos ^3\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x^m*Cos[a + b*Log[c*x^n]]^3,x]","\frac{1}{4} x^{m+1} \left(-\frac{3 \sin (b n \log (x)) \left((m+1) \sin \left(a+b \log \left(c x^n\right)-b n \log (x)\right)-b n \cos \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)}{b^2 n^2+m^2+2 m+1}+\frac{3 \cos (b n \log (x)) \left((m+1) \cos \left(a+b \log \left(c x^n\right)-b n \log (x)\right)+b n \sin \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)}{b^2 n^2+m^2+2 m+1}-\frac{\sin (3 b n \log (x)) \left((m+1) \sin \left(3 \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)-3 b n \cos \left(3 \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)\right)}{9 b^2 n^2+m^2+2 m+1}+\frac{\cos (3 b n \log (x)) \left((m+1) \cos \left(3 \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)+3 b n \sin \left(3 \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)\right)}{9 b^2 n^2+m^2+2 m+1}\right)","\frac{(m+1) x^{m+1} \cos ^3\left(a+b \log \left(c x^n\right)\right)}{9 b^2 n^2+(m+1)^2}+\frac{6 b^2 (m+1) n^2 x^{m+1} \cos \left(a+b \log \left(c x^n\right)\right)}{\left(b^2 n^2+(m+1)^2\right) \left(9 b^2 n^2+(m+1)^2\right)}+\frac{3 b n x^{m+1} \sin \left(a+b \log \left(c x^n\right)\right) \cos ^2\left(a+b \log \left(c x^n\right)\right)}{9 b^2 n^2+(m+1)^2}+\frac{6 b^3 n^3 x^{m+1} \sin \left(a+b \log \left(c x^n\right)\right)}{\left(b^2 n^2+(m+1)^2\right) \left(9 b^2 n^2+(m+1)^2\right)}",1,"(x^(1 + m)*((-3*Sin[b*n*Log[x]]*(-(b*n*Cos[a - b*n*Log[x] + b*Log[c*x^n]]) + (1 + m)*Sin[a - b*n*Log[x] + b*Log[c*x^n]]))/(1 + 2*m + m^2 + b^2*n^2) + (3*Cos[b*n*Log[x]]*((1 + m)*Cos[a - b*n*Log[x] + b*Log[c*x^n]] + b*n*Sin[a - b*n*Log[x] + b*Log[c*x^n]]))/(1 + 2*m + m^2 + b^2*n^2) - (Sin[3*b*n*Log[x]]*(-3*b*n*Cos[3*(a - b*n*Log[x] + b*Log[c*x^n])] + (1 + m)*Sin[3*(a - b*n*Log[x] + b*Log[c*x^n])]))/(1 + 2*m + m^2 + 9*b^2*n^2) + (Cos[3*b*n*Log[x]]*((1 + m)*Cos[3*(a - b*n*Log[x] + b*Log[c*x^n])] + 3*b*n*Sin[3*(a - b*n*Log[x] + b*Log[c*x^n])]))/(1 + 2*m + m^2 + 9*b^2*n^2)))/4","A",1
125,1,91,120,0.3395981,"\int x^m \cos ^2\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x^m*Cos[a + b*Log[c*x^n]]^2,x]","\frac{x^{m+1} \left(2 b (m+1) n \sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+(m+1)^2 \cos \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+4 b^2 n^2+m^2+2 m+1\right)}{2 (m+1) (-2 i b n+m+1) (2 i b n+m+1)}","\frac{(m+1) x^{m+1} \cos ^2\left(a+b \log \left(c x^n\right)\right)}{4 b^2 n^2+(m+1)^2}+\frac{2 b n x^{m+1} \sin \left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)}{4 b^2 n^2+(m+1)^2}+\frac{2 b^2 n^2 x^{m+1}}{(m+1) \left(4 b^2 n^2+(m+1)^2\right)}",1,"(x^(1 + m)*(1 + 2*m + m^2 + 4*b^2*n^2 + (1 + m)^2*Cos[2*(a + b*Log[c*x^n])] + 2*b*(1 + m)*n*Sin[2*(a + b*Log[c*x^n])]))/(2*(1 + m)*(1 + m - (2*I)*b*n)*(1 + m + (2*I)*b*n))","C",1
126,1,53,70,0.1543074,"\int x^m \cos \left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x^m*Cos[a + b*Log[c*x^n]],x]","\frac{x^{m+1} \left((m+1) \cos \left(a+b \log \left(c x^n\right)\right)+b n \sin \left(a+b \log \left(c x^n\right)\right)\right)}{b^2 n^2+m^2+2 m+1}","\frac{b n x^{m+1} \sin \left(a+b \log \left(c x^n\right)\right)}{b^2 n^2+(m+1)^2}+\frac{(m+1) x^{m+1} \cos \left(a+b \log \left(c x^n\right)\right)}{b^2 n^2+(m+1)^2}",1,"(x^(1 + m)*((1 + m)*Cos[a + b*Log[c*x^n]] + b*n*Sin[a + b*Log[c*x^n]]))/(1 + 2*m + m^2 + b^2*n^2)","A",1
127,1,204,130,2.0284617,"\int x^m \cos ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x^m*Cos[a + b*Log[c*x^n]]^(3/2),x]","\frac{x^{m+1} \left(6 b^2 n^2 \left(1+e^{2 i a} \left(c x^n\right)^{2 i b}\right) \, _2F_1\left(1,-\frac{2 i m-3 b n+2 i}{4 b n};-\frac{2 i m-5 b n+2 i}{4 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)+(i b n+2 m+2) \left(4 (m+1) \cos ^2\left(a+b \log \left(c x^n\right)\right)+3 b n \sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)\right)\right)}{(i b n+2 m+2) (-3 i b n+2 m+2) (3 i b n+2 m+2) \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)}}","\frac{2 x^{m+1} \, _2F_1\left(-\frac{3}{2},-\frac{2 i m+3 b n+2 i}{4 b n};-\frac{2 i m-b n+2 i}{4 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right) \cos ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}{(-3 i b n+2 m+2) \left(1+e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{3/2}}",1,"(x^(1 + m)*(6*b^2*n^2*(1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))*Hypergeometric2F1[1, -1/4*(2*I + (2*I)*m - 3*b*n)/(b*n), -1/4*(2*I + (2*I)*m - 5*b*n)/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))] + (2 + 2*m + I*b*n)*(4*(1 + m)*Cos[a + b*Log[c*x^n]]^2 + 3*b*n*Sin[2*(a + b*Log[c*x^n])])))/((2 + 2*m + I*b*n)*(2 + 2*m - (3*I)*b*n)*(2 + 2*m + (3*I)*b*n)*Sqrt[Cos[a + b*Log[c*x^n]]])","A",0
128,1,436,129,5.3650922,"\int x^m \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[x^m*Sqrt[Cos[a + b*Log[c*x^n]]],x]","\frac{2 x^{m+1} \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)} \cos \left(a+b \log \left(c x^n\right)-b n \log (x)\right)}{2 (m+1) \cos \left(a+b \log \left(c x^n\right)-b n \log (x)\right)-b n \sin \left(a+b \log \left(c x^n\right)-b n \log (x)\right)}-\frac{2 e^{i a} b n x^{m+1} \left(c x^n\right)^{i b} \sqrt{2+2 e^{2 i a} \left(c x^n\right)^{2 i b}} \left((3 b n-2 i m-2 i) \, _2F_1\left(\frac{1}{2},-\frac{2 i m+b n+2 i}{4 b n};-\frac{2 i m-3 b n+2 i}{4 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right)+(b n+2 i m+2 i) x^{2 i b n} \, _2F_1\left(\frac{1}{2},-\frac{i \left(m+\frac{3 i b n}{2}+1\right)}{2 b n};-\frac{2 i m-7 b n+2 i}{4 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right)\right)}{(-i b n+2 m+2) (3 i b n+2 m+2) \sqrt{e^{-i a} \left(c x^n\right)^{-i b}+e^{i a} \left(c x^n\right)^{i b}} \left(e^{2 i a} (i b n+2 m+2) \left(c x^n\right)^{2 i b}+(-i b n+2 m+2) x^{2 i b n}\right)}","\frac{2 x^{m+1} \, _2F_1\left(-\frac{1}{2},-\frac{2 i m+b n+2 i}{4 b n};-\frac{2 i m-3 b n+2 i}{4 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)}}{(-i b n+2 m+2) \sqrt{1+e^{2 i a} \left(c x^n\right)^{2 i b}}}",1,"(-2*b*E^(I*a)*n*x^(1 + m)*(c*x^n)^(I*b)*Sqrt[2 + 2*E^((2*I)*a)*(c*x^n)^((2*I)*b)]*((2*I + (2*I)*m + b*n)*x^((2*I)*b*n)*Hypergeometric2F1[1/2, ((-1/2*I)*(1 + m + ((3*I)/2)*b*n))/(b*n), -1/4*(2*I + (2*I)*m - 7*b*n)/(b*n), -(E^((2*I)*a)*(c*x^n)^((2*I)*b))] + (-2*I - (2*I)*m + 3*b*n)*Hypergeometric2F1[1/2, -1/4*(2*I + (2*I)*m + b*n)/(b*n), -1/4*(2*I + (2*I)*m - 3*b*n)/(b*n), -(E^((2*I)*a)*(c*x^n)^((2*I)*b))]))/((2 + 2*m - I*b*n)*(2 + 2*m + (3*I)*b*n)*Sqrt[1/(E^(I*a)*(c*x^n)^(I*b)) + E^(I*a)*(c*x^n)^(I*b)]*((2 + 2*m - I*b*n)*x^((2*I)*b*n) + E^((2*I)*a)*(2 + 2*m + I*b*n)*(c*x^n)^((2*I)*b))) + (2*x^(1 + m)*Sqrt[Cos[a + b*Log[c*x^n]]]*Cos[a - b*n*Log[x] + b*Log[c*x^n]])/(2*(1 + m)*Cos[a - b*n*Log[x] + b*Log[c*x^n]] - b*n*Sin[a - b*n*Log[x] + b*Log[c*x^n]])","B",0
129,1,119,130,0.5788077,"\int \frac{x^m}{\sqrt{\cos \left(a+b \log \left(c x^n\right)\right)}} \, dx","Integrate[x^m/Sqrt[Cos[a + b*Log[c*x^n]]],x]","\frac{2 x^{m+1} \left(1+e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \, _2F_1\left(1,-\frac{2 i m-3 b n+2 i}{4 b n};-\frac{2 i m-5 b n+2 i}{4 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)}{(i b n+2 m+2) \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)}}","\frac{2 x^{m+1} \sqrt{1+e^{2 i a} \left(c x^n\right)^{2 i b}} \, _2F_1\left(\frac{1}{2},-\frac{2 i m-b n+2 i}{4 b n};-\frac{2 i m-5 b n+2 i}{4 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{(i b n+2 m+2) \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)}}",1,"(2*(1 + E^((2*I)*(a + b*Log[c*x^n])))*x^(1 + m)*Hypergeometric2F1[1, -1/4*(2*I + (2*I)*m - 3*b*n)/(b*n), -1/4*(2*I + (2*I)*m - 5*b*n)/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))])/((2 + 2*m + I*b*n)*Sqrt[Cos[a + b*Log[c*x^n]]])","A",0
130,1,487,130,5.2047284,"\int \frac{x^m}{\cos ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[x^m/Cos[a + b*Log[c*x^n]]^(3/2),x]","-\frac{x^{-i b n+m+1} \left(\left(b^2 n^2+4 m^2+8 m+4\right) x^{2 i b n} \sqrt{2+2 e^{2 i a} \left(c x^n\right)^{2 i b}} \, _2F_1\left(\frac{1}{2},-\frac{i \left(m+\frac{3 i b n}{2}+1\right)}{2 b n};-\frac{2 i m-7 b n+2 i}{4 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)}+(3 b n-2 i m-2 i) \left((b n-2 i m-2 i) \sqrt{2+2 e^{2 i a} \left(c x^n\right)^{2 i b}} \, _2F_1\left(\frac{1}{2},-\frac{2 i m+b n+2 i}{4 b n};-\frac{2 i m-3 b n+2 i}{4 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)}-2 x^{i b n} \sqrt{e^{-i a} \left(c x^n\right)^{-i b}+e^{i a} \left(c x^n\right)^{i b}} (b n \cos (b n \log (x))-2 (m+1) \sin (b n \log (x)))\right)\right)}{b n (3 b n-2 i m-2 i) \sqrt{e^{-i a} \left(c x^n\right)^{-i b}+e^{i a} \left(c x^n\right)^{i b}} \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)} \left(b n \sin \left(a+b \log \left(c x^n\right)-b n \log (x)\right)-2 (m+1) \cos \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)}","\frac{2 x^{m+1} \left(1+e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{3/2} \, _2F_1\left(\frac{3}{2},-\frac{2 i m-3 b n+2 i}{4 b n};-\frac{2 i m-7 b n+2 i}{4 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{(3 i b n+2 m+2) \cos ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}",1,"-((x^(1 + m - I*b*n)*((4 + 8*m + 4*m^2 + b^2*n^2)*x^((2*I)*b*n)*Sqrt[2 + 2*E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Sqrt[Cos[a + b*Log[c*x^n]]]*Hypergeometric2F1[1/2, ((-1/2*I)*(1 + m + ((3*I)/2)*b*n))/(b*n), -1/4*(2*I + (2*I)*m - 7*b*n)/(b*n), -(E^((2*I)*a)*(c*x^n)^((2*I)*b))] + (-2*I - (2*I)*m + 3*b*n)*((-2*I - (2*I)*m + b*n)*Sqrt[2 + 2*E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Sqrt[Cos[a + b*Log[c*x^n]]]*Hypergeometric2F1[1/2, -1/4*(2*I + (2*I)*m + b*n)/(b*n), -1/4*(2*I + (2*I)*m - 3*b*n)/(b*n), -(E^((2*I)*a)*(c*x^n)^((2*I)*b))] - 2*x^(I*b*n)*Sqrt[1/(E^(I*a)*(c*x^n)^(I*b)) + E^(I*a)*(c*x^n)^(I*b)]*(b*n*Cos[b*n*Log[x]] - 2*(1 + m)*Sin[b*n*Log[x]]))))/(b*n*(-2*I - (2*I)*m + 3*b*n)*Sqrt[1/(E^(I*a)*(c*x^n)^(I*b)) + E^(I*a)*(c*x^n)^(I*b)]*Sqrt[Cos[a + b*Log[c*x^n]]]*(-2*(1 + m)*Cos[a - b*n*Log[x] + b*Log[c*x^n]] + b*n*Sin[a - b*n*Log[x] + b*Log[c*x^n]])))","B",0
131,1,205,130,2.2535454,"\int \frac{x^m}{\cos ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[x^m/Cos[a + b*Log[c*x^n]]^(5/2),x]","\frac{2 x^{m+1} \left((-i b n+2 m+2) \left(1+e^{2 i a} \left(c x^n\right)^{2 i b}\right) \cos \left(a+b \log \left(c x^n\right)\right) \, _2F_1\left(1,-\frac{2 i m-3 b n+2 i}{4 b n};-\frac{2 i m-5 b n+2 i}{4 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)+\cos \left(a+b \log \left(c x^n\right)\right) \left(b n \tan \left(a+b \log \left(c x^n\right)-b n \log (x)\right)-2 (m+1)\right)+b n \sin (b n \log (x)) \sec \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)}{3 b^2 n^2 \cos ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}","\frac{2 x^{m+1} \left(1+e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{5/2} \, _2F_1\left(\frac{5}{2},-\frac{2 i m-5 b n+2 i}{4 b n};-\frac{2 i m-9 b n+2 i}{4 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{(5 i b n+2 m+2) \cos ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right)}",1,"(2*x^(1 + m)*((2 + 2*m - I*b*n)*(1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))*Cos[a + b*Log[c*x^n]]*Hypergeometric2F1[1, -1/4*(2*I + (2*I)*m - 3*b*n)/(b*n), -1/4*(2*I + (2*I)*m - 5*b*n)/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))] + b*n*Sec[a - b*n*Log[x] + b*Log[c*x^n]]*Sin[b*n*Log[x]] + Cos[a + b*Log[c*x^n]]*(-2*(1 + m) + b*n*Tan[a - b*n*Log[x] + b*Log[c*x^n]])))/(3*b^2*n^2*Cos[a + b*Log[c*x^n]]^(3/2))","A",0
132,1,123,144,1.0249193,"\int (e x)^m \cos ^p\left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[(e*x)^m*Cos[d*(a + b*Log[c*x^n])]^p,x]","\frac{x (e x)^m \left(1+e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right) \cos ^p\left(d \left(a+b \log \left(c x^n\right)\right)\right) \, _2F_1\left(1,\frac{1}{2} \left(-\frac{i (m+1)}{b d n}+p+2\right);-\frac{i (m+1)}{2 b d n}-\frac{p}{2}+1;-e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)}{-i b d n p+m+1}","\frac{(e x)^{m+1} \left(1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)^{-p} \, _2F_1\left(-p,-\frac{i m+b d n p+i}{2 b d n};\frac{1}{2} \left(-\frac{i (m+1)}{b d n}-p+2\right);-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right) \cos ^p\left(d \left(a+b \log \left(c x^n\right)\right)\right)}{e (-i b d n p+m+1)}",1,"((1 + E^((2*I)*d*(a + b*Log[c*x^n])))*x*(e*x)^m*Cos[d*(a + b*Log[c*x^n])]^p*Hypergeometric2F1[1, (2 - (I*(1 + m))/(b*d*n) + p)/2, 1 - ((I/2)*(1 + m))/(b*d*n) - p/2, -E^((2*I)*d*(a + b*Log[c*x^n]))])/(1 + m - I*b*d*n*p)","A",0
133,1,102,114,0.6485448,"\int x \cos ^p\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x*Cos[a + b*Log[c*x^n]]^p,x]","\frac{i x^2 \left(1+e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \, _2F_1\left(1,\frac{p}{2}-\frac{i}{b n}+1;-\frac{p}{2}-\frac{i}{b n}+1;-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \cos ^p\left(a+b \log \left(c x^n\right)\right)}{b n p+2 i}","\frac{x^2 \left(1+e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{-p} \, _2F_1\left(\frac{1}{2} \left(-p-\frac{2 i}{b n}\right),-p;\frac{1}{2} \left(-p-\frac{2 i}{b n}+2\right);-e^{2 i a} \left(c x^n\right)^{2 i b}\right) \cos ^p\left(a+b \log \left(c x^n\right)\right)}{2-i b n p}",1,"(I*(1 + E^((2*I)*(a + b*Log[c*x^n])))*x^2*Cos[a + b*Log[c*x^n]]^p*Hypergeometric2F1[1, 1 - I/(b*n) + p/2, 1 - I/(b*n) - p/2, -E^((2*I)*(a + b*Log[c*x^n]))])/(2*I + b*n*p)","A",0
134,1,102,112,0.5602322,"\int \cos ^p\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[Cos[a + b*Log[c*x^n]]^p,x]","\frac{i x \left(1+e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \, _2F_1\left(1,\frac{1}{2} \left(p-\frac{i}{b n}+2\right);-\frac{p}{2}-\frac{i}{2 b n}+1;-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \cos ^p\left(a+b \log \left(c x^n\right)\right)}{b n p+i}","\frac{x \left(1+e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{-p} \, _2F_1\left(-p,-\frac{b n p+i}{2 b n};\frac{1}{2} \left(-p-\frac{i}{b n}+2\right);-e^{2 i a} \left(c x^n\right)^{2 i b}\right) \cos ^p\left(a+b \log \left(c x^n\right)\right)}{1-i b n p}",1,"(I*(1 + E^((2*I)*(a + b*Log[c*x^n])))*x*Cos[a + b*Log[c*x^n]]^p*Hypergeometric2F1[1, (2 - I/(b*n) + p)/2, 1 - (I/2)/(b*n) - p/2, -E^((2*I)*(a + b*Log[c*x^n]))])/(I + b*n*p)","A",0
135,1,132,47,0.0390609,"\int x^3 \tan (a+i \log (x)) \, dx","Integrate[x^3*Tan[a + I*Log[x]],x]","x^2 \sin (2 a)-i x^2 \cos (2 a)+\cos (4 a) \tan ^{-1}\left(\frac{\left(x^2+1\right) \cos (a)}{\sin (a)-x^2 \sin (a)}\right)+i \sin (4 a) \tan ^{-1}\left(\frac{\left(x^2+1\right) \cos (a)}{\sin (a)-x^2 \sin (a)}\right)+\frac{1}{2} i \cos (4 a) \log \left(2 x^2 \cos (2 a)+x^4+1\right)-\frac{1}{2} \sin (4 a) \log \left(2 x^2 \cos (2 a)+x^4+1\right)+\frac{i x^4}{4}","-i e^{2 i a} x^2+i e^{4 i a} \log \left(x^2+e^{2 i a}\right)+\frac{i x^4}{4}",1,"(I/4)*x^4 - I*x^2*Cos[2*a] + ArcTan[((1 + x^2)*Cos[a])/(Sin[a] - x^2*Sin[a])]*Cos[4*a] + (I/2)*Cos[4*a]*Log[1 + x^4 + 2*x^2*Cos[2*a]] + x^2*Sin[2*a] + I*ArcTan[((1 + x^2)*Cos[a])/(Sin[a] - x^2*Sin[a])]*Sin[4*a] - (Log[1 + x^4 + 2*x^2*Cos[2*a]]*Sin[4*a])/2","B",1
136,1,66,43,0.017448,"\int x^2 \tan (a+i \log (x)) \, dx","Integrate[x^2*Tan[a + I*Log[x]],x]","2 x \sin (2 a)-2 i x \cos (2 a)+2 i \cos (3 a) \tan ^{-1}(x \cos (a)-i x \sin (a))-2 \sin (3 a) \tan ^{-1}(x \cos (a)-i x \sin (a))+\frac{i x^3}{3}","-2 i e^{2 i a} x+2 i e^{3 i a} \tan ^{-1}\left(e^{-i a} x\right)+\frac{i x^3}{3}",1,"(I/3)*x^3 - (2*I)*x*Cos[2*a] + (2*I)*ArcTan[x*Cos[a] - I*x*Sin[a]]*Cos[3*a] + 2*x*Sin[2*a] - 2*ArcTan[x*Cos[a] - I*x*Sin[a]]*Sin[3*a]","A",1
137,1,114,33,0.023536,"\int x \tan (a+i \log (x)) \, dx","Integrate[x*Tan[a + I*Log[x]],x]","-\cos (2 a) \tan ^{-1}\left(\frac{\left(x^2+1\right) \cos (a)}{\sin (a)-x^2 \sin (a)}\right)-i \sin (2 a) \tan ^{-1}\left(\frac{\left(x^2+1\right) \cos (a)}{\sin (a)-x^2 \sin (a)}\right)-\frac{1}{2} i \cos (2 a) \log \left(2 x^2 \cos (2 a)+x^4+1\right)+\frac{1}{2} \sin (2 a) \log \left(2 x^2 \cos (2 a)+x^4+1\right)+\frac{i x^2}{2}","\frac{i x^2}{2}-i e^{2 i a} \log \left(x^2+e^{2 i a}\right)",1,"(I/2)*x^2 - ArcTan[((1 + x^2)*Cos[a])/(Sin[a] - x^2*Sin[a])]*Cos[2*a] - (I/2)*Cos[2*a]*Log[1 + x^4 + 2*x^2*Cos[2*a]] - I*ArcTan[((1 + x^2)*Cos[a])/(Sin[a] - x^2*Sin[a])]*Sin[2*a] + (Log[1 + x^4 + 2*x^2*Cos[2*a]]*Sin[2*a])/2","B",1
138,1,42,27,0.0098379,"\int \tan (a+i \log (x)) \, dx","Integrate[Tan[a + I*Log[x]],x]","-2 i \cos (a) \tan ^{-1}(x \cos (a)-i x \sin (a))+2 \sin (a) \tan ^{-1}(x \cos (a)-i x \sin (a))+i x","i x-2 i e^{i a} \tan ^{-1}\left(e^{-i a} x\right)",1,"I*x - (2*I)*ArcTan[x*Cos[a] - I*x*Sin[a]]*Cos[a] + 2*ArcTan[x*Cos[a] - I*x*Sin[a]]*Sin[a]","A",1
139,1,14,14,0.0221083,"\int \frac{\tan (a+i \log (x))}{x} \, dx","Integrate[Tan[a + I*Log[x]]/x,x]","i \log (\cos (a+i \log (x)))","i \log (\cos (a+i \log (x)))",1,"I*Log[Cos[a + I*Log[x]]]","A",1
140,1,44,29,0.0231842,"\int \frac{\tan (a+i \log (x))}{x^2} \, dx","Integrate[Tan[a + I*Log[x]]/x^2,x]","2 i \cos (a) \tan ^{-1}(x \cos (a)-i x \sin (a))+2 \sin (a) \tan ^{-1}(x \cos (a)-i x \sin (a))+\frac{i}{x}","2 i e^{-i a} \tan ^{-1}\left(e^{-i a} x\right)+\frac{i}{x}",1,"I/x + (2*I)*ArcTan[x*Cos[a] - I*x*Sin[a]]*Cos[a] + 2*ArcTan[x*Cos[a] - I*x*Sin[a]]*Sin[a]","A",1
141,1,132,35,0.0355522,"\int \frac{\tan (a+i \log (x))}{x^3} \, dx","Integrate[Tan[a + I*Log[x]]/x^3,x]","\cos (2 a) \left(-\tan ^{-1}\left(\frac{\left(x^2+1\right) \cos (a)}{\sin (a)-x^2 \sin (a)}\right)\right)+i \sin (2 a) \tan ^{-1}\left(\frac{\left(x^2+1\right) \cos (a)}{\sin (a)-x^2 \sin (a)}\right)-\frac{1}{2} i \cos (2 a) \log \left(2 x^2 \cos (2 a)+x^4+1\right)-\frac{1}{2} \sin (2 a) \log \left(2 x^2 \cos (2 a)+x^4+1\right)+2 \sin (2 a) \log (x)+2 i \cos (2 a) \log (x)+\frac{i}{2 x^2}","\frac{i}{2 x^2}-i e^{-2 i a} \log \left(1+\frac{e^{2 i a}}{x^2}\right)",1,"(I/2)/x^2 - ArcTan[((1 + x^2)*Cos[a])/(Sin[a] - x^2*Sin[a])]*Cos[2*a] + (2*I)*Cos[2*a]*Log[x] - (I/2)*Cos[2*a]*Log[1 + x^4 + 2*x^2*Cos[2*a]] + I*ArcTan[((1 + x^2)*Cos[a])/(Sin[a] - x^2*Sin[a])]*Sin[2*a] + 2*Log[x]*Sin[2*a] - (Log[1 + x^4 + 2*x^2*Cos[2*a]]*Sin[2*a])/2","B",1
142,1,70,45,0.0264008,"\int \frac{\tan (a+i \log (x))}{x^4} \, dx","Integrate[Tan[a + I*Log[x]]/x^4,x]","-\frac{2 \sin (2 a)}{x}-\frac{2 i \cos (2 a)}{x}-2 i \cos (3 a) \tan ^{-1}(x \cos (a)-i x \sin (a))-2 \sin (3 a) \tan ^{-1}(x \cos (a)-i x \sin (a))+\frac{i}{3 x^3}","-\frac{2 i e^{-2 i a}}{x}-2 i e^{-3 i a} \tan ^{-1}\left(e^{-i a} x\right)+\frac{i}{3 x^3}",1,"(I/3)/x^3 - ((2*I)*Cos[2*a])/x - (2*I)*ArcTan[x*Cos[a] - I*x*Sin[a]]*Cos[3*a] - (2*Sin[2*a])/x - 2*ArcTan[x*Cos[a] - I*x*Sin[a]]*Sin[3*a]","A",1
143,1,155,63,0.1808314,"\int x^3 \tan ^2(a+i \log (x)) \, dx","Integrate[x^3*Tan[a + I*Log[x]]^2,x]","2 i x^2 \sin (2 a)+2 x^2 \cos (2 a)-\frac{2 (\cos (5 a)+i \sin (5 a))}{\left(x^2+1\right) \cos (a)-i \left(x^2-1\right) \sin (a)}-4 i \cos (4 a) \tan ^{-1}\left(\frac{\left(x^2+1\right) \cot (a)}{x^2-1}\right)+4 \sin (4 a) \tan ^{-1}\left(\frac{\left(x^2+1\right) \cot (a)}{x^2-1}\right)-2 \cos (4 a) \log \left(2 x^2 \cos (2 a)+x^4+1\right)-2 i \sin (4 a) \log \left(2 x^2 \cos (2 a)+x^4+1\right)-\frac{x^4}{4}","2 e^{2 i a} x^2-\frac{2 e^{6 i a}}{x^2+e^{2 i a}}-4 e^{4 i a} \log \left(x^2+e^{2 i a}\right)-\frac{x^4}{4}",1,"-1/4*x^4 + 2*x^2*Cos[2*a] - (4*I)*ArcTan[((1 + x^2)*Cot[a])/(-1 + x^2)]*Cos[4*a] - 2*Cos[4*a]*Log[1 + x^4 + 2*x^2*Cos[2*a]] + (2*I)*x^2*Sin[2*a] + 4*ArcTan[((1 + x^2)*Cot[a])/(-1 + x^2)]*Sin[4*a] - (2*I)*Log[1 + x^4 + 2*x^2*Cos[2*a]]*Sin[4*a] - (2*(Cos[5*a] + I*Sin[5*a]))/((1 + x^2)*Cos[a] - I*(-1 + x^2)*Sin[a])","B",1
144,1,100,62,0.126177,"\int x^2 \tan ^2(a+i \log (x)) \, dx","Integrate[x^2*Tan[a + I*Log[x]]^2,x]","\frac{2 x (\cos (3 a)+i \sin (3 a))}{\left(x^2+1\right) \cos (a)-i \left(x^2-1\right) \sin (a)}+4 i x \sin (2 a)+4 x \cos (2 a)-6 \cos (3 a) \tan ^{-1}(x (\cos (a)-i \sin (a)))-6 i \sin (3 a) \tan ^{-1}(x (\cos (a)-i \sin (a)))-\frac{x^3}{3}","-\frac{2 e^{2 i a} x^3}{x^2+e^{2 i a}}+6 e^{2 i a} x-6 e^{3 i a} \tan ^{-1}\left(e^{-i a} x\right)-\frac{x^3}{3}",1,"-1/3*x^3 + 4*x*Cos[2*a] - 6*ArcTan[x*(Cos[a] - I*Sin[a])]*Cos[3*a] + (4*I)*x*Sin[2*a] + (2*x*(Cos[3*a] + I*Sin[3*a]))/((1 + x^2)*Cos[a] - I*(-1 + x^2)*Sin[a]) - (6*I)*ArcTan[x*(Cos[a] - I*Sin[a])]*Sin[3*a]","A",1
145,1,135,51,0.1238023,"\int x \tan ^2(a+i \log (x)) \, dx","Integrate[x*Tan[a + I*Log[x]]^2,x]","\frac{2 \cos (3 a)+2 i \sin (3 a)}{\left(x^2+1\right) \cos (a)-i \left(x^2-1\right) \sin (a)}+2 i \cos (2 a) \tan ^{-1}\left(\frac{\left(x^2+1\right) \cot (a)}{x^2-1}\right)-2 \sin (2 a) \tan ^{-1}\left(\frac{\left(x^2+1\right) \cot (a)}{x^2-1}\right)+\cos (2 a) \log \left(2 x^2 \cos (2 a)+x^4+1\right)+i \sin (2 a) \log \left(2 x^2 \cos (2 a)+x^4+1\right)-\frac{x^2}{2}","\frac{2 e^{4 i a}}{x^2+e^{2 i a}}+2 e^{2 i a} \log \left(x^2+e^{2 i a}\right)-\frac{x^2}{2}",1,"-1/2*x^2 + (2*I)*ArcTan[((1 + x^2)*Cot[a])/(-1 + x^2)]*Cos[2*a] + Cos[2*a]*Log[1 + x^4 + 2*x^2*Cos[2*a]] - 2*ArcTan[((1 + x^2)*Cot[a])/(-1 + x^2)]*Sin[2*a] + I*Log[1 + x^4 + 2*x^2*Cos[2*a]]*Sin[2*a] + (2*Cos[3*a] + (2*I)*Sin[3*a])/((1 + x^2)*Cos[a] - I*(-1 + x^2)*Sin[a])","B",1
146,1,70,46,0.0889977,"\int \tan ^2(a+i \log (x)) \, dx","Integrate[Tan[a + I*Log[x]]^2,x]","\frac{-x \left(x^2+3\right) \cos (a)+i x \left(x^2-3\right) \sin (a)}{\left(x^2+1\right) \cos (a)-i \left(x^2-1\right) \sin (a)}+2 (\cos (a)+i \sin (a)) \tan ^{-1}(x (\cos (a)-i \sin (a)))","-\frac{2 e^{2 i a} x}{x^2+e^{2 i a}}+2 e^{i a} \tan ^{-1}\left(e^{-i a} x\right)-x",1,"2*ArcTan[x*(Cos[a] - I*Sin[a])]*(Cos[a] + I*Sin[a]) + (-(x*(3 + x^2)*Cos[a]) + I*x*(-3 + x^2)*Sin[a])/((1 + x^2)*Cos[a] - I*(-1 + x^2)*Sin[a])","A",1
147,1,28,18,0.0385318,"\int \frac{\tan ^2(a+i \log (x))}{x} \, dx","Integrate[Tan[a + I*Log[x]]^2/x,x]","i \tan ^{-1}(\tan (a+i \log (x)))-i \tan (a+i \log (x))","-\log (x)-i \tan (a+i \log (x))",1,"I*ArcTan[Tan[a + I*Log[x]]] - I*Tan[a + I*Log[x]]","A",1
148,1,72,60,0.1104608,"\int \frac{\tan ^2(a+i \log (x))}{x^2} \, dx","Integrate[Tan[a + I*Log[x]]^2/x^2,x]","\frac{2 x (\cos (a)-i \sin (a))}{\left(x^2+1\right) \cos (a)-i \left(x^2-1\right) \sin (a)}+2 \cos (a) \tan ^{-1}(x (\cos (a)-i \sin (a)))-2 i \sin (a) \tan ^{-1}(x (\cos (a)-i \sin (a)))+\frac{1}{x}","\frac{3 x}{x^2+e^{2 i a}}+\frac{e^{2 i a}}{x \left(x^2+e^{2 i a}\right)}+2 e^{-i a} \tan ^{-1}\left(e^{-i a} x\right)",1,"x^(-1) + 2*ArcTan[x*(Cos[a] - I*Sin[a])]*Cos[a] - (2*I)*ArcTan[x*(Cos[a] - I*Sin[a])]*Sin[a] + (2*x*(Cos[a] - I*Sin[a]))/((1 + x^2)*Cos[a] - I*(-1 + x^2)*Sin[a])","A",1
149,1,150,55,0.1865976,"\int \frac{\tan ^2(a+i \log (x))}{x^3} \, dx","Integrate[Tan[a + I*Log[x]]^2/x^3,x]","\frac{2 \cos (a)-2 i \sin (a)}{\left(x^2+1\right) \cos (a)-i \left(x^2-1\right) \sin (a)}-2 i \cos (2 a) \tan ^{-1}\left(\frac{\left(x^2+1\right) \cot (a)}{x^2-1}\right)-2 \sin (2 a) \tan ^{-1}\left(\frac{\left(x^2+1\right) \cot (a)}{x^2-1}\right)-\cos (2 a) \log \left(2 x^2 \cos (2 a)+x^4+1\right)+i \sin (2 a) \log \left(2 x^2 \cos (2 a)+x^4+1\right)-4 i \sin (2 a) \log (x)+4 \cos (2 a) \log (x)+\frac{1}{2 x^2}","-\frac{2 e^{-2 i a}}{1+\frac{e^{2 i a}}{x^2}}-2 e^{-2 i a} \log \left(1+\frac{e^{2 i a}}{x^2}\right)+\frac{1}{2 x^2}",1,"1/(2*x^2) - (2*I)*ArcTan[((1 + x^2)*Cot[a])/(-1 + x^2)]*Cos[2*a] + 4*Cos[2*a]*Log[x] - Cos[2*a]*Log[1 + x^4 + 2*x^2*Cos[2*a]] + (2*Cos[a] - (2*I)*Sin[a])/((1 + x^2)*Cos[a] - I*(-1 + x^2)*Sin[a]) - 2*ArcTan[((1 + x^2)*Cot[a])/(-1 + x^2)]*Sin[2*a] - (4*I)*Log[x]*Sin[2*a] + I*Log[1 + x^4 + 2*x^2*Cos[2*a]]*Sin[2*a]","B",1
150,1,124,71,0.1996802,"\int (e x)^m \tan (a+i \log (x)) \, dx","Integrate[(e*x)^m*Tan[a + I*Log[x]],x]","\frac{x (\cos (a)-i \sin (a)) (e x)^m \left((m+1) x^2 (\sin (a)+i \cos (a)) \, _2F_1\left(1,\frac{m+3}{2};\frac{m+5}{2};-x^2 (\cos (2 a)-i \sin (2 a))\right)+(m+3) (\sin (a)-i \cos (a)) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-x^2 (\cos (2 a)-i \sin (2 a))\right)\right)}{(m+1) (m+3)}","\frac{2 i (e x)^{m+1} \, _2F_1\left(1,\frac{1}{2} (-m-1);\frac{1-m}{2};-\frac{e^{2 i a}}{x^2}\right)}{e (m+1)}-\frac{i (e x)^{m+1}}{e (m+1)}",1,"(x*(e*x)^m*(Cos[a] - I*Sin[a])*((3 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -(x^2*(Cos[2*a] - I*Sin[2*a]))]*((-I)*Cos[a] + Sin[a]) + (1 + m)*x^2*Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, -(x^2*(Cos[2*a] - I*Sin[2*a]))]*(I*Cos[a] + Sin[a])))/((1 + m)*(3 + m))","A",1
151,1,86,77,0.1622323,"\int (e x)^m \tan ^2(a+i \log (x)) \, dx","Integrate[(e*x)^m*Tan[a + I*Log[x]]^2,x]","\frac{x (e x)^m \left(4 \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-x^2 (\cos (2 a)-i \sin (2 a))\right)-4 \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-x^2 (\cos (2 a)-i \sin (2 a))\right)-1\right)}{m+1}","-2 x (e x)^m \, _2F_1\left(1,\frac{1}{2} (-m-1);\frac{1-m}{2};-\frac{e^{2 i a}}{x^2}\right)+\frac{2 x (e x)^m}{1+\frac{e^{2 i a}}{x^2}}-\frac{x (e x)^m}{m+1}",1,"(x*(e*x)^m*(-1 + 4*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -(x^2*(Cos[2*a] - I*Sin[2*a]))] - 4*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -(x^2*(Cos[2*a] - I*Sin[2*a]))]))/(1 + m)","A",1
152,1,125,184,0.2260264,"\int (e x)^m \tan ^3(a+i \log (x)) \, dx","Integrate[(e*x)^m*Tan[a + I*Log[x]]^3,x]","\frac{i x (e x)^m \left(6 \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-x^2 (\cos (2 a)-i \sin (2 a))\right)-12 \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-x^2 (\cos (2 a)-i \sin (2 a))\right)+8 \, _2F_1\left(3,\frac{m+1}{2};\frac{m+3}{2};-x^2 (\cos (2 a)-i \sin (2 a))\right)-1\right)}{m+1}","-\frac{i \left(m^2+2 m+3\right) x (e x)^m \, _2F_1\left(1,\frac{1}{2} (-m-1);\frac{1-m}{2};-\frac{e^{2 i a}}{x^2}\right)}{m+1}+\frac{i e^{-2 i a} x \left(\frac{e^{4 i a} (1-m)}{x^2}+e^{2 i a} (m+3)\right) (e x)^m}{2 \left(1+\frac{e^{2 i a}}{x^2}\right)}+\frac{i x \left(1-\frac{e^{2 i a}}{x^2}\right)^2 (e x)^m}{2 \left(1+\frac{e^{2 i a}}{x^2}\right)^2}-\frac{i (1-m) m x (e x)^m}{2 (m+1)}",1,"(I*x*(e*x)^m*(-1 + 6*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -(x^2*(Cos[2*a] - I*Sin[2*a]))] - 12*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -(x^2*(Cos[2*a] - I*Sin[2*a]))] + 8*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, -(x^2*(Cos[2*a] - I*Sin[2*a]))]))/(1 + m)","A",1
153,1,330,142,0.6940191,"\int \tan ^p(a+b \log (x)) \, dx","Integrate[Tan[a + b*Log[x]]^p,x]","\frac{(2 b-i) x \left(-\frac{i \left(-1+e^{2 i a} x^{2 i b}\right)}{1+e^{2 i a} x^{2 i b}}\right)^p F_1\left(-\frac{i}{2 b};-p,p;1-\frac{i}{2 b};e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right)}{-2 e^{2 i a} b p x^{2 i b} F_1\left(1-\frac{i}{2 b};1-p,p;2-\frac{i}{2 b};e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right)-2 e^{2 i a} b p x^{2 i b} F_1\left(1-\frac{i}{2 b};-p,p+1;2-\frac{i}{2 b};e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right)+(2 b-i) F_1\left(-\frac{i}{2 b};-p,p;1-\frac{i}{2 b};e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right)}","x \left(1-e^{2 i a} x^{2 i b}\right)^{-p} \left(\frac{i \left(1-e^{2 i a} x^{2 i b}\right)}{1+e^{2 i a} x^{2 i b}}\right)^p \left(1+e^{2 i a} x^{2 i b}\right)^p F_1\left(-\frac{i}{2 b};-p,p;1-\frac{i}{2 b};e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right)",1,"((-I + 2*b)*x*(((-I)*(-1 + E^((2*I)*a)*x^((2*I)*b)))/(1 + E^((2*I)*a)*x^((2*I)*b)))^p*AppellF1[(-1/2*I)/b, -p, p, 1 - (I/2)/b, E^((2*I)*a)*x^((2*I)*b), -(E^((2*I)*a)*x^((2*I)*b))])/(-2*b*E^((2*I)*a)*p*x^((2*I)*b)*AppellF1[1 - (I/2)/b, 1 - p, p, 2 - (I/2)/b, E^((2*I)*a)*x^((2*I)*b), -(E^((2*I)*a)*x^((2*I)*b))] - 2*b*E^((2*I)*a)*p*x^((2*I)*b)*AppellF1[1 - (I/2)/b, -p, 1 + p, 2 - (I/2)/b, E^((2*I)*a)*x^((2*I)*b), -(E^((2*I)*a)*x^((2*I)*b))] + (-I + 2*b)*AppellF1[(-1/2*I)/b, -p, p, 1 - (I/2)/b, E^((2*I)*a)*x^((2*I)*b), -(E^((2*I)*a)*x^((2*I)*b))])","B",0
154,1,157,162,0.6711425,"\int (e x)^m \tan ^p(a+b \log (x)) \, dx","Integrate[(e*x)^m*Tan[a + b*Log[x]]^p,x]","\frac{x (e x)^m \left(1-e^{2 i a} x^{2 i b}\right)^{-p} \left(-\frac{i \left(-1+e^{2 i a} x^{2 i b}\right)}{1+e^{2 i a} x^{2 i b}}\right)^p \left(1+e^{2 i a} x^{2 i b}\right)^p F_1\left(-\frac{i (m+1)}{2 b};-p,p;1-\frac{i (m+1)}{2 b};e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right)}{m+1}","\frac{(e x)^{m+1} \left(1-e^{2 i a} x^{2 i b}\right)^{-p} \left(\frac{i \left(1-e^{2 i a} x^{2 i b}\right)}{1+e^{2 i a} x^{2 i b}}\right)^p \left(1+e^{2 i a} x^{2 i b}\right)^p F_1\left(-\frac{i (m+1)}{2 b};-p,p;1-\frac{i (m+1)}{2 b};e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right)}{e (m+1)}",1,"(x*(e*x)^m*(((-I)*(-1 + E^((2*I)*a)*x^((2*I)*b)))/(1 + E^((2*I)*a)*x^((2*I)*b)))^p*(1 + E^((2*I)*a)*x^((2*I)*b))^p*AppellF1[((-1/2*I)*(1 + m))/b, -p, p, 1 - ((I/2)*(1 + m))/b, E^((2*I)*a)*x^((2*I)*b), -(E^((2*I)*a)*x^((2*I)*b))])/((1 + m)*(1 - E^((2*I)*a)*x^((2*I)*b))^p)","A",1
155,1,240,120,0.5264406,"\int \tan ^p(a+\log (x)) \, dx","Integrate[Tan[a + Log[x]]^p,x]","\frac{(1+2 i) x \left(-\frac{i \left(-1+e^{2 i a} x^{2 i}\right)}{1+e^{2 i a} x^{2 i}}\right)^p F_1\left(-\frac{i}{2};-p,p;1-\frac{i}{2};e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right)}{(1+2 i) F_1\left(-\frac{i}{2};-p,p;1-\frac{i}{2};e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right)-2 i e^{2 i a} p x^{2 i} \left(F_1\left(1-\frac{i}{2};1-p,p;2-\frac{i}{2};e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right)+F_1\left(1-\frac{i}{2};-p,p+1;2-\frac{i}{2};e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right)\right)}","x \left(1-e^{2 i a} x^{2 i}\right)^{-p} \left(\frac{i \left(1-e^{2 i a} x^{2 i}\right)}{1+e^{2 i a} x^{2 i}}\right)^p \left(1+e^{2 i a} x^{2 i}\right)^p F_1\left(-\frac{i}{2};-p,p;1-\frac{i}{2};e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right)",1,"((1 + 2*I)*(((-I)*(-1 + E^((2*I)*a)*x^(2*I)))/(1 + E^((2*I)*a)*x^(2*I)))^p*x*AppellF1[-1/2*I, -p, p, 1 - I/2, E^((2*I)*a)*x^(2*I), -(E^((2*I)*a)*x^(2*I))])/((1 + 2*I)*AppellF1[-1/2*I, -p, p, 1 - I/2, E^((2*I)*a)*x^(2*I), -(E^((2*I)*a)*x^(2*I))] - (2*I)*E^((2*I)*a)*p*x^(2*I)*(AppellF1[1 - I/2, 1 - p, p, 2 - I/2, E^((2*I)*a)*x^(2*I), -(E^((2*I)*a)*x^(2*I))] + AppellF1[1 - I/2, -p, 1 + p, 2 - I/2, E^((2*I)*a)*x^(2*I), -(E^((2*I)*a)*x^(2*I))]))","A",0
156,1,240,120,0.5135969,"\int \tan ^p(a+2 \log (x)) \, dx","Integrate[Tan[a + 2*Log[x]]^p,x]","\frac{(1+4 i) x \left(-\frac{i \left(-1+e^{2 i a} x^{4 i}\right)}{1+e^{2 i a} x^{4 i}}\right)^p F_1\left(-\frac{i}{4};-p,p;1-\frac{i}{4};e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right)}{(1+4 i) F_1\left(-\frac{i}{4};-p,p;1-\frac{i}{4};e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right)-4 i e^{2 i a} p x^{4 i} \left(F_1\left(1-\frac{i}{4};1-p,p;2-\frac{i}{4};e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right)+F_1\left(1-\frac{i}{4};-p,p+1;2-\frac{i}{4};e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right)\right)}","x \left(1-e^{2 i a} x^{4 i}\right)^{-p} \left(\frac{i \left(1-e^{2 i a} x^{4 i}\right)}{1+e^{2 i a} x^{4 i}}\right)^p \left(1+e^{2 i a} x^{4 i}\right)^p F_1\left(-\frac{i}{4};-p,p;1-\frac{i}{4};e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right)",1,"((1 + 4*I)*(((-I)*(-1 + E^((2*I)*a)*x^(4*I)))/(1 + E^((2*I)*a)*x^(4*I)))^p*x*AppellF1[-1/4*I, -p, p, 1 - I/4, E^((2*I)*a)*x^(4*I), -(E^((2*I)*a)*x^(4*I))])/((1 + 4*I)*AppellF1[-1/4*I, -p, p, 1 - I/4, E^((2*I)*a)*x^(4*I), -(E^((2*I)*a)*x^(4*I))] - (4*I)*E^((2*I)*a)*p*x^(4*I)*(AppellF1[1 - I/4, 1 - p, p, 2 - I/4, E^((2*I)*a)*x^(4*I), -(E^((2*I)*a)*x^(4*I))] + AppellF1[1 - I/4, -p, 1 + p, 2 - I/4, E^((2*I)*a)*x^(4*I), -(E^((2*I)*a)*x^(4*I))]))","A",0
157,1,240,120,0.5031612,"\int \tan ^p(a+3 \log (x)) \, dx","Integrate[Tan[a + 3*Log[x]]^p,x]","\frac{(1+6 i) x \left(-\frac{i \left(-1+e^{2 i a} x^{6 i}\right)}{1+e^{2 i a} x^{6 i}}\right)^p F_1\left(-\frac{i}{6};-p,p;1-\frac{i}{6};e^{2 i a} x^{6 i},-e^{2 i a} x^{6 i}\right)}{(1+6 i) F_1\left(-\frac{i}{6};-p,p;1-\frac{i}{6};e^{2 i a} x^{6 i},-e^{2 i a} x^{6 i}\right)-6 i e^{2 i a} p x^{6 i} \left(F_1\left(1-\frac{i}{6};1-p,p;2-\frac{i}{6};e^{2 i a} x^{6 i},-e^{2 i a} x^{6 i}\right)+F_1\left(1-\frac{i}{6};-p,p+1;2-\frac{i}{6};e^{2 i a} x^{6 i},-e^{2 i a} x^{6 i}\right)\right)}","x \left(1-e^{2 i a} x^{6 i}\right)^{-p} \left(\frac{i \left(1-e^{2 i a} x^{6 i}\right)}{1+e^{2 i a} x^{6 i}}\right)^p \left(1+e^{2 i a} x^{6 i}\right)^p F_1\left(-\frac{i}{6};-p,p;1-\frac{i}{6};e^{2 i a} x^{6 i},-e^{2 i a} x^{6 i}\right)",1,"((1 + 6*I)*(((-I)*(-1 + E^((2*I)*a)*x^(6*I)))/(1 + E^((2*I)*a)*x^(6*I)))^p*x*AppellF1[-1/6*I, -p, p, 1 - I/6, E^((2*I)*a)*x^(6*I), -(E^((2*I)*a)*x^(6*I))])/((1 + 6*I)*AppellF1[-1/6*I, -p, p, 1 - I/6, E^((2*I)*a)*x^(6*I), -(E^((2*I)*a)*x^(6*I))] - (6*I)*E^((2*I)*a)*p*x^(6*I)*(AppellF1[1 - I/6, 1 - p, p, 2 - I/6, E^((2*I)*a)*x^(6*I), -(E^((2*I)*a)*x^(6*I))] + AppellF1[1 - I/6, -p, 1 + p, 2 - I/6, E^((2*I)*a)*x^(6*I), -(E^((2*I)*a)*x^(6*I))]))","A",0
158,1,146,71,6.3439453,"\int x^3 \tan \left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[x^3*Tan[d*(a + b*Log[c*x^n])],x]","\frac{x^4 \left(2 i e^{2 i d \left(a+b \log \left(c x^n\right)\right)} \, _2F_1\left(1,1-\frac{2 i}{b d n};2-\frac{2 i}{b d n};-e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)+(b d n-2 i) \, _2F_1\left(1,-\frac{2 i}{b d n};1-\frac{2 i}{b d n};-e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)\right)}{-8-4 i b d n}","\frac{1}{2} i x^4 \, _2F_1\left(1,-\frac{2 i}{b d n};1-\frac{2 i}{b d n};-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)-\frac{i x^4}{4}",1,"(x^4*((2*I)*E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 - (2*I)/(b*d*n), 2 - (2*I)/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))] + (-2*I + b*d*n)*Hypergeometric2F1[1, (-2*I)/(b*d*n), 1 - (2*I)/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))]))/(-8 - (4*I)*b*d*n)","B",1
159,1,155,75,5.9139017,"\int x^2 \tan \left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[x^2*Tan[d*(a + b*Log[c*x^n])],x]","\frac{x^3 \left(3 i e^{2 i d \left(a+b \log \left(c x^n\right)\right)} \, _2F_1\left(1,1-\frac{3 i}{2 b d n};2-\frac{3 i}{2 b d n};-e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)+(2 b d n-3 i) \, _2F_1\left(1,-\frac{3 i}{2 b d n};1-\frac{3 i}{2 b d n};-e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)\right)}{-9-6 i b d n}","\frac{2}{3} i x^3 \, _2F_1\left(1,-\frac{3 i}{2 b d n};1-\frac{3 i}{2 b d n};-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)-\frac{i x^3}{3}",1,"(x^3*((3*I)*E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 - ((3*I)/2)/(b*d*n), 2 - ((3*I)/2)/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))] + (-3*I + 2*b*d*n)*Hypergeometric2F1[1, ((-3*I)/2)/(b*d*n), 1 - ((3*I)/2)/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))]))/(-9 - (6*I)*b*d*n)","B",1
160,1,146,69,6.0074276,"\int x \tan \left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[x*Tan[d*(a + b*Log[c*x^n])],x]","\frac{x^2 \left(i e^{2 i d \left(a+b \log \left(c x^n\right)\right)} \, _2F_1\left(1,1-\frac{i}{b d n};2-\frac{i}{b d n};-e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)+(b d n-i) \, _2F_1\left(1,-\frac{i}{b d n};1-\frac{i}{b d n};-e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)\right)}{-2-2 i b d n}","i x^2 \, _2F_1\left(1,-\frac{i}{b d n};1-\frac{i}{b d n};-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)-\frac{i x^2}{2}",1,"(x^2*(I*E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 - I/(b*d*n), 2 - I/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))] + (-I + b*d*n)*Hypergeometric2F1[1, (-I)/(b*d*n), 1 - I/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))]))/(-2 - (2*I)*b*d*n)","B",1
161,1,151,67,11.2203165,"\int \tan \left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[Tan[d*(a + b*Log[c*x^n])],x]","\frac{x \left((1+2 i b d n) \, _2F_1\left(1,-\frac{i}{2 b d n};1-\frac{i}{2 b d n};-e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)-e^{2 i d \left(a+b \log \left(c x^n\right)\right)} \, _2F_1\left(1,1-\frac{i}{2 b d n};2-\frac{i}{2 b d n};-e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)\right)}{2 b d n-i}","2 i x \, _2F_1\left(1,-\frac{i}{2 b d n};1-\frac{i}{2 b d n};-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)-i x",1,"(x*(-(E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 - (I/2)/(b*d*n), 2 - (I/2)/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))]) + (1 + (2*I)*b*d*n)*Hypergeometric2F1[1, (-1/2*I)/(b*d*n), 1 - (I/2)/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))]))/(-I + 2*b*d*n)","B",1
162,1,25,26,0.0548132,"\int \frac{\tan \left(d \left(a+b \log \left(c x^n\right)\right)\right)}{x} \, dx","Integrate[Tan[d*(a + b*Log[c*x^n])]/x,x]","-\frac{\log \left(\cos \left(d \left(a+b \log \left(c x^n\right)\right)\right)\right)}{b d n}","-\frac{\log \left(\cos \left(a d+b d \log \left(c x^n\right)\right)\right)}{b d n}",1,"-(Log[Cos[d*(a + b*Log[c*x^n])]]/(b*d*n))","A",1
163,1,153,71,4.1354768,"\int \frac{\tan \left(d \left(a+b \log \left(c x^n\right)\right)\right)}{x^2} \, dx","Integrate[Tan[d*(a + b*Log[c*x^n])]/x^2,x]","\frac{(1-2 i b d n) \, _2F_1\left(1,\frac{i}{2 b d n};1+\frac{i}{2 b d n};-e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)-e^{2 i d \left(a+b \log \left(c x^n\right)\right)} \, _2F_1\left(1,1+\frac{i}{2 b d n};2+\frac{i}{2 b d n};-e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)}{x (2 b d n+i)}","\frac{i}{x}-\frac{2 i \, _2F_1\left(1,\frac{i}{2 b d n};1+\frac{i}{2 b d n};-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{x}",1,"(-(E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 + (I/2)/(b*d*n), 2 + (I/2)/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))]) + (1 - (2*I)*b*d*n)*Hypergeometric2F1[1, (I/2)/(b*d*n), 1 + (I/2)/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))])/((I + 2*b*d*n)*x)","B",1
164,1,147,69,3.7619765,"\int \frac{\tan \left(d \left(a+b \log \left(c x^n\right)\right)\right)}{x^3} \, dx","Integrate[Tan[d*(a + b*Log[c*x^n])]/x^3,x]","\frac{(1-i b d n) \, _2F_1\left(1,\frac{i}{b d n};1+\frac{i}{b d n};-e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)-e^{2 i d \left(a+b \log \left(c x^n\right)\right)} \, _2F_1\left(1,1+\frac{i}{b d n};2+\frac{i}{b d n};-e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)}{2 x^2 (b d n+i)}","\frac{i}{2 x^2}-\frac{i \, _2F_1\left(1,\frac{i}{b d n};1+\frac{i}{b d n};-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{x^2}",1,"(-(E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 + I/(b*d*n), 2 + I/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))]) + (1 - I*b*d*n)*Hypergeometric2F1[1, I/(b*d*n), 1 + I/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))])/(2*(I + b*d*n)*x^2)","B",1
165,1,179,159,6.5180683,"\int x^3 \tan ^2\left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[x^3*Tan[d*(a + b*Log[c*x^n])]^2,x]","-\frac{x^4 \left((b d n-2 i) \left(4 i \, _2F_1\left(1,-\frac{2 i}{b d n};1-\frac{2 i}{b d n};-e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)-4 \tan \left(d \left(a+b \log \left(c x^n\right)\right)\right)+b d n\right)-8 e^{2 i d \left(a+b \log \left(c x^n\right)\right)} \, _2F_1\left(1,1-\frac{2 i}{b d n};2-\frac{2 i}{b d n};-e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)\right)}{4 b d n (b d n-2 i)}","-\frac{2 i x^4 \, _2F_1\left(1,-\frac{2 i}{b d n};1-\frac{2 i}{b d n};-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{b d n}+\frac{i x^4 \left(1-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{b d n \left(1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}+\frac{x^4 (-b d n+4 i)}{4 b d n}",1,"-1/4*(x^4*(-8*E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 - (2*I)/(b*d*n), 2 - (2*I)/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))] + (-2*I + b*d*n)*(b*d*n + (4*I)*Hypergeometric2F1[1, (-2*I)/(b*d*n), 1 - (2*I)/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))] - 4*Tan[d*(a + b*Log[c*x^n])])))/(b*d*n*(-2*I + b*d*n))","A",1
166,1,189,163,6.4133492,"\int x^2 \tan ^2\left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[x^2*Tan[d*(a + b*Log[c*x^n])]^2,x]","-\frac{x^3 \left((2 b d n-3 i) \left(3 i \, _2F_1\left(1,-\frac{3 i}{2 b d n};1-\frac{3 i}{2 b d n};-e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)-3 \tan \left(d \left(a+b \log \left(c x^n\right)\right)\right)+b d n\right)-9 e^{2 i d \left(a+b \log \left(c x^n\right)\right)} \, _2F_1\left(1,1-\frac{3 i}{2 b d n};2-\frac{3 i}{2 b d n};-e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)\right)}{3 b d n (2 b d n-3 i)}","-\frac{2 i x^3 \, _2F_1\left(1,-\frac{3 i}{2 b d n};1-\frac{3 i}{2 b d n};-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{b d n}+\frac{i x^3 \left(1-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{b d n \left(1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}+\frac{x^3 (-b d n+3 i)}{3 b d n}",1,"-1/3*(x^3*(-9*E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 - ((3*I)/2)/(b*d*n), 2 - ((3*I)/2)/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))] + (-3*I + 2*b*d*n)*(b*d*n + (3*I)*Hypergeometric2F1[1, ((-3*I)/2)/(b*d*n), 1 - ((3*I)/2)/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))] - 3*Tan[d*(a + b*Log[c*x^n])])))/(b*d*n*(-3*I + 2*b*d*n))","A",1
167,1,179,159,6.4291813,"\int x \tan ^2\left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[x*Tan[d*(a + b*Log[c*x^n])]^2,x]","-\frac{x^2 \left((b d n-i) \left(2 i \, _2F_1\left(1,-\frac{i}{b d n};1-\frac{i}{b d n};-e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)-2 \tan \left(d \left(a+b \log \left(c x^n\right)\right)\right)+b d n\right)-2 e^{2 i d \left(a+b \log \left(c x^n\right)\right)} \, _2F_1\left(1,1-\frac{i}{b d n};2-\frac{i}{b d n};-e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)\right)}{2 b d n (b d n-i)}","-\frac{2 i x^2 \, _2F_1\left(1,-\frac{i}{b d n};1-\frac{i}{b d n};-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{b d n}+\frac{i x^2 \left(1-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{b d n \left(1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}+\frac{x^2 (-b d n+2 i)}{2 b d n}",1,"-1/2*(x^2*(-2*E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 - I/(b*d*n), 2 - I/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))] + (-I + b*d*n)*(b*d*n + (2*I)*Hypergeometric2F1[1, (-I)/(b*d*n), 1 - I/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))] - 2*Tan[d*(a + b*Log[c*x^n])])))/(b*d*n*(-I + b*d*n))","A",1
168,1,185,154,11.730083,"\int \tan ^2\left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[Tan[d*(a + b*Log[c*x^n])]^2,x]","\frac{x e^{2 i d \left(a+b \log \left(c x^n\right)\right)} \, _2F_1\left(1,1-\frac{i}{2 b d n};2-\frac{i}{2 b d n};-e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)-x (2 b d n-i) \left(i \, _2F_1\left(1,-\frac{i}{2 b d n};1-\frac{i}{2 b d n};-e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)-\tan \left(d \left(a+b \log \left(c x^n\right)\right)\right)+b d n\right)}{b d n (2 b d n-i)}","-\frac{2 i x \, _2F_1\left(1,-\frac{i}{2 b d n};1-\frac{i}{2 b d n};-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{b d n}+\frac{i x \left(1-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{b d n \left(1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}+\frac{x (-b d n+i)}{b d n}",1,"(E^((2*I)*d*(a + b*Log[c*x^n]))*x*Hypergeometric2F1[1, 1 - (I/2)/(b*d*n), 2 - (I/2)/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))] - (-I + 2*b*d*n)*x*(b*d*n + I*Hypergeometric2F1[1, (-1/2*I)/(b*d*n), 1 - (I/2)/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))] - Tan[d*(a + b*Log[c*x^n])]))/(b*d*n*(-I + 2*b*d*n))","A",1
169,1,51,29,0.0796272,"\int \frac{\tan ^2\left(d \left(a+b \log \left(c x^n\right)\right)\right)}{x} \, dx","Integrate[Tan[d*(a + b*Log[c*x^n])]^2/x,x]","\frac{\tan \left(a d+b d \log \left(c x^n\right)\right)}{b d n}-\frac{\tan ^{-1}\left(\tan \left(a d+b d \log \left(c x^n\right)\right)\right)}{b d n}","\frac{\tan \left(a d+b d \log \left(c x^n\right)\right)}{b d n}-\log (x)",1,"-(ArcTan[Tan[a*d + b*d*Log[c*x^n]]]/(b*d*n)) + Tan[a*d + b*d*Log[c*x^n]]/(b*d*n)","A",1
170,1,184,157,4.2921777,"\int \frac{\tan ^2\left(d \left(a+b \log \left(c x^n\right)\right)\right)}{x^2} \, dx","Integrate[Tan[d*(a + b*Log[c*x^n])]^2/x^2,x]","\frac{(2 b d n+i) \left(-i \, _2F_1\left(1,\frac{i}{2 b d n};1+\frac{i}{2 b d n};-e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)+\tan \left(d \left(a+b \log \left(c x^n\right)\right)\right)+b d n\right)-e^{2 i d \left(a+b \log \left(c x^n\right)\right)} \, _2F_1\left(1,1+\frac{i}{2 b d n};2+\frac{i}{2 b d n};-e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)}{b d n x (2 b d n+i)}","-\frac{2 i \, _2F_1\left(1,\frac{i}{2 b d n};1+\frac{i}{2 b d n};-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{b d n x}+\frac{i \left(1-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{b d n x \left(1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}+\frac{1+\frac{i}{b d n}}{x}",1,"(-(E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 + (I/2)/(b*d*n), 2 + (I/2)/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))]) + (I + 2*b*d*n)*(b*d*n - I*Hypergeometric2F1[1, (I/2)/(b*d*n), 1 + (I/2)/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))] + Tan[d*(a + b*Log[c*x^n])]))/(b*d*n*(I + 2*b*d*n)*x)","A",1
171,1,179,156,3.9141743,"\int \frac{\tan ^2\left(d \left(a+b \log \left(c x^n\right)\right)\right)}{x^3} \, dx","Integrate[Tan[d*(a + b*Log[c*x^n])]^2/x^3,x]","\frac{(b d n+i) \left(-2 i \, _2F_1\left(1,\frac{i}{b d n};1+\frac{i}{b d n};-e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)+2 \tan \left(d \left(a+b \log \left(c x^n\right)\right)\right)+b d n\right)-2 e^{2 i d \left(a+b \log \left(c x^n\right)\right)} \, _2F_1\left(1,1+\frac{i}{b d n};2+\frac{i}{b d n};-e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)}{2 b d n x^2 (b d n+i)}","-\frac{2 i \, _2F_1\left(1,\frac{i}{b d n};1+\frac{i}{b d n};-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{b d n x^2}+\frac{i \left(1-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{b d n x^2 \left(1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}+\frac{1+\frac{2 i}{b d n}}{2 x^2}",1,"(-2*E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 + I/(b*d*n), 2 + I/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))] + (I + b*d*n)*(b*d*n - (2*I)*Hypergeometric2F1[1, I/(b*d*n), 1 + I/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))] + 2*Tan[d*(a + b*Log[c*x^n])]))/(2*b*d*n*(I + b*d*n)*x^2)","A",1
172,1,38,43,0.1537798,"\int \frac{\tan ^3\left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Tan[a + b*Log[c*x^n]]^3/x,x]","\frac{\tan ^2\left(a+b \log \left(c x^n\right)\right)+2 \log \left(\cos \left(a+b \log \left(c x^n\right)\right)\right)}{2 b n}","\frac{\tan ^2\left(a+b \log \left(c x^n\right)\right)}{2 b n}+\frac{\log \left(\cos \left(a+b \log \left(c x^n\right)\right)\right)}{b n}",1,"(2*Log[Cos[a + b*Log[c*x^n]]] + Tan[a + b*Log[c*x^n]]^2)/(2*b*n)","A",1
173,1,62,45,0.0928193,"\int \frac{\tan ^4\left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Tan[a + b*Log[c*x^n]]^4/x,x]","\frac{\tan ^{-1}\left(\tan \left(a+b \log \left(c x^n\right)\right)\right)}{b n}+\frac{\tan ^3\left(a+b \log \left(c x^n\right)\right)}{3 b n}-\frac{\tan \left(a+b \log \left(c x^n\right)\right)}{b n}","\frac{\tan ^3\left(a+b \log \left(c x^n\right)\right)}{3 b n}-\frac{\tan \left(a+b \log \left(c x^n\right)\right)}{b n}+\log (x)",1,"ArcTan[Tan[a + b*Log[c*x^n]]]/(b*n) - Tan[a + b*Log[c*x^n]]/(b*n) + Tan[a + b*Log[c*x^n]]^3/(3*b*n)","A",1
174,1,55,67,0.1595184,"\int \frac{\tan ^5\left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Tan[a + b*Log[c*x^n]]^5/x,x]","-\frac{-\tan ^4\left(a+b \log \left(c x^n\right)\right)+2 \tan ^2\left(a+b \log \left(c x^n\right)\right)+4 \log \left(\cos \left(a+b \log \left(c x^n\right)\right)\right)}{4 b n}","\frac{\tan ^4\left(a+b \log \left(c x^n\right)\right)}{4 b n}-\frac{\tan ^2\left(a+b \log \left(c x^n\right)\right)}{2 b n}-\frac{\log \left(\cos \left(a+b \log \left(c x^n\right)\right)\right)}{b n}",1,"-1/4*(4*Log[Cos[a + b*Log[c*x^n]]] + 2*Tan[a + b*Log[c*x^n]]^2 - Tan[a + b*Log[c*x^n]]^4)/(b*n)","A",1
175,1,186,101,14.6918456,"\int (e x)^m \tan \left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[(e*x)^m*Tan[d*(a + b*Log[c*x^n])],x]","\frac{i x (e x)^m \left(\, _2F_1\left(1,-\frac{i (m+1)}{2 b d n};1-\frac{i (m+1)}{2 b d n};-e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)-\frac{(m+1) e^{2 i a d} \left(c x^n\right)^{2 i b d} \, _2F_1\left(1,-\frac{i (m+2 i b d n+1)}{2 b d n};-\frac{i (m+4 i b d n+1)}{2 b d n};-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{2 i b d n+m+1}\right)}{m+1}","\frac{2 i (e x)^{m+1} \, _2F_1\left(1,-\frac{i (m+1)}{2 b d n};1-\frac{i (m+1)}{2 b d n};-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{e (m+1)}-\frac{i (e x)^{m+1}}{e (m+1)}",1,"(I*x*(e*x)^m*(Hypergeometric2F1[1, ((-1/2*I)*(1 + m))/(b*d*n), 1 - ((I/2)*(1 + m))/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))] - (E^((2*I)*a*d)*(1 + m)*(c*x^n)^((2*I)*b*d)*Hypergeometric2F1[1, ((-1/2*I)*(1 + m + (2*I)*b*d*n))/(b*d*n), ((-1/2*I)*(1 + m + (4*I)*b*d*n))/(b*d*n), -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))])/(1 + m + (2*I)*b*d*n)))/(1 + m)","A",1
176,1,550,196,17.5530081,"\int (e x)^m \tan ^2\left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[(e*x)^m*Tan[d*(a + b*Log[c*x^n])]^2,x]","-\frac{(m+1) x^{-m} (e x)^m \sec \left(d \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right) \left(\frac{x^{m+1} \sin (b d n \log (x)) \sec \left(d \left(a+b \log \left(c x^n\right)\right)\right)}{m+1}-\frac{i \cos \left(d \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right) \exp \left(-\frac{(2 m+1) \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)}{b n}\right) \left((2 i b d n+m+1) \left(-\exp \left(\frac{2 a m+a+b (2 m+1) \left(\log \left(c x^n\right)-n \log (x)\right)+b (m+1) n \log (x)}{b n}\right)\right) \, _2F_1\left(1,-\frac{i (m+1)}{2 b d n};1-\frac{i (m+1)}{2 b d n};-e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)+(m+1) \exp \left(\frac{a (2 i b d n+2 m+1)}{b n}+\frac{(2 i b d n+2 m+1) \left(\log \left(c x^n\right)-n \log (x)\right)}{n}+\log (x) (2 i b d n+m+1)\right) \, _2F_1\left(1,-\frac{i (m+2 i b d n+1)}{2 b d n};-\frac{i (m+4 i b d n+1)}{2 b d n};-e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)-i (2 i b d n+m+1) \tan \left(d \left(a+b \log \left(c x^n\right)\right)\right) \exp \left(\frac{2 a m+a+b (2 m+1) \left(\log \left(c x^n\right)-n \log (x)\right)+b (m+1) n \log (x)}{b n}\right)\right)}{(m+1) (2 i b d n+m+1)}\right)}{b d n}+\frac{x (e x)^m \sin (b d n \log (x)) \sec \left(d \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right) \sec \left(d \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)+b d n \log (x)\right)}{b d n}-\frac{x (e x)^m}{m+1}","-\frac{2 i (e x)^{m+1} \, _2F_1\left(1,-\frac{i (m+1)}{2 b d n};1-\frac{i (m+1)}{2 b d n};-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{b d e n}+\frac{i (e x)^{m+1} \left(1-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{b d e n \left(1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}+\frac{(e x)^{m+1} (-b d n+i (m+1))}{b d e (m+1) n}",1,"-((x*(e*x)^m)/(1 + m)) + (x*(e*x)^m*Sec[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]*Sec[b*d*n*Log[x] + d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]*Sin[b*d*n*Log[x]])/(b*d*n) - ((1 + m)*(e*x)^m*Sec[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]*((x^(1 + m)*Sec[d*(a + b*Log[c*x^n])]*Sin[b*d*n*Log[x]])/(1 + m) - (I*Cos[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]*(-(E^((a + 2*a*m + b*(1 + m)*n*Log[x] + b*(1 + 2*m)*(-(n*Log[x]) + Log[c*x^n]))/(b*n))*(1 + m + (2*I)*b*d*n)*Hypergeometric2F1[1, ((-1/2*I)*(1 + m))/(b*d*n), 1 - ((I/2)*(1 + m))/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))]) + E^((a*(1 + 2*m + (2*I)*b*d*n))/(b*n) + (1 + m + (2*I)*b*d*n)*Log[x] + ((1 + 2*m + (2*I)*b*d*n)*(-(n*Log[x]) + Log[c*x^n]))/n)*(1 + m)*Hypergeometric2F1[1, ((-1/2*I)*(1 + m + (2*I)*b*d*n))/(b*d*n), ((-1/2*I)*(1 + m + (4*I)*b*d*n))/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))] - I*E^((a + 2*a*m + b*(1 + m)*n*Log[x] + b*(1 + 2*m)*(-(n*Log[x]) + Log[c*x^n]))/(b*n))*(1 + m + (2*I)*b*d*n)*Tan[d*(a + b*Log[c*x^n])]))/(E^(((1 + 2*m)*(a + b*(-(n*Log[x]) + Log[c*x^n])))/(b*n))*(1 + m)*(1 + m + (2*I)*b*d*n))))/(b*d*n*x^m)","B",1
177,1,642,351,17.9895588,"\int (e x)^m \tan ^3\left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[(e*x)^m*Tan[d*(a + b*Log[c*x^n])]^3,x]","-\frac{x^{-m} (e x)^m \left(2 b^2 d^2 n^2-m^2-2 m-1\right) \sec \left(d \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right) \left(\frac{x^{m+1} \sin (b d n \log (x)) \sec \left(d \left(a+b \log \left(c x^n\right)\right)\right)}{m+1}-\frac{i \cos \left(d \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right) \exp \left(-\frac{(2 m+1) \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)}{b n}\right) \left((2 i b d n+m+1) \left(-\exp \left(\frac{2 a m+a+b (2 m+1) \left(\log \left(c x^n\right)-n \log (x)\right)+b (m+1) n \log (x)}{b n}\right)\right) \, _2F_1\left(1,-\frac{i (m+1)}{2 b d n};1-\frac{i (m+1)}{2 b d n};-e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)+(m+1) \exp \left(\frac{a (2 i b d n+2 m+1)}{b n}+\frac{(2 i b d n+2 m+1) \left(\log \left(c x^n\right)-n \log (x)\right)}{n}+\log (x) (2 i b d n+m+1)\right) \, _2F_1\left(1,-\frac{i (m+2 i b d n+1)}{2 b d n};-\frac{i (m+4 i b d n+1)}{2 b d n};-e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)-i (2 i b d n+m+1) \tan \left(d \left(a+b \log \left(c x^n\right)\right)\right) \exp \left(\frac{2 a m+a+b (2 m+1) \left(\log \left(c x^n\right)-n \log (x)\right)+b (m+1) n \log (x)}{b n}\right)\right)}{(m+1) (2 i b d n+m+1)}\right)}{2 b^2 d^2 n^2}-\frac{(m+1) x (e x)^m \sin (b d n \log (x)) \sec \left(d \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right) \sec \left(d \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)+b d n \log (x)\right)}{2 b^2 d^2 n^2}-\frac{x (e x)^m \tan \left(d \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right)}{m+1}+\frac{x (e x)^m \sec ^2\left(d \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)+b d n \log (x)\right)}{2 b d n}","\frac{i (e x)^{m+1} \left(-2 b^2 d^2 n^2+m^2+2 m+1\right) \, _2F_1\left(1,-\frac{i (m+1)}{2 b d n};1-\frac{i (m+1)}{2 b d n};-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{b^2 d^2 e (m+1) n^2}-\frac{i e^{-2 i a d} (e x)^{m+1} \left(\frac{e^{2 i a d} (-2 i b d n+m+1)}{n}-\frac{e^{4 i a d} (2 i b d n+m+1) \left(c x^n\right)^{2 i b d}}{n}\right)}{2 b^2 d^2 e n \left(1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}-\frac{(e x)^{m+1} \left(1-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)^2}{2 b d e n \left(1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)^2}-\frac{(e x)^{m+1} (-b d n+i (m+1)) (2 i b d n+m+1)}{2 b^2 d^2 e (m+1) n^2}",1,"(x*(e*x)^m*Sec[b*d*n*Log[x] + d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]^2)/(2*b*d*n) - ((1 + m)*x*(e*x)^m*Sec[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]*Sec[b*d*n*Log[x] + d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]*Sin[b*d*n*Log[x]])/(2*b^2*d^2*n^2) - ((-1 - 2*m - m^2 + 2*b^2*d^2*n^2)*(e*x)^m*Sec[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]*((x^(1 + m)*Sec[d*(a + b*Log[c*x^n])]*Sin[b*d*n*Log[x]])/(1 + m) - (I*Cos[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]*(-(E^((a + 2*a*m + b*(1 + m)*n*Log[x] + b*(1 + 2*m)*(-(n*Log[x]) + Log[c*x^n]))/(b*n))*(1 + m + (2*I)*b*d*n)*Hypergeometric2F1[1, ((-1/2*I)*(1 + m))/(b*d*n), 1 - ((I/2)*(1 + m))/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))]) + E^((a*(1 + 2*m + (2*I)*b*d*n))/(b*n) + (1 + m + (2*I)*b*d*n)*Log[x] + ((1 + 2*m + (2*I)*b*d*n)*(-(n*Log[x]) + Log[c*x^n]))/n)*(1 + m)*Hypergeometric2F1[1, ((-1/2*I)*(1 + m + (2*I)*b*d*n))/(b*d*n), ((-1/2*I)*(1 + m + (4*I)*b*d*n))/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))] - I*E^((a + 2*a*m + b*(1 + m)*n*Log[x] + b*(1 + 2*m)*(-(n*Log[x]) + Log[c*x^n]))/(b*n))*(1 + m + (2*I)*b*d*n)*Tan[d*(a + b*Log[c*x^n])]))/(E^(((1 + 2*m)*(a + b*(-(n*Log[x]) + Log[c*x^n])))/(b*n))*(1 + m)*(1 + m + (2*I)*b*d*n))))/(2*b^2*d^2*n^2*x^m) - (x*(e*x)^m*Tan[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))])/(1 + m)","A",0
178,1,458,190,1.4104867,"\int \tan ^p\left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[Tan[d*(a + b*Log[c*x^n])]^p,x]","\frac{x (2 b d n-i) \left(-\frac{i \left(-1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{1+e^{2 i a d} \left(c x^n\right)^{2 i b d}}\right)^p F_1\left(-\frac{i}{2 b d n};-p,p;1-\frac{i}{2 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d},-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{-2 b d n p e^{2 i a d} \left(c x^n\right)^{2 i b d} F_1\left(1-\frac{i}{2 b d n};1-p,p;2-\frac{i}{2 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d},-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)-2 b d n p e^{2 i a d} \left(c x^n\right)^{2 i b d} F_1\left(1-\frac{i}{2 b d n};-p,p+1;2-\frac{i}{2 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d},-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)+(2 b d n-i) F_1\left(-\frac{i}{2 b d n};-p,p;1-\frac{i}{2 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d},-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}","x \left(1-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)^{-p} \left(\frac{i \left(1-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{1+e^{2 i a d} \left(c x^n\right)^{2 i b d}}\right)^p \left(1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)^p F_1\left(-\frac{i}{2 b d n};-p,p;1-\frac{i}{2 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d},-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)",1,"((-I + 2*b*d*n)*x*(((-I)*(-1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)))/(1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)))^p*AppellF1[(-1/2*I)/(b*d*n), -p, p, 1 - (I/2)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d), -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))])/(-2*b*d*E^((2*I)*a*d)*n*p*(c*x^n)^((2*I)*b*d)*AppellF1[1 - (I/2)/(b*d*n), 1 - p, p, 2 - (I/2)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d), -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))] - 2*b*d*E^((2*I)*a*d)*n*p*(c*x^n)^((2*I)*b*d)*AppellF1[1 - (I/2)/(b*d*n), -p, 1 + p, 2 - (I/2)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d), -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))] + (-I + 2*b*d*n)*AppellF1[(-1/2*I)/(b*d*n), -p, p, 1 - (I/2)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d), -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))])","B",0
179,1,205,210,1.1481591,"\int (e x)^m \tan ^p\left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[(e*x)^m*Tan[d*(a + b*Log[c*x^n])]^p,x]","\frac{x (e x)^m \left(1-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)^{-p} \left(-\frac{i \left(-1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{1+e^{2 i a d} \left(c x^n\right)^{2 i b d}}\right)^p \left(1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)^p F_1\left(-\frac{i (m+1)}{2 b d n};-p,p;1-\frac{i (m+1)}{2 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d},-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{m+1}","\frac{(e x)^{m+1} \left(1-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)^{-p} \left(\frac{i \left(1-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{1+e^{2 i a d} \left(c x^n\right)^{2 i b d}}\right)^p \left(1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)^p F_1\left(-\frac{i (m+1)}{2 b d n};-p,p;1-\frac{i (m+1)}{2 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d},-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{e (m+1)}",1,"(x*(e*x)^m*(((-I)*(-1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)))/(1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)))^p*(1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))^p*AppellF1[((-1/2*I)*(1 + m))/(b*d*n), -p, p, 1 - ((I/2)*(1 + m))/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d), -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))])/((1 + m)*(1 - E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))^p)","A",1
180,1,50,201,0.2530523,"\int \frac{\tan ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Tan[a + b*Log[c*x^n]]^(5/2)/x,x]","-\frac{2 \tan ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right) \left(\, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2\left(a+b \log \left(c x^n\right)\right)\right)-1\right)}{3 b n}","\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}\right)}{\sqrt{2} b n}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}+1\right)}{\sqrt{2} b n}+\frac{2 \tan ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}{3 b n}-\frac{\log \left(\tan \left(a+b \log \left(c x^n\right)\right)-\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}+1\right)}{2 \sqrt{2} b n}+\frac{\log \left(\tan \left(a+b \log \left(c x^n\right)\right)+\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}+1\right)}{2 \sqrt{2} b n}",1,"(-2*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Tan[a + b*Log[c*x^n]]^2])*Tan[a + b*Log[c*x^n]]^(3/2))/(3*b*n)","C",1
181,1,175,199,0.2420544,"\int \frac{\tan ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Tan[a + b*Log[c*x^n]]^(3/2)/x,x]","\frac{2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}+1\right)+\sqrt{2} \log \left(\tan \left(a+b \log \left(c x^n\right)\right)-\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}+1\right)-\sqrt{2} \log \left(\tan \left(a+b \log \left(c x^n\right)\right)+\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}+1\right)+8 \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}}{4 b n}","\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}\right)}{\sqrt{2} b n}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}+1\right)}{\sqrt{2} b n}+\frac{\log \left(\tan \left(a+b \log \left(c x^n\right)\right)-\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}+1\right)}{2 \sqrt{2} b n}-\frac{\log \left(\tan \left(a+b \log \left(c x^n\right)\right)+\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}+1\right)}{2 \sqrt{2} b n}+\frac{2 \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}}{b n}",1,"(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]] + Tan[a + b*Log[c*x^n]]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]] + Tan[a + b*Log[c*x^n]]] + 8*Sqrt[Tan[a + b*Log[c*x^n]]])/(4*b*n)","A",1
182,1,48,176,0.0968637,"\int \frac{\sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}}{x} \, dx","Integrate[Sqrt[Tan[a + b*Log[c*x^n]]]/x,x]","\frac{2 \tan ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2\left(a+b \log \left(c x^n\right)\right)\right)}{3 b n}","-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}\right)}{\sqrt{2} b n}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}+1\right)}{\sqrt{2} b n}+\frac{\log \left(\tan \left(a+b \log \left(c x^n\right)\right)-\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}+1\right)}{2 \sqrt{2} b n}-\frac{\log \left(\tan \left(a+b \log \left(c x^n\right)\right)+\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}+1\right)}{2 \sqrt{2} b n}",1,"(2*Hypergeometric2F1[3/4, 1, 7/4, -Tan[a + b*Log[c*x^n]]^2]*Tan[a + b*Log[c*x^n]]^(3/2))/(3*b*n)","C",1
183,1,142,176,0.1303345,"\int \frac{1}{x \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}} \, dx","Integrate[1/(x*Sqrt[Tan[a + b*Log[c*x^n]]]),x]","\frac{-2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}\right)+2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}+1\right)-\log \left(\tan \left(a+b \log \left(c x^n\right)\right)-\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}+1\right)+\log \left(\tan \left(a+b \log \left(c x^n\right)\right)+\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}+1\right)}{2 \sqrt{2} b n}","-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}\right)}{\sqrt{2} b n}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}+1\right)}{\sqrt{2} b n}-\frac{\log \left(\tan \left(a+b \log \left(c x^n\right)\right)-\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}+1\right)}{2 \sqrt{2} b n}+\frac{\log \left(\tan \left(a+b \log \left(c x^n\right)\right)+\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}+1\right)}{2 \sqrt{2} b n}",1,"(-2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]]] + 2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]]] - Log[1 - Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]] + Tan[a + b*Log[c*x^n]]] + Log[1 + Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]] + Tan[a + b*Log[c*x^n]]])/(2*Sqrt[2]*b*n)","A",1
184,1,46,199,0.1097839,"\int \frac{1}{x \tan ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[1/(x*Tan[a + b*Log[c*x^n]]^(3/2)),x]","-\frac{2 \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\tan ^2\left(a+b \log \left(c x^n\right)\right)\right)}{b n \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}}","\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}\right)}{\sqrt{2} b n}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}+1\right)}{\sqrt{2} b n}-\frac{\log \left(\tan \left(a+b \log \left(c x^n\right)\right)-\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}+1\right)}{2 \sqrt{2} b n}+\frac{\log \left(\tan \left(a+b \log \left(c x^n\right)\right)+\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}+1\right)}{2 \sqrt{2} b n}-\frac{2}{b n \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}}",1,"(-2*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[a + b*Log[c*x^n]]^2])/(b*n*Sqrt[Tan[a + b*Log[c*x^n]]])","C",1
185,1,48,201,0.2033826,"\int \frac{1}{x \tan ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[1/(x*Tan[a + b*Log[c*x^n]]^(5/2)),x]","-\frac{2 \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\tan ^2\left(a+b \log \left(c x^n\right)\right)\right)}{3 b n \tan ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}","\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}\right)}{\sqrt{2} b n}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}+1\right)}{\sqrt{2} b n}-\frac{2}{3 b n \tan ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}+\frac{\log \left(\tan \left(a+b \log \left(c x^n\right)\right)-\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}+1\right)}{2 \sqrt{2} b n}-\frac{\log \left(\tan \left(a+b \log \left(c x^n\right)\right)+\sqrt{2} \sqrt{\tan \left(a+b \log \left(c x^n\right)\right)}+1\right)}{2 \sqrt{2} b n}",1,"(-2*Hypergeometric2F1[-3/4, 1, 1/4, -Tan[a + b*Log[c*x^n]]^2])/(3*b*n*Tan[a + b*Log[c*x^n]]^(3/2))","C",1
186,1,137,49,0.0406869,"\int x^3 \cot (a+i \log (x)) \, dx","Integrate[x^3*Cot[a + I*Log[x]],x]","x^2 \sin (2 a)-i x^2 \cos (2 a)-\cos (4 a) \tan ^{-1}\left(\frac{\left(x^2-1\right) \cos (a)}{x^2 (-\sin (a))-\sin (a)}\right)-i \sin (4 a) \tan ^{-1}\left(\frac{\left(x^2-1\right) \cos (a)}{x^2 (-\sin (a))-\sin (a)}\right)-\frac{1}{2} i \cos (4 a) \log \left(-2 x^2 \cos (2 a)+x^4+1\right)+\frac{1}{2} \sin (4 a) \log \left(-2 x^2 \cos (2 a)+x^4+1\right)-\frac{i x^4}{4}","-i e^{2 i a} x^2-i e^{4 i a} \log \left(-x^2+e^{2 i a}\right)-\frac{i x^4}{4}",1,"(-1/4*I)*x^4 - I*x^2*Cos[2*a] - ArcTan[((-1 + x^2)*Cos[a])/(-Sin[a] - x^2*Sin[a])]*Cos[4*a] - (I/2)*Cos[4*a]*Log[1 + x^4 - 2*x^2*Cos[2*a]] + x^2*Sin[2*a] - I*ArcTan[((-1 + x^2)*Cos[a])/(-Sin[a] - x^2*Sin[a])]*Sin[4*a] + (Log[1 + x^4 - 2*x^2*Cos[2*a]]*Sin[4*a])/2","B",1
187,1,66,43,0.0179161,"\int x^2 \cot (a+i \log (x)) \, dx","Integrate[x^2*Cot[a + I*Log[x]],x]","2 x \sin (2 a)-2 i x \cos (2 a)+2 i \cos (3 a) \tanh ^{-1}(x \cos (a)-i x \sin (a))-2 \sin (3 a) \tanh ^{-1}(x \cos (a)-i x \sin (a))-\frac{i x^3}{3}","-2 i e^{2 i a} x+2 i e^{3 i a} \tanh ^{-1}\left(e^{-i a} x\right)-\frac{i x^3}{3}",1,"(-1/3*I)*x^3 - (2*I)*x*Cos[2*a] + (2*I)*ArcTanh[x*Cos[a] - I*x*Sin[a]]*Cos[3*a] + 2*x*Sin[2*a] - 2*ArcTanh[x*Cos[a] - I*x*Sin[a]]*Sin[3*a]","A",1
188,1,118,35,0.0225408,"\int x \cot (a+i \log (x)) \, dx","Integrate[x*Cot[a + I*Log[x]],x]","-\cos (2 a) \tan ^{-1}\left(\frac{\left(x^2-1\right) \cos (a)}{x^2 (-\sin (a))-\sin (a)}\right)-i \sin (2 a) \tan ^{-1}\left(\frac{\left(x^2-1\right) \cos (a)}{x^2 (-\sin (a))-\sin (a)}\right)-\frac{1}{2} i \cos (2 a) \log \left(-2 x^2 \cos (2 a)+x^4+1\right)+\frac{1}{2} \sin (2 a) \log \left(-2 x^2 \cos (2 a)+x^4+1\right)-\frac{i x^2}{2}","-i e^{2 i a} \log \left(-x^2+e^{2 i a}\right)-\frac{i x^2}{2}",1,"(-1/2*I)*x^2 - ArcTan[((-1 + x^2)*Cos[a])/(-Sin[a] - x^2*Sin[a])]*Cos[2*a] - (I/2)*Cos[2*a]*Log[1 + x^4 - 2*x^2*Cos[2*a]] - I*ArcTan[((-1 + x^2)*Cos[a])/(-Sin[a] - x^2*Sin[a])]*Sin[2*a] + (Log[1 + x^4 - 2*x^2*Cos[2*a]]*Sin[2*a])/2","B",1
189,1,42,27,0.0090693,"\int \cot (a+i \log (x)) \, dx","Integrate[Cot[a + I*Log[x]],x]","2 i \cos (a) \tanh ^{-1}(x \cos (a)-i x \sin (a))-2 \sin (a) \tanh ^{-1}(x \cos (a)-i x \sin (a))-i x","2 i e^{i a} \tanh ^{-1}\left(e^{-i a} x\right)-i x",1,"(-I)*x + (2*I)*ArcTanh[x*Cos[a] - I*x*Sin[a]]*Cos[a] - 2*ArcTanh[x*Cos[a] - I*x*Sin[a]]*Sin[a]","A",1
190,1,25,14,0.0256843,"\int \frac{\cot (a+i \log (x))}{x} \, dx","Integrate[Cot[a + I*Log[x]]/x,x]","-i (\log (\tan (a+i \log (x)))+\log (\cos (a+i \log (x))))","-i \log (\sin (a+i \log (x)))",1,"(-I)*(Log[Cos[a + I*Log[x]]] + Log[Tan[a + I*Log[x]]])","A",1
191,1,44,29,0.0211508,"\int \frac{\cot (a+i \log (x))}{x^2} \, dx","Integrate[Cot[a + I*Log[x]]/x^2,x]","2 i \cos (a) \tanh ^{-1}(x \cos (a)-i x \sin (a))+2 \sin (a) \tanh ^{-1}(x \cos (a)-i x \sin (a))-\frac{i}{x}","2 i e^{-i a} \tanh ^{-1}\left(e^{-i a} x\right)-\frac{i}{x}",1,"(-I)/x + (2*I)*ArcTanh[x*Cos[a] - I*x*Sin[a]]*Cos[a] + 2*ArcTanh[x*Cos[a] - I*x*Sin[a]]*Sin[a]","A",1
192,1,136,36,0.0302094,"\int \frac{\cot (a+i \log (x))}{x^3} \, dx","Integrate[Cot[a + I*Log[x]]/x^3,x]","\cos (2 a) \left(-\tan ^{-1}\left(\frac{\left(x^2-1\right) \cos (a)}{x^2 (-\sin (a))-\sin (a)}\right)\right)+i \sin (2 a) \tan ^{-1}\left(\frac{\left(x^2-1\right) \cos (a)}{x^2 (-\sin (a))-\sin (a)}\right)-\frac{1}{2} i \cos (2 a) \log \left(-2 x^2 \cos (2 a)+x^4+1\right)-\frac{1}{2} \sin (2 a) \log \left(-2 x^2 \cos (2 a)+x^4+1\right)+2 \sin (2 a) \log (x)+2 i \cos (2 a) \log (x)-\frac{i}{2 x^2}","-i e^{-2 i a} \log \left(1-\frac{e^{2 i a}}{x^2}\right)-\frac{i}{2 x^2}",1,"(-1/2*I)/x^2 - ArcTan[((-1 + x^2)*Cos[a])/(-Sin[a] - x^2*Sin[a])]*Cos[2*a] + (2*I)*Cos[2*a]*Log[x] - (I/2)*Cos[2*a]*Log[1 + x^4 - 2*x^2*Cos[2*a]] + I*ArcTan[((-1 + x^2)*Cos[a])/(-Sin[a] - x^2*Sin[a])]*Sin[2*a] + 2*Log[x]*Sin[2*a] - (Log[1 + x^4 - 2*x^2*Cos[2*a]]*Sin[2*a])/2","B",1
193,1,70,45,0.021249,"\int \frac{\cot (a+i \log (x))}{x^4} \, dx","Integrate[Cot[a + I*Log[x]]/x^4,x]","-\frac{2 \sin (2 a)}{x}-\frac{2 i \cos (2 a)}{x}+2 i \cos (3 a) \tanh ^{-1}(x \cos (a)-i x \sin (a))+2 \sin (3 a) \tanh ^{-1}(x \cos (a)-i x \sin (a))-\frac{i}{3 x^3}","-\frac{2 i e^{-2 i a}}{x}+2 i e^{-3 i a} \tanh ^{-1}\left(e^{-i a} x\right)-\frac{i}{3 x^3}",1,"(-1/3*I)/x^3 - ((2*I)*Cos[2*a])/x + (2*I)*ArcTanh[x*Cos[a] - I*x*Sin[a]]*Cos[3*a] - (2*Sin[2*a])/x + 2*ArcTanh[x*Cos[a] - I*x*Sin[a]]*Sin[3*a]","A",1
194,1,162,67,0.1792414,"\int x^3 \cot ^2(a+i \log (x)) \, dx","Integrate[x^3*Cot[a + I*Log[x]]^2,x]","-2 i x^2 \sin (2 a)-2 x^2 \cos (2 a)+\frac{2 \cos (5 a)+2 i \sin (5 a)}{\left(x^2-1\right) \cos (a)-i \left(x^2+1\right) \sin (a)}+4 i \cos (4 a) \tan ^{-1}\left(\frac{\cot (a)-x^2 \cot (a)}{x^2+1}\right)-4 \sin (4 a) \tan ^{-1}\left(\frac{\cot (a)-x^2 \cot (a)}{x^2+1}\right)-2 \cos (4 a) \log \left(-2 x^2 \cos (2 a)+x^4+1\right)-2 i \sin (4 a) \log \left(-2 x^2 \cos (2 a)+x^4+1\right)-\frac{x^4}{4}","-2 e^{2 i a} x^2-\frac{2 e^{6 i a}}{-x^2+e^{2 i a}}-4 e^{4 i a} \log \left(-x^2+e^{2 i a}\right)-\frac{x^4}{4}",1,"-1/4*x^4 - 2*x^2*Cos[2*a] + (4*I)*ArcTan[(Cot[a] - x^2*Cot[a])/(1 + x^2)]*Cos[4*a] - 2*Cos[4*a]*Log[1 + x^4 - 2*x^2*Cos[2*a]] - (2*I)*x^2*Sin[2*a] - 4*ArcTan[(Cot[a] - x^2*Cot[a])/(1 + x^2)]*Sin[4*a] - (2*I)*Log[1 + x^4 - 2*x^2*Cos[2*a]]*Sin[4*a] + (2*Cos[5*a] + (2*I)*Sin[5*a])/((-1 + x^2)*Cos[a] - I*(1 + x^2)*Sin[a])","B",1
195,1,100,64,0.1252987,"\int x^2 \cot ^2(a+i \log (x)) \, dx","Integrate[x^2*Cot[a + I*Log[x]]^2,x]","\frac{2 x (\cos (3 a)+i \sin (3 a))}{\left(x^2-1\right) \cos (a)-i \left(x^2+1\right) \sin (a)}-4 i x \sin (2 a)-4 x \cos (2 a)+6 \cos (3 a) \tanh ^{-1}(x (\cos (a)-i \sin (a)))+6 i \sin (3 a) \tanh ^{-1}(x (\cos (a)-i \sin (a)))-\frac{x^3}{3}","-\frac{2 e^{2 i a} x^3}{-x^2+e^{2 i a}}-6 e^{2 i a} x+6 e^{3 i a} \tanh ^{-1}\left(e^{-i a} x\right)-\frac{x^3}{3}",1,"-1/3*x^3 - 4*x*Cos[2*a] + 6*ArcTanh[x*(Cos[a] - I*Sin[a])]*Cos[3*a] - (4*I)*x*Sin[2*a] + (2*x*(Cos[3*a] + I*Sin[3*a]))/((-1 + x^2)*Cos[a] - I*(1 + x^2)*Sin[a]) + (6*I)*ArcTanh[x*(Cos[a] - I*Sin[a])]*Sin[3*a]","A",1
196,1,142,55,0.1286369,"\int x \cot ^2(a+i \log (x)) \, dx","Integrate[x*Cot[a + I*Log[x]]^2,x]","\frac{2 \cos (3 a)+2 i \sin (3 a)}{\left(x^2-1\right) \cos (a)-i \left(x^2+1\right) \sin (a)}+2 i \cos (2 a) \tan ^{-1}\left(\frac{\cot (a)-x^2 \cot (a)}{x^2+1}\right)-4 \sin (a) \cos (a) \tan ^{-1}\left(\frac{\cot (a)-x^2 \cot (a)}{x^2+1}\right)-\cos (2 a) \log \left(-2 x^2 \cos (2 a)+x^4+1\right)-i \sin (2 a) \log \left(-2 x^2 \cos (2 a)+x^4+1\right)-\frac{x^2}{2}","-\frac{2 e^{4 i a}}{-x^2+e^{2 i a}}-2 e^{2 i a} \log \left(-x^2+e^{2 i a}\right)-\frac{x^2}{2}",1,"-1/2*x^2 + (2*I)*ArcTan[(Cot[a] - x^2*Cot[a])/(1 + x^2)]*Cos[2*a] - Cos[2*a]*Log[1 + x^4 - 2*x^2*Cos[2*a]] - 4*ArcTan[(Cot[a] - x^2*Cot[a])/(1 + x^2)]*Cos[a]*Sin[a] - I*Log[1 + x^4 - 2*x^2*Cos[2*a]]*Sin[2*a] + (2*Cos[3*a] + (2*I)*Sin[3*a])/((-1 + x^2)*Cos[a] - I*(1 + x^2)*Sin[a])","B",1
197,1,70,48,0.0823359,"\int \cot ^2(a+i \log (x)) \, dx","Integrate[Cot[a + I*Log[x]]^2,x]","\frac{-x \left(x^2-3\right) \cos (a)+i x \left(x^2+3\right) \sin (a)}{\left(x^2-1\right) \cos (a)-i \left(x^2+1\right) \sin (a)}+2 (\cos (a)+i \sin (a)) \tanh ^{-1}(x (\cos (a)-i \sin (a)))","-\frac{2 e^{2 i a} x}{-x^2+e^{2 i a}}+2 e^{i a} \tanh ^{-1}\left(e^{-i a} x\right)-x",1,"2*ArcTanh[x*(Cos[a] - I*Sin[a])]*(Cos[a] + I*Sin[a]) + (-(x*(-3 + x^2)*Cos[a]) + I*x*(3 + x^2)*Sin[a])/((-1 + x^2)*Cos[a] - I*(1 + x^2)*Sin[a])","A",1
198,1,34,18,0.0492745,"\int \frac{\cot ^2(a+i \log (x))}{x} \, dx","Integrate[Cot[a + I*Log[x]]^2/x,x]","i \cot (a+i \log (x)) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(a+i \log (x))\right)","-\log (x)+i \cot (a+i \log (x))",1,"I*Cot[a + I*Log[x]]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[a + I*Log[x]]^2]","C",1
199,1,72,64,0.1219437,"\int \frac{\cot ^2(a+i \log (x))}{x^2} \, dx","Integrate[Cot[a + I*Log[x]]^2/x^2,x]","\frac{2 x (\cos (a)-i \sin (a))}{\left(x^2-1\right) \cos (a)-i \left(x^2+1\right) \sin (a)}-2 \cos (a) \tanh ^{-1}(x (\cos (a)-i \sin (a)))+2 i \sin (a) \tanh ^{-1}(x (\cos (a)-i \sin (a)))+\frac{1}{x}","-\frac{3 x}{-x^2+e^{2 i a}}+\frac{e^{2 i a}}{x \left(-x^2+e^{2 i a}\right)}-2 e^{-i a} \tanh ^{-1}\left(e^{-i a} x\right)",1,"x^(-1) - 2*ArcTanh[x*(Cos[a] - I*Sin[a])]*Cos[a] + (2*I)*ArcTanh[x*(Cos[a] - I*Sin[a])]*Sin[a] + (2*x*(Cos[a] - I*Sin[a]))/((-1 + x^2)*Cos[a] - I*(1 + x^2)*Sin[a])","A",1
200,1,153,57,0.2324061,"\int \frac{\cot ^2(a+i \log (x))}{x^3} \, dx","Integrate[Cot[a + I*Log[x]]^2/x^3,x]","\frac{2 \cos (a)}{\left(x^2-1\right) \cos (a)-i \left(x^2+1\right) \sin (a)}+\frac{2 \sin (a)}{\left(x^2+1\right) \sin (a)+i \left(x^2-1\right) \cos (a)}+(-4 \sin (a) \cos (a)-2 i \cos (2 a)) \tan ^{-1}\left(\frac{\cot (a)-x^2 \cot (a)}{x^2+1}\right)+\cos (2 a) \left(\log \left(-2 x^2 \cos (2 a)+x^4+1\right)-4 \log (x)\right)-i \sin (2 a) \log \left(-2 x^2 \cos (2 a)+x^4+1\right)+4 i \sin (2 a) \log (x)+\frac{1}{2 x^2}","\frac{2 e^{-2 i a}}{1-\frac{e^{2 i a}}{x^2}}+2 e^{-2 i a} \log \left(1-\frac{e^{2 i a}}{x^2}\right)+\frac{1}{2 x^2}",1,"1/(2*x^2) + Cos[2*a]*(-4*Log[x] + Log[1 + x^4 - 2*x^2*Cos[2*a]]) + (2*Cos[a])/((-1 + x^2)*Cos[a] - I*(1 + x^2)*Sin[a]) + (2*Sin[a])/(I*(-1 + x^2)*Cos[a] + (1 + x^2)*Sin[a]) + ArcTan[(Cot[a] - x^2*Cot[a])/(1 + x^2)]*((-2*I)*Cos[2*a] - 4*Cos[a]*Sin[a]) + (4*I)*Log[x]*Sin[2*a] - I*Log[1 + x^4 - 2*x^2*Cos[2*a]]*Sin[2*a]","B",1
201,1,103,70,0.2556117,"\int (e x)^m \cot (a+i \log (x)) \, dx","Integrate[(e*x)^m*Cot[a + I*Log[x]],x]","i x (e x)^m \left(\frac{x^2 (\cos (a)-i \sin (a))^2 \, _2F_1\left(1,\frac{m+3}{2};\frac{m+5}{2};x^2 (\cos (2 a)-i \sin (2 a))\right)}{m+3}+\frac{\, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};x^2 (\cos (2 a)-i \sin (2 a))\right)}{m+1}\right)","\frac{i (e x)^{m+1}}{e (m+1)}-\frac{2 i (e x)^{m+1} \, _2F_1\left(1,\frac{1}{2} (-m-1);\frac{1-m}{2};\frac{e^{2 i a}}{x^2}\right)}{e (m+1)}",1,"I*x*(e*x)^m*(Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, x^2*(Cos[2*a] - I*Sin[2*a])]/(1 + m) + (x^2*Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, x^2*(Cos[2*a] - I*Sin[2*a])]*(Cos[a] - I*Sin[a])^2)/(3 + m))","A",1
202,1,84,77,0.1665617,"\int (e x)^m \cot ^2(a+i \log (x)) \, dx","Integrate[(e*x)^m*Cot[a + I*Log[x]]^2,x]","\frac{x (e x)^m \left(4 \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};x^2 (\cos (2 a)-i \sin (2 a))\right)-4 \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};x^2 (\cos (2 a)-i \sin (2 a))\right)-1\right)}{m+1}","-2 x (e x)^m \, _2F_1\left(1,\frac{1}{2} (-m-1);\frac{1-m}{2};\frac{e^{2 i a}}{x^2}\right)+\frac{2 x (e x)^m}{1-\frac{e^{2 i a}}{x^2}}-\frac{x (e x)^m}{m+1}",1,"(x*(e*x)^m*(-1 + 4*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, x^2*(Cos[2*a] - I*Sin[2*a])] - 4*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, x^2*(Cos[2*a] - I*Sin[2*a])]))/(1 + m)","A",1
203,1,122,169,0.2263689,"\int (e x)^m \cot ^3(a+i \log (x)) \, dx","Integrate[(e*x)^m*Cot[a + I*Log[x]]^3,x]","-\frac{i x (e x)^m \left(6 \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};x^2 (\cos (2 a)-i \sin (2 a))\right)-12 \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};x^2 (\cos (2 a)-i \sin (2 a))\right)+8 \, _2F_1\left(3,\frac{m+1}{2};\frac{m+3}{2};x^2 (\cos (2 a)-i \sin (2 a))\right)-1\right)}{m+1}","\frac{i \left(m^2+2 m+3\right) x (e x)^m \, _2F_1\left(1,\frac{1}{2} (-m-1);\frac{1-m}{2};\frac{e^{2 i a}}{x^2}\right)}{m+1}-\frac{i x \left(1+\frac{e^{2 i a}}{x^2}\right)^2 (e x)^m}{2 \left(1-\frac{e^{2 i a}}{x^2}\right)^2}-\frac{i x \left(-\frac{e^{2 i a} (1-m)}{x^2}+m+3\right) (e x)^m}{2 \left(1-\frac{e^{2 i a}}{x^2}\right)}+\frac{i (1-m) m x (e x)^m}{2 (m+1)}",1,"((-I)*x*(e*x)^m*(-1 + 6*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, x^2*(Cos[2*a] - I*Sin[2*a])] - 12*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, x^2*(Cos[2*a] - I*Sin[2*a])] + 8*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, x^2*(Cos[2*a] - I*Sin[2*a])]))/(1 + m)","A",1
204,1,330,142,0.611788,"\int \cot ^p(a+b \log (x)) \, dx","Integrate[Cot[a + b*Log[x]]^p,x]","\frac{(2 b-i) x \left(\frac{i \left(1+e^{2 i a} x^{2 i b}\right)}{-1+e^{2 i a} x^{2 i b}}\right)^p F_1\left(-\frac{i}{2 b};p,-p;1-\frac{i}{2 b};e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right)}{2 e^{2 i a} b p x^{2 i b} F_1\left(1-\frac{i}{2 b};p,1-p;2-\frac{i}{2 b};e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right)+2 e^{2 i a} b p x^{2 i b} F_1\left(1-\frac{i}{2 b};p+1,-p;2-\frac{i}{2 b};e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right)+(2 b-i) F_1\left(-\frac{i}{2 b};p,-p;1-\frac{i}{2 b};e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right)}","x \left(1-e^{2 i a} x^{2 i b}\right)^p \left(1+e^{2 i a} x^{2 i b}\right)^{-p} \left(-\frac{i \left(1+e^{2 i a} x^{2 i b}\right)}{1-e^{2 i a} x^{2 i b}}\right)^p F_1\left(-\frac{i}{2 b};p,-p;1-\frac{i}{2 b};e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right)",1,"((-I + 2*b)*x*((I*(1 + E^((2*I)*a)*x^((2*I)*b)))/(-1 + E^((2*I)*a)*x^((2*I)*b)))^p*AppellF1[(-1/2*I)/b, p, -p, 1 - (I/2)/b, E^((2*I)*a)*x^((2*I)*b), -(E^((2*I)*a)*x^((2*I)*b))])/(2*b*E^((2*I)*a)*p*x^((2*I)*b)*AppellF1[1 - (I/2)/b, p, 1 - p, 2 - (I/2)/b, E^((2*I)*a)*x^((2*I)*b), -(E^((2*I)*a)*x^((2*I)*b))] + 2*b*E^((2*I)*a)*p*x^((2*I)*b)*AppellF1[1 - (I/2)/b, 1 + p, -p, 2 - (I/2)/b, E^((2*I)*a)*x^((2*I)*b), -(E^((2*I)*a)*x^((2*I)*b))] + (-I + 2*b)*AppellF1[(-1/2*I)/b, p, -p, 1 - (I/2)/b, E^((2*I)*a)*x^((2*I)*b), -(E^((2*I)*a)*x^((2*I)*b))])","B",0
205,1,157,162,0.6501457,"\int (e x)^m \cot ^p(a+b \log (x)) \, dx","Integrate[(e*x)^m*Cot[a + b*Log[x]]^p,x]","\frac{x (e x)^m \left(1-e^{2 i a} x^{2 i b}\right)^p \left(1+e^{2 i a} x^{2 i b}\right)^{-p} \left(\frac{i \left(1+e^{2 i a} x^{2 i b}\right)}{-1+e^{2 i a} x^{2 i b}}\right)^p F_1\left(-\frac{i (m+1)}{2 b};p,-p;1-\frac{i (m+1)}{2 b};e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right)}{m+1}","\frac{(e x)^{m+1} \left(1-e^{2 i a} x^{2 i b}\right)^p \left(1+e^{2 i a} x^{2 i b}\right)^{-p} \left(-\frac{i \left(1+e^{2 i a} x^{2 i b}\right)}{1-e^{2 i a} x^{2 i b}}\right)^p F_1\left(-\frac{i (m+1)}{2 b};p,-p;1-\frac{i (m+1)}{2 b};e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right)}{e (m+1)}",1,"(x*(e*x)^m*(1 - E^((2*I)*a)*x^((2*I)*b))^p*((I*(1 + E^((2*I)*a)*x^((2*I)*b)))/(-1 + E^((2*I)*a)*x^((2*I)*b)))^p*AppellF1[((-1/2*I)*(1 + m))/b, p, -p, 1 - ((I/2)*(1 + m))/b, E^((2*I)*a)*x^((2*I)*b), -(E^((2*I)*a)*x^((2*I)*b))])/((1 + m)*(1 + E^((2*I)*a)*x^((2*I)*b))^p)","A",1
206,1,238,120,0.4836779,"\int \cot ^p(a+\log (x)) \, dx","Integrate[Cot[a + Log[x]]^p,x]","\frac{(2-i) x \left(\frac{i \left(1+e^{2 i a} x^{2 i}\right)}{-1+e^{2 i a} x^{2 i}}\right)^p F_1\left(-\frac{i}{2};p,-p;1-\frac{i}{2};e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right)}{(2-i) F_1\left(-\frac{i}{2};p,-p;1-\frac{i}{2};e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right)+2 e^{2 i a} p x^{2 i} \left(F_1\left(1-\frac{i}{2};p,1-p;2-\frac{i}{2};e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right)+F_1\left(1-\frac{i}{2};p+1,-p;2-\frac{i}{2};e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right)\right)}","x \left(1-e^{2 i a} x^{2 i}\right)^p \left(1+e^{2 i a} x^{2 i}\right)^{-p} \left(-\frac{i \left(1+e^{2 i a} x^{2 i}\right)}{1-e^{2 i a} x^{2 i}}\right)^p F_1\left(-\frac{i}{2};p,-p;1-\frac{i}{2};e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right)",1,"((2 - I)*((I*(1 + E^((2*I)*a)*x^(2*I)))/(-1 + E^((2*I)*a)*x^(2*I)))^p*x*AppellF1[-1/2*I, p, -p, 1 - I/2, E^((2*I)*a)*x^(2*I), -(E^((2*I)*a)*x^(2*I))])/((2 - I)*AppellF1[-1/2*I, p, -p, 1 - I/2, E^((2*I)*a)*x^(2*I), -(E^((2*I)*a)*x^(2*I))] + 2*E^((2*I)*a)*p*x^(2*I)*(AppellF1[1 - I/2, p, 1 - p, 2 - I/2, E^((2*I)*a)*x^(2*I), -(E^((2*I)*a)*x^(2*I))] + AppellF1[1 - I/2, 1 + p, -p, 2 - I/2, E^((2*I)*a)*x^(2*I), -(E^((2*I)*a)*x^(2*I))]))","A",0
207,1,238,120,0.4679554,"\int \cot ^p(a+2 \log (x)) \, dx","Integrate[Cot[a + 2*Log[x]]^p,x]","\frac{(4-i) x \left(\frac{i \left(1+e^{2 i a} x^{4 i}\right)}{-1+e^{2 i a} x^{4 i}}\right)^p F_1\left(-\frac{i}{4};p,-p;1-\frac{i}{4};e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right)}{(4-i) F_1\left(-\frac{i}{4};p,-p;1-\frac{i}{4};e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right)+4 e^{2 i a} p x^{4 i} \left(F_1\left(1-\frac{i}{4};p,1-p;2-\frac{i}{4};e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right)+F_1\left(1-\frac{i}{4};p+1,-p;2-\frac{i}{4};e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right)\right)}","x \left(1-e^{2 i a} x^{4 i}\right)^p \left(1+e^{2 i a} x^{4 i}\right)^{-p} \left(-\frac{i \left(1+e^{2 i a} x^{4 i}\right)}{1-e^{2 i a} x^{4 i}}\right)^p F_1\left(-\frac{i}{4};p,-p;1-\frac{i}{4};e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right)",1,"((4 - I)*((I*(1 + E^((2*I)*a)*x^(4*I)))/(-1 + E^((2*I)*a)*x^(4*I)))^p*x*AppellF1[-1/4*I, p, -p, 1 - I/4, E^((2*I)*a)*x^(4*I), -(E^((2*I)*a)*x^(4*I))])/((4 - I)*AppellF1[-1/4*I, p, -p, 1 - I/4, E^((2*I)*a)*x^(4*I), -(E^((2*I)*a)*x^(4*I))] + 4*E^((2*I)*a)*p*x^(4*I)*(AppellF1[1 - I/4, p, 1 - p, 2 - I/4, E^((2*I)*a)*x^(4*I), -(E^((2*I)*a)*x^(4*I))] + AppellF1[1 - I/4, 1 + p, -p, 2 - I/4, E^((2*I)*a)*x^(4*I), -(E^((2*I)*a)*x^(4*I))]))","A",0
208,1,238,120,0.469223,"\int \cot ^p(a+3 \log (x)) \, dx","Integrate[Cot[a + 3*Log[x]]^p,x]","\frac{(6-i) x \left(\frac{i \left(1+e^{2 i a} x^{6 i}\right)}{-1+e^{2 i a} x^{6 i}}\right)^p F_1\left(-\frac{i}{6};p,-p;1-\frac{i}{6};e^{2 i a} x^{6 i},-e^{2 i a} x^{6 i}\right)}{(6-i) F_1\left(-\frac{i}{6};p,-p;1-\frac{i}{6};e^{2 i a} x^{6 i},-e^{2 i a} x^{6 i}\right)+6 e^{2 i a} p x^{6 i} \left(F_1\left(1-\frac{i}{6};p,1-p;2-\frac{i}{6};e^{2 i a} x^{6 i},-e^{2 i a} x^{6 i}\right)+F_1\left(1-\frac{i}{6};p+1,-p;2-\frac{i}{6};e^{2 i a} x^{6 i},-e^{2 i a} x^{6 i}\right)\right)}","x \left(1-e^{2 i a} x^{6 i}\right)^p \left(1+e^{2 i a} x^{6 i}\right)^{-p} \left(-\frac{i \left(1+e^{2 i a} x^{6 i}\right)}{1-e^{2 i a} x^{6 i}}\right)^p F_1\left(-\frac{i}{6};p,-p;1-\frac{i}{6};e^{2 i a} x^{6 i},-e^{2 i a} x^{6 i}\right)",1,"((6 - I)*((I*(1 + E^((2*I)*a)*x^(6*I)))/(-1 + E^((2*I)*a)*x^(6*I)))^p*x*AppellF1[-1/6*I, p, -p, 1 - I/6, E^((2*I)*a)*x^(6*I), -(E^((2*I)*a)*x^(6*I))])/((6 - I)*AppellF1[-1/6*I, p, -p, 1 - I/6, E^((2*I)*a)*x^(6*I), -(E^((2*I)*a)*x^(6*I))] + 6*E^((2*I)*a)*p*x^(6*I)*(AppellF1[1 - I/6, p, 1 - p, 2 - I/6, E^((2*I)*a)*x^(6*I), -(E^((2*I)*a)*x^(6*I))] + AppellF1[1 - I/6, 1 + p, -p, 2 - I/6, E^((2*I)*a)*x^(6*I), -(E^((2*I)*a)*x^(6*I))]))","A",0
209,1,220,70,5.300034,"\int x^3 \cot \left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[x^3*Cot[d*(a + b*Log[c*x^n])],x]","-\frac{x^4 \left(2 e^{2 i d \left(a+b \log \left(c x^n\right)\right)} \, _2F_1\left(1,1-\frac{2 i}{b d n};2-\frac{2 i}{b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)+(b d n-2 i) \left(i \, _2F_1\left(1,-\frac{2 i}{b d n};1-\frac{2 i}{b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)+\cot \left(d \left(a+b \log \left(c x^n\right)\right)\right)-\cot \left(d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)+\sin (b d n \log (x)) \csc \left(d \left(a+b \log \left(c x^n\right)\right)\right) \csc \left(d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)\right)\right)}{4 b d n-8 i}","\frac{i x^4}{4}-\frac{1}{2} i x^4 \, _2F_1\left(1,-\frac{2 i}{b d n};1-\frac{2 i}{b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)",1,"-((x^4*(2*E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 - (2*I)/(b*d*n), 2 - (2*I)/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] + (-2*I + b*d*n)*(Cot[d*(a + b*Log[c*x^n])] - Cot[d*(a - b*n*Log[x] + b*Log[c*x^n])] + I*Hypergeometric2F1[1, (-2*I)/(b*d*n), 1 - (2*I)/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] + Csc[d*(a + b*Log[c*x^n])]*Csc[d*(a - b*n*Log[x] + b*Log[c*x^n])]*Sin[b*d*n*Log[x]])))/(-8*I + 4*b*d*n))","B",1
210,1,229,74,5.5770351,"\int x^2 \cot \left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[x^2*Cot[d*(a + b*Log[c*x^n])],x]","-\frac{x^3 \left(3 e^{2 i d \left(a+b \log \left(c x^n\right)\right)} \, _2F_1\left(1,1-\frac{3 i}{2 b d n};2-\frac{3 i}{2 b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)+(2 b d n-3 i) \left(i \, _2F_1\left(1,-\frac{3 i}{2 b d n};1-\frac{3 i}{2 b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)+\cot \left(d \left(a+b \log \left(c x^n\right)\right)\right)-\cot \left(d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)+\sin (b d n \log (x)) \csc \left(d \left(a+b \log \left(c x^n\right)\right)\right) \csc \left(d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)\right)\right)}{6 b d n-9 i}","\frac{i x^3}{3}-\frac{2}{3} i x^3 \, _2F_1\left(1,-\frac{3 i}{2 b d n};1-\frac{3 i}{2 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)",1,"-((x^3*(3*E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 - ((3*I)/2)/(b*d*n), 2 - ((3*I)/2)/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] + (-3*I + 2*b*d*n)*(Cot[d*(a + b*Log[c*x^n])] - Cot[d*(a - b*n*Log[x] + b*Log[c*x^n])] + I*Hypergeometric2F1[1, ((-3*I)/2)/(b*d*n), 1 - ((3*I)/2)/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] + Csc[d*(a + b*Log[c*x^n])]*Csc[d*(a - b*n*Log[x] + b*Log[c*x^n])]*Sin[b*d*n*Log[x]])))/(-9*I + 6*b*d*n))","B",1
211,1,219,68,5.5231031,"\int x \cot \left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[x*Cot[d*(a + b*Log[c*x^n])],x]","-\frac{x^2 \left(e^{2 i d \left(a+b \log \left(c x^n\right)\right)} \, _2F_1\left(1,1-\frac{i}{b d n};2-\frac{i}{b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)+(b d n-i) \left(i \, _2F_1\left(1,-\frac{i}{b d n};1-\frac{i}{b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)+\cot \left(d \left(a+b \log \left(c x^n\right)\right)\right)-\cot \left(d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)+\sin (b d n \log (x)) \csc \left(d \left(a+b \log \left(c x^n\right)\right)\right) \csc \left(d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)\right)\right)}{2 b d n-2 i}","\frac{i x^2}{2}-i x^2 \, _2F_1\left(1,-\frac{i}{b d n};1-\frac{i}{b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)",1,"-((x^2*(E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 - I/(b*d*n), 2 - I/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] + (-I + b*d*n)*(Cot[d*(a + b*Log[c*x^n])] - Cot[d*(a - b*n*Log[x] + b*Log[c*x^n])] + I*Hypergeometric2F1[1, (-I)/(b*d*n), 1 - I/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] + Csc[d*(a + b*Log[c*x^n])]*Csc[d*(a - b*n*Log[x] + b*Log[c*x^n])]*Sin[b*d*n*Log[x]])))/(-2*I + 2*b*d*n))","B",1
212,1,141,66,10.3185606,"\int \cot \left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[Cot[d*(a + b*Log[c*x^n])],x]","x \left(-\frac{e^{2 i d \left(a+b \log \left(c x^n\right)\right)} \, _2F_1\left(1,1-\frac{i}{2 b d n};2-\frac{i}{2 b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)}{2 b d n-i}-i \, _2F_1\left(1,-\frac{i}{2 b d n};1-\frac{i}{2 b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)\right)","i x-2 i x \, _2F_1\left(1,-\frac{i}{2 b d n};1-\frac{i}{2 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)",1,"x*(-((E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 - (I/2)/(b*d*n), 2 - (I/2)/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))])/(-I + 2*b*d*n)) - I*Hypergeometric2F1[1, (-1/2*I)/(b*d*n), 1 - (I/2)/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))])","B",1
213,1,40,25,0.0621781,"\int \frac{\cot \left(d \left(a+b \log \left(c x^n\right)\right)\right)}{x} \, dx","Integrate[Cot[d*(a + b*Log[c*x^n])]/x,x]","\frac{\log \left(\tan \left(a d+b d \log \left(c x^n\right)\right)\right)+\log \left(\cos \left(d \left(a+b \log \left(c x^n\right)\right)\right)\right)}{b d n}","\frac{\log \left(\sin \left(a d+b d \log \left(c x^n\right)\right)\right)}{b d n}",1,"(Log[Cos[d*(a + b*Log[c*x^n])]] + Log[Tan[a*d + b*d*Log[c*x^n]]])/(b*d*n)","A",1
214,1,217,70,4.5900431,"\int \frac{\cot \left(d \left(a+b \log \left(c x^n\right)\right)\right)}{x^2} \, dx","Integrate[Cot[d*(a + b*Log[c*x^n])]/x^2,x]","\frac{-\frac{e^{2 i d \left(a+b \log \left(c x^n\right)\right)} \, _2F_1\left(1,1+\frac{i}{2 b d n};2+\frac{i}{2 b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)}{2 b d n+i}+i \, _2F_1\left(1,\frac{i}{2 b d n};1+\frac{i}{2 b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)+\cot \left(d \left(a+b \log \left(c x^n\right)\right)\right)-\cot \left(d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)+\sin (b d n \log (x)) \csc \left(d \left(a+b \log \left(c x^n\right)\right)\right) \csc \left(d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)}{x}","\frac{2 i \, _2F_1\left(1,\frac{i}{2 b d n};1+\frac{i}{2 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{x}-\frac{i}{x}",1,"(Cot[d*(a + b*Log[c*x^n])] - Cot[d*(a - b*n*Log[x] + b*Log[c*x^n])] - (E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 + (I/2)/(b*d*n), 2 + (I/2)/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))])/(I + 2*b*d*n) + I*Hypergeometric2F1[1, (I/2)/(b*d*n), 1 + (I/2)/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] + Csc[d*(a + b*Log[c*x^n])]*Csc[d*(a - b*n*Log[x] + b*Log[c*x^n])]*Sin[b*d*n*Log[x]])/x","B",1
215,1,211,68,4.1581192,"\int \frac{\cot \left(d \left(a+b \log \left(c x^n\right)\right)\right)}{x^3} \, dx","Integrate[Cot[d*(a + b*Log[c*x^n])]/x^3,x]","\frac{-\frac{e^{2 i d \left(a+b \log \left(c x^n\right)\right)} \, _2F_1\left(1,1+\frac{i}{b d n};2+\frac{i}{b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)}{b d n+i}+i \, _2F_1\left(1,\frac{i}{b d n};1+\frac{i}{b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)+\cot \left(d \left(a+b \log \left(c x^n\right)\right)\right)-\cot \left(d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)+\sin (b d n \log (x)) \csc \left(d \left(a+b \log \left(c x^n\right)\right)\right) \csc \left(d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)}{2 x^2}","\frac{i \, _2F_1\left(1,\frac{i}{b d n};1+\frac{i}{b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{x^2}-\frac{i}{2 x^2}",1,"(Cot[d*(a + b*Log[c*x^n])] - Cot[d*(a - b*n*Log[x] + b*Log[c*x^n])] - (E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 + I/(b*d*n), 2 + I/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))])/(I + b*d*n) + I*Hypergeometric2F1[1, I/(b*d*n), 1 + I/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] + Csc[d*(a + b*Log[c*x^n])]*Csc[d*(a - b*n*Log[x] + b*Log[c*x^n])]*Sin[b*d*n*Log[x]])/(2*x^2)","B",1
216,1,175,158,4.6359652,"\int x^3 \cot ^2\left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[x^3*Cot[d*(a + b*Log[c*x^n])]^2,x]","-\frac{x^4 \left(8 e^{2 i d \left(a+b \log \left(c x^n\right)\right)} \, _2F_1\left(1,1-\frac{2 i}{b d n};2-\frac{2 i}{b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)+(b d n-2 i) \left(4 i \, _2F_1\left(1,-\frac{2 i}{b d n};1-\frac{2 i}{b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)+4 \cot \left(d \left(a+b \log \left(c x^n\right)\right)\right)+b d n\right)\right)}{4 b d n (b d n-2 i)}","-\frac{2 i x^4 \, _2F_1\left(1,-\frac{2 i}{b d n};1-\frac{2 i}{b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{b d n}+\frac{i x^4 \left(1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{b d n \left(1-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}+\frac{x^4 (-b d n+4 i)}{4 b d n}",1,"-1/4*(x^4*(8*E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 - (2*I)/(b*d*n), 2 - (2*I)/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] + (-2*I + b*d*n)*(b*d*n + 4*Cot[d*(a + b*Log[c*x^n])] + (4*I)*Hypergeometric2F1[1, (-2*I)/(b*d*n), 1 - (2*I)/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))])))/(b*d*n*(-2*I + b*d*n))","A",1
217,1,185,162,5.322263,"\int x^2 \cot ^2\left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[x^2*Cot[d*(a + b*Log[c*x^n])]^2,x]","-\frac{x^3 \left(9 e^{2 i d \left(a+b \log \left(c x^n\right)\right)} \, _2F_1\left(1,1-\frac{3 i}{2 b d n};2-\frac{3 i}{2 b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)+(2 b d n-3 i) \left(3 i \, _2F_1\left(1,-\frac{3 i}{2 b d n};1-\frac{3 i}{2 b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)+3 \cot \left(d \left(a+b \log \left(c x^n\right)\right)\right)+b d n\right)\right)}{3 b d n (2 b d n-3 i)}","-\frac{2 i x^3 \, _2F_1\left(1,-\frac{3 i}{2 b d n};1-\frac{3 i}{2 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{b d n}+\frac{i x^3 \left(1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{b d n \left(1-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}+\frac{x^3 (-b d n+3 i)}{3 b d n}",1,"-1/3*(x^3*(9*E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 - ((3*I)/2)/(b*d*n), 2 - ((3*I)/2)/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] + (-3*I + 2*b*d*n)*(b*d*n + 3*Cot[d*(a + b*Log[c*x^n])] + (3*I)*Hypergeometric2F1[1, ((-3*I)/2)/(b*d*n), 1 - ((3*I)/2)/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))])))/(b*d*n*(-3*I + 2*b*d*n))","A",1
218,1,175,158,5.2041261,"\int x \cot ^2\left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[x*Cot[d*(a + b*Log[c*x^n])]^2,x]","-\frac{x^2 \left(2 e^{2 i d \left(a+b \log \left(c x^n\right)\right)} \, _2F_1\left(1,1-\frac{i}{b d n};2-\frac{i}{b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)+(b d n-i) \left(2 i \, _2F_1\left(1,-\frac{i}{b d n};1-\frac{i}{b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)+2 \cot \left(d \left(a+b \log \left(c x^n\right)\right)\right)+b d n\right)\right)}{2 b d n (b d n-i)}","-\frac{2 i x^2 \, _2F_1\left(1,-\frac{i}{b d n};1-\frac{i}{b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{b d n}+\frac{i x^2 \left(1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{b d n \left(1-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}+\frac{x^2 (-b d n+2 i)}{2 b d n}",1,"-1/2*(x^2*(2*E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 - I/(b*d*n), 2 - I/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] + (-I + b*d*n)*(b*d*n + 2*Cot[d*(a + b*Log[c*x^n])] + (2*I)*Hypergeometric2F1[1, (-I)/(b*d*n), 1 - I/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))])))/(b*d*n*(-I + b*d*n))","A",1
219,1,178,153,11.530627,"\int \cot ^2\left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[Cot[d*(a + b*Log[c*x^n])]^2,x]","-\frac{x \left(e^{2 i d \left(a+b \log \left(c x^n\right)\right)} \, _2F_1\left(1,1-\frac{i}{2 b d n};2-\frac{i}{2 b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)+(2 b d n-i) \left(i \, _2F_1\left(1,-\frac{i}{2 b d n};1-\frac{i}{2 b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)+\cot \left(d \left(a+b \log \left(c x^n\right)\right)\right)+b d n\right)\right)}{b d n (2 b d n-i)}","-\frac{2 i x \, _2F_1\left(1,-\frac{i}{2 b d n};1-\frac{i}{2 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{b d n}+\frac{i x \left(1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{b d n \left(1-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}+\frac{x (-b d n+i)}{b d n}",1,"-((x*(E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 - (I/2)/(b*d*n), 2 - (I/2)/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] + (-I + 2*b*d*n)*(b*d*n + Cot[d*(a + b*Log[c*x^n])] + I*Hypergeometric2F1[1, (-1/2*I)/(b*d*n), 1 - (I/2)/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))])))/(b*d*n*(-I + 2*b*d*n)))","A",1
220,1,51,30,0.1181584,"\int \frac{\cot ^2\left(d \left(a+b \log \left(c x^n\right)\right)\right)}{x} \, dx","Integrate[Cot[d*(a + b*Log[c*x^n])]^2/x,x]","-\frac{\cot \left(a d+b d \log \left(c x^n\right)\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2\left(a d+b \log \left(c x^n\right) d\right)\right)}{b d n}","-\frac{\cot \left(a d+b d \log \left(c x^n\right)\right)}{b d n}-\log (x)",1,"-((Cot[a*d + b*d*Log[c*x^n]]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[a*d + b*d*Log[c*x^n]]^2])/(b*d*n))","C",1
221,1,181,156,4.3835641,"\int \frac{\cot ^2\left(d \left(a+b \log \left(c x^n\right)\right)\right)}{x^2} \, dx","Integrate[Cot[d*(a + b*Log[c*x^n])]^2/x^2,x]","\frac{e^{2 i d \left(a+b \log \left(c x^n\right)\right)} \, _2F_1\left(1,1+\frac{i}{2 b d n};2+\frac{i}{2 b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)+(2 b d n+i) \left(-i \, _2F_1\left(1,\frac{i}{2 b d n};1+\frac{i}{2 b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)-\cot \left(d \left(a+b \log \left(c x^n\right)\right)\right)+b d n\right)}{b d n x (2 b d n+i)}","-\frac{2 i \, _2F_1\left(1,\frac{i}{2 b d n};1+\frac{i}{2 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{b d n x}+\frac{i \left(1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{b d n x \left(1-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}+\frac{1+\frac{i}{b d n}}{x}",1,"(E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 + (I/2)/(b*d*n), 2 + (I/2)/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] + (I + 2*b*d*n)*(b*d*n - Cot[d*(a + b*Log[c*x^n])] - I*Hypergeometric2F1[1, (I/2)/(b*d*n), 1 + (I/2)/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))]))/(b*d*n*(I + 2*b*d*n)*x)","A",1
222,1,175,155,3.9110783,"\int \frac{\cot ^2\left(d \left(a+b \log \left(c x^n\right)\right)\right)}{x^3} \, dx","Integrate[Cot[d*(a + b*Log[c*x^n])]^2/x^3,x]","\frac{2 e^{2 i d \left(a+b \log \left(c x^n\right)\right)} \, _2F_1\left(1,1+\frac{i}{b d n};2+\frac{i}{b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)+(b d n+i) \left(-2 i \, _2F_1\left(1,\frac{i}{b d n};1+\frac{i}{b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)-2 \cot \left(d \left(a+b \log \left(c x^n\right)\right)\right)+b d n\right)}{2 b d n x^2 (b d n+i)}","-\frac{2 i \, _2F_1\left(1,\frac{i}{b d n};1+\frac{i}{b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{b d n x^2}+\frac{i \left(1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{b d n x^2 \left(1-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}+\frac{1+\frac{2 i}{b d n}}{2 x^2}",1,"(2*E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 + I/(b*d*n), 2 + I/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] + (I + b*d*n)*(b*d*n - 2*Cot[d*(a + b*Log[c*x^n])] - (2*I)*Hypergeometric2F1[1, I/(b*d*n), 1 + I/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))]))/(2*b*d*n*(I + b*d*n)*x^2)","A",1
223,1,52,44,0.2206199,"\int \frac{\cot ^3\left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Cot[a + b*Log[c*x^n]]^3/x,x]","-\frac{2 \log \left(\tan \left(a+b \log \left(c x^n\right)\right)\right)+2 \log \left(\cos \left(a+b \log \left(c x^n\right)\right)\right)+\cot ^2\left(a+b \log \left(c x^n\right)\right)}{2 b n}","-\frac{\log \left(\sin \left(a+b \log \left(c x^n\right)\right)\right)}{b n}-\frac{\cot ^2\left(a+b \log \left(c x^n\right)\right)}{2 b n}",1,"-1/2*(Cot[a + b*Log[c*x^n]]^2 + 2*Log[Cos[a + b*Log[c*x^n]]] + 2*Log[Tan[a + b*Log[c*x^n]]])/(b*n)","A",1
224,1,46,44,0.1089387,"\int \frac{\cot ^4\left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Cot[a + b*Log[c*x^n]]^4/x,x]","-\frac{\cot ^3\left(a+b \log \left(c x^n\right)\right) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\tan ^2\left(a+b \log \left(c x^n\right)\right)\right)}{3 b n}","-\frac{\cot ^3\left(a+b \log \left(c x^n\right)\right)}{3 b n}+\frac{\cot \left(a+b \log \left(c x^n\right)\right)}{b n}+\log (x)",1,"-1/3*(Cot[a + b*Log[c*x^n]]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[a + b*Log[c*x^n]]^2])/(b*n)","C",1
225,1,69,66,0.2237065,"\int \frac{\cot ^5\left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Cot[a + b*Log[c*x^n]]^5/x,x]","\frac{4 \log \left(\tan \left(a+b \log \left(c x^n\right)\right)\right)+4 \log \left(\cos \left(a+b \log \left(c x^n\right)\right)\right)-\cot ^4\left(a+b \log \left(c x^n\right)\right)+2 \cot ^2\left(a+b \log \left(c x^n\right)\right)}{4 b n}","\frac{\log \left(\sin \left(a+b \log \left(c x^n\right)\right)\right)}{b n}-\frac{\cot ^4\left(a+b \log \left(c x^n\right)\right)}{4 b n}+\frac{\cot ^2\left(a+b \log \left(c x^n\right)\right)}{2 b n}",1,"(2*Cot[a + b*Log[c*x^n]]^2 - Cot[a + b*Log[c*x^n]]^4 + 4*Log[Cos[a + b*Log[c*x^n]]] + 4*Log[Tan[a + b*Log[c*x^n]]])/(4*b*n)","A",1
226,1,182,100,13.6707579,"\int (e x)^m \cot \left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[(e*x)^m*Cot[d*(a + b*Log[c*x^n])],x]","-\frac{i x (e x)^m \left(\frac{(m+1) e^{2 i a d} \left(c x^n\right)^{2 i b d} \, _2F_1\left(1,-\frac{i (m+2 i b d n+1)}{2 b d n};-\frac{i (m+4 i b d n+1)}{2 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{2 i b d n+m+1}+\, _2F_1\left(1,-\frac{i (m+1)}{2 b d n};1-\frac{i (m+1)}{2 b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)\right)}{m+1}","\frac{i (e x)^{m+1}}{e (m+1)}-\frac{2 i (e x)^{m+1} \, _2F_1\left(1,-\frac{i (m+1)}{2 b d n};1-\frac{i (m+1)}{2 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{e (m+1)}",1,"((-I)*x*(e*x)^m*(Hypergeometric2F1[1, ((-1/2*I)*(1 + m))/(b*d*n), 1 - ((I/2)*(1 + m))/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] + (E^((2*I)*a*d)*(1 + m)*(c*x^n)^((2*I)*b*d)*Hypergeometric2F1[1, ((-1/2*I)*(1 + m + (2*I)*b*d*n))/(b*d*n), ((-1/2*I)*(1 + m + (4*I)*b*d*n))/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)])/(1 + m + (2*I)*b*d*n)))/(1 + m)","A",1
227,1,547,195,16.5748264,"\int (e x)^m \cot ^2\left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[(e*x)^m*Cot[d*(a + b*Log[c*x^n])]^2,x]","-\frac{(m+1) x^{-m} (e x)^m \csc \left(d \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right) \left(\frac{x^{m+1} \sin (b d n \log (x)) \csc \left(d \left(a+b \log \left(c x^n\right)\right)\right)}{m+1}-\frac{i \sin \left(d \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right) \exp \left(-\frac{(2 m+1) \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)}{b n}\right) \left(-(2 i b d n+m+1) \exp \left(\frac{2 a m+a+b (2 m+1) \left(\log \left(c x^n\right)-n \log (x)\right)+b (m+1) n \log (x)}{b n}\right) \, _2F_1\left(1,-\frac{i (m+1)}{2 b d n};1-\frac{i (m+1)}{2 b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)-(m+1) \exp \left(\frac{a (2 i b d n+2 m+1)}{b n}+\frac{(2 i b d n+2 m+1) \left(\log \left(c x^n\right)-n \log (x)\right)}{n}+\log (x) (2 i b d n+m+1)\right) \, _2F_1\left(1,-\frac{i (m+2 i b d n+1)}{2 b d n};-\frac{i (m+4 i b d n+1)}{2 b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)+i (2 i b d n+m+1) \cot \left(d \left(a+b \log \left(c x^n\right)\right)\right) \exp \left(\frac{2 a m+a+b (2 m+1) \left(\log \left(c x^n\right)-n \log (x)\right)+b (m+1) n \log (x)}{b n}\right)\right)}{(m+1) (2 i b d n+m+1)}\right)}{b d n}+\frac{x (e x)^m \sin (b d n \log (x)) \csc \left(d \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right) \csc \left(d \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)+b d n \log (x)\right)}{b d n}-\frac{x (e x)^m}{m+1}","-\frac{2 i (e x)^{m+1} \, _2F_1\left(1,-\frac{i (m+1)}{2 b d n};1-\frac{i (m+1)}{2 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{b d e n}+\frac{i (e x)^{m+1} \left(1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{b d e n \left(1-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}+\frac{(e x)^{m+1} (-b d n+i (m+1))}{b d e (m+1) n}",1,"-((x*(e*x)^m)/(1 + m)) + (x*(e*x)^m*Csc[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]*Csc[b*d*n*Log[x] + d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]*Sin[b*d*n*Log[x]])/(b*d*n) - ((1 + m)*(e*x)^m*Csc[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]*((x^(1 + m)*Csc[d*(a + b*Log[c*x^n])]*Sin[b*d*n*Log[x]])/(1 + m) - (I*(I*E^((a + 2*a*m + b*(1 + m)*n*Log[x] + b*(1 + 2*m)*(-(n*Log[x]) + Log[c*x^n]))/(b*n))*(1 + m + (2*I)*b*d*n)*Cot[d*(a + b*Log[c*x^n])] - E^((a + 2*a*m + b*(1 + m)*n*Log[x] + b*(1 + 2*m)*(-(n*Log[x]) + Log[c*x^n]))/(b*n))*(1 + m + (2*I)*b*d*n)*Hypergeometric2F1[1, ((-1/2*I)*(1 + m))/(b*d*n), 1 - ((I/2)*(1 + m))/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] - E^((a*(1 + 2*m + (2*I)*b*d*n))/(b*n) + (1 + m + (2*I)*b*d*n)*Log[x] + ((1 + 2*m + (2*I)*b*d*n)*(-(n*Log[x]) + Log[c*x^n]))/n)*(1 + m)*Hypergeometric2F1[1, ((-1/2*I)*(1 + m + (2*I)*b*d*n))/(b*d*n), ((-1/2*I)*(1 + m + (4*I)*b*d*n))/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))])*Sin[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))])/(E^(((1 + 2*m)*(a + b*(-(n*Log[x]) + Log[c*x^n])))/(b*n))*(1 + m)*(1 + m + (2*I)*b*d*n))))/(b*d*n*x^m)","B",1
228,1,639,350,16.9779116,"\int (e x)^m \cot ^3\left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[(e*x)^m*Cot[d*(a + b*Log[c*x^n])]^3,x]","\frac{x^{-m} (e x)^m \left(2 b^2 d^2 n^2-m^2-2 m-1\right) \csc \left(d \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right) \left(\frac{x^{m+1} \sin (b d n \log (x)) \csc \left(d \left(a+b \log \left(c x^n\right)\right)\right)}{m+1}-\frac{i \sin \left(d \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right) \exp \left(-\frac{(2 m+1) \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)}{b n}\right) \left(-(2 i b d n+m+1) \exp \left(\frac{2 a m+a+b (2 m+1) \left(\log \left(c x^n\right)-n \log (x)\right)+b (m+1) n \log (x)}{b n}\right) \, _2F_1\left(1,-\frac{i (m+1)}{2 b d n};1-\frac{i (m+1)}{2 b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)-(m+1) \exp \left(\frac{a (2 i b d n+2 m+1)}{b n}+\frac{(2 i b d n+2 m+1) \left(\log \left(c x^n\right)-n \log (x)\right)}{n}+\log (x) (2 i b d n+m+1)\right) \, _2F_1\left(1,-\frac{i (m+2 i b d n+1)}{2 b d n};-\frac{i (m+4 i b d n+1)}{2 b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)+i (2 i b d n+m+1) \cot \left(d \left(a+b \log \left(c x^n\right)\right)\right) \exp \left(\frac{2 a m+a+b (2 m+1) \left(\log \left(c x^n\right)-n \log (x)\right)+b (m+1) n \log (x)}{b n}\right)\right)}{(m+1) (2 i b d n+m+1)}\right)}{2 b^2 d^2 n^2}+\frac{(m+1) x (e x)^m \sin (b d n \log (x)) \csc \left(d \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right) \csc \left(d \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)+b d n \log (x)\right)}{2 b^2 d^2 n^2}-\frac{x (e x)^m \cot \left(d \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right)}{m+1}-\frac{x (e x)^m \csc ^2\left(d \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)+b d n \log (x)\right)}{2 b d n}","-\frac{i (e x)^{m+1} \left(-2 b^2 d^2 n^2+m^2+2 m+1\right) \, _2F_1\left(1,-\frac{i (m+1)}{2 b d n};1-\frac{i (m+1)}{2 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{b^2 d^2 e (m+1) n^2}+\frac{i e^{-2 i a d} (e x)^{m+1} \left(\frac{e^{4 i a d} (2 i b d n+m+1) \left(c x^n\right)^{2 i b d}}{n}+\frac{e^{2 i a d} (-2 i b d n+m+1)}{n}\right)}{2 b^2 d^2 e n \left(1-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}+\frac{(e x)^{m+1} \left(1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)^2}{2 b d e n \left(1-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)^2}+\frac{(e x)^{m+1} (-b d n+i (m+1)) (2 i b d n+m+1)}{2 b^2 d^2 e (m+1) n^2}",1,"-((x*(e*x)^m*Cot[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))])/(1 + m)) - (x*(e*x)^m*Csc[b*d*n*Log[x] + d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]^2)/(2*b*d*n) + ((1 + m)*x*(e*x)^m*Csc[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]*Csc[b*d*n*Log[x] + d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]*Sin[b*d*n*Log[x]])/(2*b^2*d^2*n^2) + ((-1 - 2*m - m^2 + 2*b^2*d^2*n^2)*(e*x)^m*Csc[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]*((x^(1 + m)*Csc[d*(a + b*Log[c*x^n])]*Sin[b*d*n*Log[x]])/(1 + m) - (I*(I*E^((a + 2*a*m + b*(1 + m)*n*Log[x] + b*(1 + 2*m)*(-(n*Log[x]) + Log[c*x^n]))/(b*n))*(1 + m + (2*I)*b*d*n)*Cot[d*(a + b*Log[c*x^n])] - E^((a + 2*a*m + b*(1 + m)*n*Log[x] + b*(1 + 2*m)*(-(n*Log[x]) + Log[c*x^n]))/(b*n))*(1 + m + (2*I)*b*d*n)*Hypergeometric2F1[1, ((-1/2*I)*(1 + m))/(b*d*n), 1 - ((I/2)*(1 + m))/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] - E^((a*(1 + 2*m + (2*I)*b*d*n))/(b*n) + (1 + m + (2*I)*b*d*n)*Log[x] + ((1 + 2*m + (2*I)*b*d*n)*(-(n*Log[x]) + Log[c*x^n]))/n)*(1 + m)*Hypergeometric2F1[1, ((-1/2*I)*(1 + m + (2*I)*b*d*n))/(b*d*n), ((-1/2*I)*(1 + m + (4*I)*b*d*n))/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))])*Sin[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))])/(E^(((1 + 2*m)*(a + b*(-(n*Log[x]) + Log[c*x^n])))/(b*n))*(1 + m)*(1 + m + (2*I)*b*d*n))))/(2*b^2*d^2*n^2*x^m)","A",0
229,1,458,190,1.2909281,"\int \cot ^p\left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[Cot[d*(a + b*Log[c*x^n])]^p,x]","\frac{x (2 b d n-i) \left(\frac{i \left(1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{-1+e^{2 i a d} \left(c x^n\right)^{2 i b d}}\right)^p F_1\left(-\frac{i}{2 b d n};p,-p;1-\frac{i}{2 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d},-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{2 b d n p e^{2 i a d} \left(c x^n\right)^{2 i b d} F_1\left(1-\frac{i}{2 b d n};p,1-p;2-\frac{i}{2 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d},-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)+2 b d n p e^{2 i a d} \left(c x^n\right)^{2 i b d} F_1\left(1-\frac{i}{2 b d n};p+1,-p;2-\frac{i}{2 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d},-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)+(2 b d n-i) F_1\left(-\frac{i}{2 b d n};p,-p;1-\frac{i}{2 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d},-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}","x \left(1-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)^p \left(1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)^{-p} \left(-\frac{i \left(1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{1-e^{2 i a d} \left(c x^n\right)^{2 i b d}}\right)^p F_1\left(-\frac{i}{2 b d n};p,-p;1-\frac{i}{2 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d},-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)",1,"((-I + 2*b*d*n)*x*((I*(1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)))/(-1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)))^p*AppellF1[(-1/2*I)/(b*d*n), p, -p, 1 - (I/2)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d), -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))])/(2*b*d*E^((2*I)*a*d)*n*p*(c*x^n)^((2*I)*b*d)*AppellF1[1 - (I/2)/(b*d*n), p, 1 - p, 2 - (I/2)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d), -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))] + 2*b*d*E^((2*I)*a*d)*n*p*(c*x^n)^((2*I)*b*d)*AppellF1[1 - (I/2)/(b*d*n), 1 + p, -p, 2 - (I/2)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d), -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))] + (-I + 2*b*d*n)*AppellF1[(-1/2*I)/(b*d*n), p, -p, 1 - (I/2)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d), -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))])","B",0
230,1,205,210,1.1102119,"\int (e x)^m \cot ^p\left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[(e*x)^m*Cot[d*(a + b*Log[c*x^n])]^p,x]","\frac{x (e x)^m \left(1-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)^p \left(1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)^{-p} \left(\frac{i \left(1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{-1+e^{2 i a d} \left(c x^n\right)^{2 i b d}}\right)^p F_1\left(-\frac{i (m+1)}{2 b d n};p,-p;1-\frac{i (m+1)}{2 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d},-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{m+1}","\frac{(e x)^{m+1} \left(1-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)^p \left(1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)^{-p} \left(-\frac{i \left(1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{1-e^{2 i a d} \left(c x^n\right)^{2 i b d}}\right)^p F_1\left(-\frac{i (m+1)}{2 b d n};p,-p;1-\frac{i (m+1)}{2 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d},-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{e (m+1)}",1,"(x*(e*x)^m*(1 - E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))^p*((I*(1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)))/(-1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)))^p*AppellF1[((-1/2*I)*(1 + m))/(b*d*n), p, -p, 1 - ((I/2)*(1 + m))/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d), -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))])/((1 + m)*(1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))^p)","A",1
231,1,50,201,0.2451593,"\int \frac{\cot ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Cot[a + b*Log[c*x^n]]^(5/2)/x,x]","\frac{2 \cot ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right) \left(\, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2\left(a+b \log \left(c x^n\right)\right)\right)-1\right)}{3 b n}","-\frac{2 \cot ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}{3 b n}+\frac{\log \left(\cot \left(a+b \log \left(c x^n\right)\right)-\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}+1\right)}{2 \sqrt{2} b n}-\frac{\log \left(\cot \left(a+b \log \left(c x^n\right)\right)+\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}+1\right)}{2 \sqrt{2} b n}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}\right)}{\sqrt{2} b n}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}+1\right)}{\sqrt{2} b n}",1,"(2*Cot[a + b*Log[c*x^n]]^(3/2)*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Cot[a + b*Log[c*x^n]]^2]))/(3*b*n)","C",1
232,1,175,199,0.2751779,"\int \frac{\cot ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Cot[a + b*Log[c*x^n]]^(3/2)/x,x]","-\frac{\sqrt{2} \log \left(\cot \left(a+b \log \left(c x^n\right)\right)-\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}+1\right)-\sqrt{2} \log \left(\cot \left(a+b \log \left(c x^n\right)\right)+\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}+1\right)+8 \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}+2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}+1\right)}{4 b n}","-\frac{\log \left(\cot \left(a+b \log \left(c x^n\right)\right)-\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}+1\right)}{2 \sqrt{2} b n}+\frac{\log \left(\cot \left(a+b \log \left(c x^n\right)\right)+\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}+1\right)}{2 \sqrt{2} b n}-\frac{2 \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}}{b n}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}\right)}{\sqrt{2} b n}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}+1\right)}{\sqrt{2} b n}",1,"-1/4*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]]] + 8*Sqrt[Cot[a + b*Log[c*x^n]]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]] + Cot[a + b*Log[c*x^n]]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]] + Cot[a + b*Log[c*x^n]]])/(b*n)","A",1
233,1,48,176,0.0953747,"\int \frac{\sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}}{x} \, dx","Integrate[Sqrt[Cot[a + b*Log[c*x^n]]]/x,x]","-\frac{2 \cot ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2\left(a+b \log \left(c x^n\right)\right)\right)}{3 b n}","-\frac{\log \left(\cot \left(a+b \log \left(c x^n\right)\right)-\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}+1\right)}{2 \sqrt{2} b n}+\frac{\log \left(\cot \left(a+b \log \left(c x^n\right)\right)+\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}+1\right)}{2 \sqrt{2} b n}+\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}\right)}{\sqrt{2} b n}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}+1\right)}{\sqrt{2} b n}",1,"(-2*Cot[a + b*Log[c*x^n]]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[a + b*Log[c*x^n]]^2])/(3*b*n)","C",1
234,1,142,176,0.1414669,"\int \frac{1}{x \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}} \, dx","Integrate[1/(x*Sqrt[Cot[a + b*Log[c*x^n]]]),x]","\frac{\log \left(\cot \left(a+b \log \left(c x^n\right)\right)-\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}+1\right)-\log \left(\cot \left(a+b \log \left(c x^n\right)\right)+\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}+1\right)+2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}+1\right)}{2 \sqrt{2} b n}","\frac{\log \left(\cot \left(a+b \log \left(c x^n\right)\right)-\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}+1\right)}{2 \sqrt{2} b n}-\frac{\log \left(\cot \left(a+b \log \left(c x^n\right)\right)+\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}+1\right)}{2 \sqrt{2} b n}+\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}\right)}{\sqrt{2} b n}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}+1\right)}{\sqrt{2} b n}",1,"(2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]]] + Log[1 - Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]] + Cot[a + b*Log[c*x^n]]] - Log[1 + Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]] + Cot[a + b*Log[c*x^n]]])/(2*Sqrt[2]*b*n)","A",1
235,1,46,199,0.135637,"\int \frac{1}{x \cot ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[1/(x*Cot[a + b*Log[c*x^n]]^(3/2)),x]","\frac{2 \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2\left(a+b \log \left(c x^n\right)\right)\right)}{b n \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}}","\frac{\log \left(\cot \left(a+b \log \left(c x^n\right)\right)-\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}+1\right)}{2 \sqrt{2} b n}-\frac{\log \left(\cot \left(a+b \log \left(c x^n\right)\right)+\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}+1\right)}{2 \sqrt{2} b n}+\frac{2}{b n \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}\right)}{\sqrt{2} b n}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}+1\right)}{\sqrt{2} b n}",1,"(2*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[a + b*Log[c*x^n]]^2])/(b*n*Sqrt[Cot[a + b*Log[c*x^n]]])","C",1
236,1,48,201,0.2013976,"\int \frac{1}{x \cot ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[1/(x*Cot[a + b*Log[c*x^n]]^(5/2)),x]","\frac{2 \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\cot ^2\left(a+b \log \left(c x^n\right)\right)\right)}{3 b n \cot ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}","\frac{2}{3 b n \cot ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}-\frac{\log \left(\cot \left(a+b \log \left(c x^n\right)\right)-\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}+1\right)}{2 \sqrt{2} b n}+\frac{\log \left(\cot \left(a+b \log \left(c x^n\right)\right)+\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}+1\right)}{2 \sqrt{2} b n}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}\right)}{\sqrt{2} b n}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\cot \left(a+b \log \left(c x^n\right)\right)}+1\right)}{\sqrt{2} b n}",1,"(2*Hypergeometric2F1[-3/4, 1, 1/4, -Cot[a + b*Log[c*x^n]]^2])/(3*b*n*Cot[a + b*Log[c*x^n]]^(3/2))","C",1
237,1,86,87,0.157647,"\int x^2 \sec \left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x^2*Sec[a + b*Log[c*x^n]],x]","-\frac{2 i e^{i a} x^3 \left(c x^n\right)^{i b} \, _2F_1\left(1,\frac{1}{2}-\frac{3 i}{2 b n};\frac{3}{2}-\frac{3 i}{2 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)}{b n-3 i}","\frac{2 e^{i a} x^3 \left(c x^n\right)^{i b} \, _2F_1\left(1,\frac{1}{2} \left(1-\frac{3 i}{b n}\right);\frac{3}{2} \left(1-\frac{i}{b n}\right);-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{3+i b n}",1,"((-2*I)*E^(I*a)*x^3*(c*x^n)^(I*b)*Hypergeometric2F1[1, 1/2 - ((3*I)/2)/(b*n), 3/2 - ((3*I)/2)/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))])/(-3*I + b*n)","A",1
238,1,82,87,0.1273584,"\int x \sec \left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x*Sec[a + b*Log[c*x^n]],x]","-\frac{2 i e^{i a} x^2 \left(c x^n\right)^{i b} \, _2F_1\left(1,\frac{1}{2}-\frac{i}{b n};\frac{3}{2}-\frac{i}{b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)}{b n-2 i}","\frac{2 e^{i a} x^2 \left(c x^n\right)^{i b} \, _2F_1\left(1,\frac{1}{2} \left(1-\frac{2 i}{b n}\right);\frac{1}{2} \left(3-\frac{2 i}{b n}\right);-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{2+i b n}",1,"((-2*I)*E^(I*a)*x^2*(c*x^n)^(I*b)*Hypergeometric2F1[1, 1/2 - I/(b*n), 3/2 - I/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))])/(-2*I + b*n)","A",1
239,1,84,85,0.1109856,"\int \sec \left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[Sec[a + b*Log[c*x^n]],x]","-\frac{2 i e^{i a} x \left(c x^n\right)^{i b} \, _2F_1\left(1,\frac{1}{2}-\frac{i}{2 b n};\frac{3}{2}-\frac{i}{2 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)}{b n-i}","\frac{2 e^{i a} x \left(c x^n\right)^{i b} \, _2F_1\left(1,\frac{1}{2} \left(1-\frac{i}{b n}\right);\frac{1}{2} \left(3-\frac{i}{b n}\right);-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{1+i b n}",1,"((-2*I)*E^(I*a)*x*(c*x^n)^(I*b)*Hypergeometric2F1[1, 1/2 - (I/2)/(b*n), 3/2 - (I/2)/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))])/(-I + b*n)","A",1
240,1,19,19,0.0357283,"\int \frac{\sec \left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Sec[a + b*Log[c*x^n]]/x,x]","\frac{\tanh ^{-1}\left(\sin \left(a+b \log \left(c x^n\right)\right)\right)}{b n}","\frac{\tanh ^{-1}\left(\sin \left(a+b \log \left(c x^n\right)\right)\right)}{b n}",1,"ArcTanh[Sin[a + b*Log[c*x^n]]]/(b*n)","A",1
241,1,85,87,0.1440252,"\int \frac{\sec \left(a+b \log \left(c x^n\right)\right)}{x^2} \, dx","Integrate[Sec[a + b*Log[c*x^n]]/x^2,x]","\frac{2 e^{i a} \left(c x^n\right)^{i b} \, _2F_1\left(1,\frac{1}{2}+\frac{i}{2 b n};\frac{3}{2}+\frac{i}{2 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)}{x (-1+i b n)}","-\frac{2 e^{i a} \left(c x^n\right)^{i b} \, _2F_1\left(1,\frac{1}{2} \left(1+\frac{i}{b n}\right);\frac{1}{2} \left(3+\frac{i}{b n}\right);-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{x (1-i b n)}",1,"(2*E^(I*a)*(c*x^n)^(I*b)*Hypergeometric2F1[1, 1/2 + (I/2)/(b*n), 3/2 + (I/2)/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))])/((-1 + I*b*n)*x)","A",1
242,1,81,87,0.1398844,"\int \frac{\sec \left(a+b \log \left(c x^n\right)\right)}{x^3} \, dx","Integrate[Sec[a + b*Log[c*x^n]]/x^3,x]","\frac{2 e^{i a} \left(c x^n\right)^{i b} \, _2F_1\left(1,\frac{1}{2}+\frac{i}{b n};\frac{3}{2}+\frac{i}{b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)}{x^2 (-2+i b n)}","-\frac{2 e^{i a} \left(c x^n\right)^{i b} \, _2F_1\left(1,\frac{1}{2} \left(1+\frac{2 i}{b n}\right);\frac{1}{2} \left(3+\frac{2 i}{b n}\right);-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{x^2 (2-i b n)}",1,"(2*E^(I*a)*(c*x^n)^(I*b)*Hypergeometric2F1[1, 1/2 + I/(b*n), 3/2 + I/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))])/((-2 + I*b*n)*x^2)","A",1
243,1,160,87,5.4107881,"\int x^2 \sec ^2\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x^2*Sec[a + b*Log[c*x^n]]^2,x]","\frac{x^3 \left(3 e^{2 i a} \left(c x^n\right)^{2 i b} \, _2F_1\left(1,1-\frac{3 i}{2 b n};2-\frac{3 i}{2 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)+(2 b n-3 i) \left(\tan \left(a+b \log \left(c x^n\right)\right)-i \, _2F_1\left(1,-\frac{3 i}{2 b n};1-\frac{3 i}{2 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)\right)\right)}{b n (2 b n-3 i)}","\frac{4 e^{2 i a} x^3 \left(c x^n\right)^{2 i b} \, _2F_1\left(2,\frac{1}{2} \left(2-\frac{3 i}{b n}\right);\frac{1}{2} \left(4-\frac{3 i}{b n}\right);-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{3+2 i b n}",1,"(x^3*(3*E^((2*I)*a)*(c*x^n)^((2*I)*b)*Hypergeometric2F1[1, 1 - ((3*I)/2)/(b*n), 2 - ((3*I)/2)/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))] + (-3*I + 2*b*n)*((-I)*Hypergeometric2F1[1, ((-3*I)/2)/(b*n), 1 - ((3*I)/2)/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))] + Tan[a + b*Log[c*x^n]])))/(b*n*(-3*I + 2*b*n))","A",0
244,1,149,79,5.2013932,"\int x \sec ^2\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x*Sec[a + b*Log[c*x^n]]^2,x]","\frac{x^2 \left(e^{2 i a} \left(c x^n\right)^{2 i b} \, _2F_1\left(1,1-\frac{i}{b n};2-\frac{i}{b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)+(b n-i) \left(\tan \left(a+b \log \left(c x^n\right)\right)-i \, _2F_1\left(1,-\frac{i}{b n};1-\frac{i}{b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)\right)\right)}{b n (b n-i)}","\frac{2 e^{2 i a} x^2 \left(c x^n\right)^{2 i b} \, _2F_1\left(2,1-\frac{i}{b n};2-\frac{i}{b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{1+i b n}",1,"(x^2*(E^((2*I)*a)*(c*x^n)^((2*I)*b)*Hypergeometric2F1[1, 1 - I/(b*n), 2 - I/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))] + (-I + b*n)*((-I)*Hypergeometric2F1[1, (-I)/(b*n), 1 - I/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))] + Tan[a + b*Log[c*x^n]])))/(b*n*(-I + b*n))","A",0
245,1,147,85,6.1830331,"\int \sec ^2\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[Sec[a + b*Log[c*x^n]]^2,x]","\frac{x \left(\frac{e^{2 i a} \left(c x^n\right)^{2 i b} \, _2F_1\left(1,1-\frac{i}{2 b n};2-\frac{i}{2 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)}{2 b n-i}-i \, _2F_1\left(1,-\frac{i}{2 b n};1-\frac{i}{2 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)+\tan \left(a+b \log \left(c x^n\right)\right)\right)}{b n}","\frac{4 e^{2 i a} x \left(c x^n\right)^{2 i b} \, _2F_1\left(2,\frac{1}{2} \left(2-\frac{i}{b n}\right);\frac{1}{2} \left(4-\frac{i}{b n}\right);-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{1+2 i b n}",1,"(x*((E^((2*I)*a)*(c*x^n)^((2*I)*b)*Hypergeometric2F1[1, 1 - (I/2)/(b*n), 2 - (I/2)/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))])/(-I + 2*b*n) - I*Hypergeometric2F1[1, (-1/2*I)/(b*n), 1 - (I/2)/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))] + Tan[a + b*Log[c*x^n]]))/(b*n)","A",0
246,1,18,18,0.0762422,"\int \frac{\sec ^2\left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Sec[a + b*Log[c*x^n]]^2/x,x]","\frac{\tan \left(a+b \log \left(c x^n\right)\right)}{b n}","\frac{\tan \left(a+b \log \left(c x^n\right)\right)}{b n}",1,"Tan[a + b*Log[c*x^n]]/(b*n)","A",1
247,1,160,87,3.7426072,"\int \frac{\sec ^2\left(a+b \log \left(c x^n\right)\right)}{x^2} \, dx","Integrate[Sec[a + b*Log[c*x^n]]^2/x^2,x]","\frac{(1-2 i b n) \left(\, _2F_1\left(1,\frac{i}{2 b n};1+\frac{i}{2 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)+i \tan \left(a+b \log \left(c x^n\right)\right)\right)-e^{2 i a} \left(c x^n\right)^{2 i b} \, _2F_1\left(1,1+\frac{i}{2 b n};2+\frac{i}{2 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)}{b n x (2 b n+i)}","-\frac{4 e^{2 i a} \left(c x^n\right)^{2 i b} \, _2F_1\left(2,\frac{1}{2} \left(2+\frac{i}{b n}\right);\frac{1}{2} \left(4+\frac{i}{b n}\right);-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{x (1-2 i b n)}",1,"(-(E^((2*I)*a)*(c*x^n)^((2*I)*b)*Hypergeometric2F1[1, 1 + (I/2)/(b*n), 2 + (I/2)/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))]) + (1 - (2*I)*b*n)*(Hypergeometric2F1[1, (I/2)/(b*n), 1 + (I/2)/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))] + I*Tan[a + b*Log[c*x^n]]))/(b*n*(I + 2*b*n)*x)","A",0
248,1,150,79,3.6019743,"\int \frac{\sec ^2\left(a+b \log \left(c x^n\right)\right)}{x^3} \, dx","Integrate[Sec[a + b*Log[c*x^n]]^2/x^3,x]","\frac{(b n+i) \left(\tan \left(a+b \log \left(c x^n\right)\right)-i \, _2F_1\left(1,\frac{i}{b n};1+\frac{i}{b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)\right)-e^{2 i a} \left(c x^n\right)^{2 i b} \, _2F_1\left(1,1+\frac{i}{b n};2+\frac{i}{b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)}{b n x^2 (b n+i)}","-\frac{2 e^{2 i a} \left(c x^n\right)^{2 i b} \, _2F_1\left(2,1+\frac{i}{b n};2+\frac{i}{b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{x^2 (1-i b n)}",1,"(-(E^((2*I)*a)*(c*x^n)^((2*I)*b)*Hypergeometric2F1[1, 1 + I/(b*n), 2 + I/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))]) + (I + b*n)*((-I)*Hypergeometric2F1[1, I/(b*n), 1 + I/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))] + Tan[a + b*Log[c*x^n]]))/(b*n*(I + b*n)*x^2)","A",0
249,1,118,87,4.7129033,"\int x \sec ^3\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x*Sec[a + b*Log[c*x^n]]^3,x]","\frac{x^2 \left(\left(b n \tan \left(a+b \log \left(c x^n\right)\right)-2\right) \sec \left(a+b \log \left(c x^n\right)\right)+2 e^{i a} (2-i b n) \left(c x^n\right)^{i b} \, _2F_1\left(1,\frac{1}{2}-\frac{i}{b n};\frac{3}{2}-\frac{i}{b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)\right)}{2 b^2 n^2}","\frac{8 e^{3 i a} x^2 \left(c x^n\right)^{3 i b} \, _2F_1\left(3,\frac{1}{2} \left(3-\frac{2 i}{b n}\right);\frac{1}{2} \left(5-\frac{2 i}{b n}\right);-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{2+3 i b n}",1,"(x^2*(2*E^(I*a)*(2 - I*b*n)*(c*x^n)^(I*b)*Hypergeometric2F1[1, 1/2 - I/(b*n), 3/2 - I/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))] + Sec[a + b*Log[c*x^n]]*(-2 + b*n*Tan[a + b*Log[c*x^n]])))/(2*b^2*n^2)","A",0
250,1,120,85,4.4129678,"\int \sec ^3\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[Sec[a + b*Log[c*x^n]]^3,x]","\frac{x \left(\left(b n \tan \left(a+b \log \left(c x^n\right)\right)-1\right) \sec \left(a+b \log \left(c x^n\right)\right)+2 e^{i a} (1-i b n) \left(c x^n\right)^{i b} \, _2F_1\left(1,\frac{1}{2}-\frac{i}{2 b n};\frac{3}{2}-\frac{i}{2 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)\right)}{2 b^2 n^2}","\frac{8 e^{3 i a} x \left(c x^n\right)^{3 i b} \, _2F_1\left(3,\frac{1}{2} \left(3-\frac{i}{b n}\right);\frac{1}{2} \left(5-\frac{i}{b n}\right);-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{1+3 i b n}",1,"(x*(2*E^(I*a)*(1 - I*b*n)*(c*x^n)^(I*b)*Hypergeometric2F1[1, 1/2 - (I/2)/(b*n), 3/2 - (I/2)/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))] + Sec[a + b*Log[c*x^n]]*(-1 + b*n*Tan[a + b*Log[c*x^n]])))/(2*b^2*n^2)","A",0
251,1,55,55,0.0707695,"\int \frac{\sec ^3\left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Sec[a + b*Log[c*x^n]]^3/x,x]","\frac{\tanh ^{-1}\left(\sin \left(a+b \log \left(c x^n\right)\right)\right)}{2 b n}+\frac{\tan \left(a+b \log \left(c x^n\right)\right) \sec \left(a+b \log \left(c x^n\right)\right)}{2 b n}","\frac{\tanh ^{-1}\left(\sin \left(a+b \log \left(c x^n\right)\right)\right)}{2 b n}+\frac{\tan \left(a+b \log \left(c x^n\right)\right) \sec \left(a+b \log \left(c x^n\right)\right)}{2 b n}",1,"ArcTanh[Sin[a + b*Log[c*x^n]]]/(2*b*n) + (Sec[a + b*Log[c*x^n]]*Tan[a + b*Log[c*x^n]])/(2*b*n)","A",1
252,1,123,87,4.6106257,"\int \frac{\sec ^3\left(a+b \log \left(c x^n\right)\right)}{x^2} \, dx","Integrate[Sec[a + b*Log[c*x^n]]^3/x^2,x]","\frac{\left(b n \tan \left(a+b \log \left(c x^n\right)\right)+1\right) \sec \left(a+b \log \left(c x^n\right)\right)-2 i e^{i a} (b n-i) \left(c x^n\right)^{i b} \, _2F_1\left(1,\frac{1}{2}+\frac{i}{2 b n};\frac{3}{2}+\frac{i}{2 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)}{2 b^2 n^2 x}","-\frac{8 e^{3 i a} \left(c x^n\right)^{3 i b} \, _2F_1\left(3,\frac{1}{2} \left(3+\frac{i}{b n}\right);\frac{1}{2} \left(5+\frac{i}{b n}\right);-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{x (1-3 i b n)}",1,"((-2*I)*E^(I*a)*(-I + b*n)*(c*x^n)^(I*b)*Hypergeometric2F1[1, 1/2 + (I/2)/(b*n), 3/2 + (I/2)/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))] + Sec[a + b*Log[c*x^n]]*(1 + b*n*Tan[a + b*Log[c*x^n]]))/(2*b^2*n^2*x)","A",0
253,1,119,87,4.6282676,"\int \frac{\sec ^3\left(a+b \log \left(c x^n\right)\right)}{x^3} \, dx","Integrate[Sec[a + b*Log[c*x^n]]^3/x^3,x]","\frac{\left(b n \tan \left(a+b \log \left(c x^n\right)\right)+2\right) \sec \left(a+b \log \left(c x^n\right)\right)-2 i e^{i a} (b n-2 i) \left(c x^n\right)^{i b} \, _2F_1\left(1,\frac{1}{2}+\frac{i}{b n};\frac{3}{2}+\frac{i}{b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)}{2 b^2 n^2 x^2}","-\frac{8 e^{3 i a} \left(c x^n\right)^{3 i b} \, _2F_1\left(3,\frac{1}{2} \left(3+\frac{2 i}{b n}\right);\frac{1}{2} \left(5+\frac{2 i}{b n}\right);-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{x^2 (2-3 i b n)}",1,"((-2*I)*E^(I*a)*(-2*I + b*n)*(c*x^n)^(I*b)*Hypergeometric2F1[1, 1/2 + I/(b*n), 3/2 + I/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))] + Sec[a + b*Log[c*x^n]]*(2 + b*n*Tan[a + b*Log[c*x^n]]))/(2*b^2*n^2*x^2)","A",0
254,1,204,79,12.4087289,"\int x \sec ^4\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x*Sec[a + b*Log[c*x^n]]^4,x]","\frac{x^2 \left(-2 i \left(b^2 n^2+1\right) \, _2F_1\left(1,-\frac{i}{b n};1-\frac{i}{b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)+\sec ^2\left(a+b \log \left(c x^n\right)\right) \left(\tan \left(a+b \log \left(c x^n\right)\right) \left(\left(b^2 n^2+1\right) \cos \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+2 b^2 n^2+1\right)-b n\right)+2 e^{2 i a} (b n+i) \left(c x^n\right)^{2 i b} \, _2F_1\left(1,1-\frac{i}{b n};2-\frac{i}{b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)\right)}{3 b^3 n^3}","\frac{8 e^{4 i a} x^2 \left(c x^n\right)^{4 i b} \, _2F_1\left(4,2-\frac{i}{b n};3-\frac{i}{b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{1+2 i b n}",1,"(x^2*(2*E^((2*I)*a)*(I + b*n)*(c*x^n)^((2*I)*b)*Hypergeometric2F1[1, 1 - I/(b*n), 2 - I/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))] - (2*I)*(1 + b^2*n^2)*Hypergeometric2F1[1, (-I)/(b*n), 1 - I/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))] + Sec[a + b*Log[c*x^n]]^2*(-(b*n) + (1 + 2*b^2*n^2 + (1 + b^2*n^2)*Cos[2*(a + b*Log[c*x^n])])*Tan[a + b*Log[c*x^n]])))/(3*b^3*n^3)","B",0
255,1,213,85,10.5473239,"\int \sec ^4\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[Sec[a + b*Log[c*x^n]]^4,x]","\frac{x \left(-2 i \left(4 b^2 n^2+1\right) \, _2F_1\left(1,-\frac{i}{2 b n};1-\frac{i}{2 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)+\sec ^2\left(a+b \log \left(c x^n\right)\right) \left(\tan \left(a+b \log \left(c x^n\right)\right) \left(\left(4 b^2 n^2+1\right) \cos \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+8 b^2 n^2+1\right)-2 b n\right)+2 e^{2 i a} (2 b n+i) \left(c x^n\right)^{2 i b} \, _2F_1\left(1,1-\frac{i}{2 b n};2-\frac{i}{2 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)\right)}{12 b^3 n^3}","\frac{16 e^{4 i a} x \left(c x^n\right)^{4 i b} \, _2F_1\left(4,\frac{1}{2} \left(4-\frac{i}{b n}\right);\frac{1}{2} \left(6-\frac{i}{b n}\right);-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{1+4 i b n}",1,"(x*(2*E^((2*I)*a)*(I + 2*b*n)*(c*x^n)^((2*I)*b)*Hypergeometric2F1[1, 1 - (I/2)/(b*n), 2 - (I/2)/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))] - (2*I)*(1 + 4*b^2*n^2)*Hypergeometric2F1[1, (-1/2*I)/(b*n), 1 - (I/2)/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))] + Sec[a + b*Log[c*x^n]]^2*(-2*b*n + (1 + 8*b^2*n^2 + (1 + 4*b^2*n^2)*Cos[2*(a + b*Log[c*x^n])])*Tan[a + b*Log[c*x^n]])))/(12*b^3*n^3)","B",0
256,1,36,42,0.1085872,"\int \frac{\sec ^4\left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Sec[a + b*Log[c*x^n]]^4/x,x]","\frac{\frac{1}{3} \tan ^3\left(a+b \log \left(c x^n\right)\right)+\tan \left(a+b \log \left(c x^n\right)\right)}{b n}","\frac{\tan ^3\left(a+b \log \left(c x^n\right)\right)}{3 b n}+\frac{\tan \left(a+b \log \left(c x^n\right)\right)}{b n}",1,"(Tan[a + b*Log[c*x^n]] + Tan[a + b*Log[c*x^n]]^3/3)/(b*n)","A",1
257,1,215,87,9.3970698,"\int \frac{\sec ^4\left(a+b \log \left(c x^n\right)\right)}{x^2} \, dx","Integrate[Sec[a + b*Log[c*x^n]]^4/x^2,x]","\frac{-2 i \left(4 b^2 n^2+1\right) \, _2F_1\left(1,\frac{i}{2 b n};1+\frac{i}{2 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)+\sec ^2\left(a+b \log \left(c x^n\right)\right) \left(\tan \left(a+b \log \left(c x^n\right)\right) \left(\left(4 b^2 n^2+1\right) \cos \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+8 b^2 n^2+1\right)+2 b n\right)-2 e^{2 i a} (2 b n-i) \left(c x^n\right)^{2 i b} \, _2F_1\left(1,1+\frac{i}{2 b n};2+\frac{i}{2 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)}{12 b^3 n^3 x}","-\frac{16 e^{4 i a} \left(c x^n\right)^{4 i b} \, _2F_1\left(4,\frac{1}{2} \left(4+\frac{i}{b n}\right);\frac{1}{2} \left(6+\frac{i}{b n}\right);-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{x (1-4 i b n)}",1,"(-2*E^((2*I)*a)*(-I + 2*b*n)*(c*x^n)^((2*I)*b)*Hypergeometric2F1[1, 1 + (I/2)/(b*n), 2 + (I/2)/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))] - (2*I)*(1 + 4*b^2*n^2)*Hypergeometric2F1[1, (I/2)/(b*n), 1 + (I/2)/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))] + Sec[a + b*Log[c*x^n]]^2*(2*b*n + (1 + 8*b^2*n^2 + (1 + 4*b^2*n^2)*Cos[2*(a + b*Log[c*x^n])])*Tan[a + b*Log[c*x^n]]))/(12*b^3*n^3*x)","B",0
258,1,203,79,9.2447954,"\int \frac{\sec ^4\left(a+b \log \left(c x^n\right)\right)}{x^3} \, dx","Integrate[Sec[a + b*Log[c*x^n]]^4/x^3,x]","\frac{-2 i \left(b^2 n^2+1\right) \, _2F_1\left(1,\frac{i}{b n};1+\frac{i}{b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)+\sec ^2\left(a+b \log \left(c x^n\right)\right) \left(\tan \left(a+b \log \left(c x^n\right)\right) \left(\left(b^2 n^2+1\right) \cos \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+2 b^2 n^2+1\right)+b n\right)-2 e^{2 i a} (b n-i) \left(c x^n\right)^{2 i b} \, _2F_1\left(1,1+\frac{i}{b n};2+\frac{i}{b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)}{3 b^3 n^3 x^2}","-\frac{8 e^{4 i a} \left(c x^n\right)^{4 i b} \, _2F_1\left(4,2+\frac{i}{b n};3+\frac{i}{b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{x^2 (1-2 i b n)}",1,"(-2*E^((2*I)*a)*(-I + b*n)*(c*x^n)^((2*I)*b)*Hypergeometric2F1[1, 1 + I/(b*n), 2 + I/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))] - (2*I)*(1 + b^2*n^2)*Hypergeometric2F1[1, I/(b*n), 1 + I/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))] + Sec[a + b*Log[c*x^n]]^2*(b*n + (1 + 2*b^2*n^2 + (1 + b^2*n^2)*Cos[2*(a + b*Log[c*x^n])])*Tan[a + b*Log[c*x^n]]))/(3*b^3*n^3*x^2)","B",0
259,1,29,41,0.4564389,"\int \left(-\left(\left(1+b^2 n^2\right) \sec \left(a+b \log \left(c x^n\right)\right)\right)+2 b^2 n^2 \sec ^3\left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[-((1 + b^2*n^2)*Sec[a + b*Log[c*x^n]]) + 2*b^2*n^2*Sec[a + b*Log[c*x^n]]^3,x]","x \left(b n \tan \left(a+b \log \left(c x^n\right)\right)-1\right) \sec \left(a+b \log \left(c x^n\right)\right)","b n x \tan \left(a+b \log \left(c x^n\right)\right) \sec \left(a+b \log \left(c x^n\right)\right)-x \sec \left(a+b \log \left(c x^n\right)\right)",1,"x*Sec[a + b*Log[c*x^n]]*(-1 + b*n*Tan[a + b*Log[c*x^n]])","A",1
260,1,198,110,2.0730609,"\int x^m \sec ^3\left(a+2 \log \left(c x^{\frac{1}{2} \sqrt{-(1+m)^2}}\right)\right) \, dx","Integrate[x^m*Sec[a + 2*Log[c*x^(Sqrt[-(1 + m)^2]/2)]]^3,x]","\frac{x^{m+1} \left((m+1) \cos \left(a+2 \log \left(c x^{\frac{1}{2} \sqrt{-(m+1)^2}}\right)\right)-\sqrt{-(m+1)^2} \sin \left(a+2 \log \left(c x^{\frac{1}{2} \sqrt{-(m+1)^2}}\right)\right)\right)}{2 (m+1)^2 \left(\cos \left(\frac{a}{2}+\log \left(c x^{\frac{1}{2} \sqrt{-(m+1)^2}}\right)\right)-\sin \left(\frac{a}{2}+\log \left(c x^{\frac{1}{2} \sqrt{-(m+1)^2}}\right)\right)\right)^2 \left(\sin \left(\frac{a}{2}+\log \left(c x^{\frac{1}{2} \sqrt{-(m+1)^2}}\right)\right)+\cos \left(\frac{a}{2}+\log \left(c x^{\frac{1}{2} \sqrt{-(m+1)^2}}\right)\right)\right)^2}","\frac{x^{m+1} \sec \left(a+2 \log \left(c x^{\frac{1}{2} \sqrt{-(m+1)^2}}\right)\right)}{2 (m+1)}+\frac{x^{m+1} \tan \left(a+2 \log \left(c x^{\frac{1}{2} \sqrt{-(m+1)^2}}\right)\right) \sec \left(a+2 \log \left(c x^{\frac{1}{2} \sqrt{-(m+1)^2}}\right)\right)}{2 \sqrt{-(m+1)^2}}",1,"(x^(1 + m)*((1 + m)*Cos[a + 2*Log[c*x^(Sqrt[-(1 + m)^2]/2)]] - Sqrt[-(1 + m)^2]*Sin[a + 2*Log[c*x^(Sqrt[-(1 + m)^2]/2)]]))/(2*(1 + m)^2*(Cos[a/2 + Log[c*x^(Sqrt[-(1 + m)^2]/2)]] - Sin[a/2 + Log[c*x^(Sqrt[-(1 + m)^2]/2)]])^2*(Cos[a/2 + Log[c*x^(Sqrt[-(1 + m)^2]/2)]] + Sin[a/2 + Log[c*x^(Sqrt[-(1 + m)^2]/2)]])^2)","A",1
261,1,127,45,0.1742707,"\int x \sec ^3\left(a+2 \log \left(c x^i\right)\right) \, dx","Integrate[x*Sec[a + 2*Log[c*x^I]]^3,x]","-\frac{\sec ^2\left(a+2 \log \left(c x^i\right)\right) \left(i \left(1-2 x^4\right) \sin \left(a+2 \log \left(c x^i\right)-2 i \log (x)\right)+\left(2 x^4+1\right) \cos \left(a+2 \log \left(c x^i\right)-2 i \log (x)\right)\right) \left(i \sin \left(2 \left(a+2 \log \left(c x^i\right)-2 i \log (x)\right)\right)+\cos \left(2 \left(a+2 \log \left(c x^i\right)-2 i \log (x)\right)\right)\right)}{4 x^4}","\frac{e^{i a} x^2 \left(c x^i\right)^{2 i}}{\left(1+e^{2 i a} \left(c x^i\right)^{4 i}\right)^2}",1,"-1/4*(Sec[a + 2*Log[c*x^I]]^2*((1 + 2*x^4)*Cos[a + 2*Log[c*x^I] - (2*I)*Log[x]] + I*(1 - 2*x^4)*Sin[a + 2*Log[c*x^I] - (2*I)*Log[x]])*(Cos[2*(a + 2*Log[c*x^I] - (2*I)*Log[x])] + I*Sin[2*(a + 2*Log[c*x^I] - (2*I)*Log[x])]))/x^4","B",1
262,1,137,58,0.1352002,"\int \sec ^3\left(a+2 \log \left(c x^{\frac{i}{2}}\right)\right) \, dx","Integrate[Sec[a + 2*Log[c*x^(I/2)]]^3,x]","-\frac{\sec ^2\left(a+2 \log \left(c x^{\frac{i}{2}}\right)\right) \left(i \left(1-2 x^2\right) \sin \left(a+2 \log \left(c x^{\frac{i}{2}}\right)-i \log (x)\right)+\left(2 x^2+1\right) \cos \left(a+2 \log \left(c x^{\frac{i}{2}}\right)-i \log (x)\right)\right) \left(i \sin \left(2 \left(a+2 \log \left(c x^{\frac{i}{2}}\right)-i \log (x)\right)\right)+\cos \left(2 \left(a+2 \log \left(c x^{\frac{i}{2}}\right)-i \log (x)\right)\right)\right)}{2 x^2}","\frac{1}{2} x \sec \left(a+2 \log \left(c x^{\frac{i}{2}}\right)\right)-\frac{1}{2} i x \tan \left(a+2 \log \left(c x^{\frac{i}{2}}\right)\right) \sec \left(a+2 \log \left(c x^{\frac{i}{2}}\right)\right)",1,"-1/2*(Sec[a + 2*Log[c*x^(I/2)]]^2*((1 + 2*x^2)*Cos[a + 2*Log[c*x^(I/2)] - I*Log[x]] + I*(1 - 2*x^2)*Sin[a + 2*Log[c*x^(I/2)] - I*Log[x]])*(Cos[2*(a + 2*Log[c*x^(I/2)] - I*Log[x])] + I*Sin[2*(a + 2*Log[c*x^(I/2)] - I*Log[x])]))/x^2","B",1
263,1,139,48,0.1658241,"\int \sec ^3\left(a+2 \log \left(c x^{-\frac{i}{2}}\right)\right) \, dx","Integrate[Sec[a + 2*Log[c/x^(I/2)]]^3,x]","\frac{\sec ^2\left(a+2 \log \left(c x^{-\frac{i}{2}}\right)\right) \left(i \left(2 x^2-1\right) \sin \left(a+2 \log \left(c x^{-\frac{i}{2}}\right)+i \log (x)\right)+\left(2 x^2+1\right) \cos \left(a+2 \log \left(c x^{-\frac{i}{2}}\right)+i \log (x)\right)\right) \left(2 i \sin \left(2 \left(a+2 \log \left(c x^{-\frac{i}{2}}\right)+i \log (x)\right)\right)-2 \cos \left(2 \left(a+2 \log \left(c x^{-\frac{i}{2}}\right)+i \log (x)\right)\right)\right)}{4 x^2}","\frac{2 e^{3 i a} x \left(c x^{-\frac{i}{2}}\right)^{6 i}}{\left(1+e^{2 i a} \left(c x^{-\frac{i}{2}}\right)^{4 i}\right)^2}",1,"(Sec[a + 2*Log[c/x^(I/2)]]^2*((1 + 2*x^2)*Cos[a + 2*Log[c/x^(I/2)] + I*Log[x]] + I*(-1 + 2*x^2)*Sin[a + 2*Log[c/x^(I/2)] + I*Log[x]])*(-2*Cos[2*(a + 2*Log[c/x^(I/2)] + I*Log[x])] + (2*I)*Sin[2*(a + 2*Log[c/x^(I/2)] + I*Log[x])]))/(4*x^2)","B",1
264,1,67,95,0.8649,"\int \sec ^p\left(a+\frac{i \log \left(c x^n\right)}{n (-2+p)}\right) \, dx","Integrate[Sec[a + (I*Log[c*x^n])/(n*(-2 + p))]^p,x]","\frac{e^{-2 i a} (p-2) x \left(\left(c x^n\right)^{\frac{2}{n (p-2)}}+e^{2 i a}\right) \sec ^p\left(a+\frac{i \log \left(c x^n\right)}{n (p-2)}\right)}{2 (p-1)}","\frac{e^{-2 i a} (2-p) x \left(c x^n\right)^{-\frac{2}{n (2-p)}} \left(1+e^{2 i a} \left(c x^n\right)^{\frac{2}{n (2-p)}}\right) \sec ^p\left(a-\frac{i \log \left(c x^n\right)}{n (2-p)}\right)}{2 (1-p)}",1,"((-2 + p)*x*(E^((2*I)*a) + (c*x^n)^(2/(n*(-2 + p))))*Sec[a + (I*Log[c*x^n])/(n*(-2 + p))]^p)/(2*E^((2*I)*a)*(-1 + p))","A",1
265,1,62,70,0.8436099,"\int \sec ^p\left(a-\frac{i \log \left(c x^n\right)}{n (-2+p)}\right) \, dx","Integrate[Sec[a - (I*Log[c*x^n])/(n*(-2 + p))]^p,x]","\frac{(p-2) x \left(1+e^{2 i a} \left(c x^n\right)^{\frac{2}{n (p-2)}}\right) \sec ^p\left(a-\frac{i \log \left(c x^n\right)}{n (p-2)}\right)}{2 (p-1)}","\frac{(2-p) x \left(1+e^{2 i a} \left(c x^n\right)^{-\frac{2}{n (2-p)}}\right) \sec ^p\left(a+\frac{i \log \left(c x^n\right)}{n (2-p)}\right)}{2 (1-p)}",1,"((-2 + p)*x*(1 + E^((2*I)*a)*(c*x^n)^(2/(n*(-2 + p))))*Sec[a - (I*Log[c*x^n])/(n*(-2 + p))]^p)/(2*(-1 + p))","A",1
266,1,99,109,0.4640847,"\int \sqrt{\sec \left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[Sqrt[Sec[a + b*Log[c*x^n]]],x]","-\frac{2 i x \left(1+e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \, _2F_1\left(1,\frac{3}{4}-\frac{i}{2 b n};\frac{5}{4}-\frac{i}{2 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \sqrt{\sec \left(a+b \log \left(c x^n\right)\right)}}{b n-2 i}","\frac{2 x \sqrt{1+e^{2 i a} \left(c x^n\right)^{2 i b}} \, _2F_1\left(\frac{1}{2},\frac{1}{4} \left(1-\frac{2 i}{b n}\right);\frac{1}{4} \left(5-\frac{2 i}{b n}\right);-e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sqrt{\sec \left(a+b \log \left(c x^n\right)\right)}}{2+i b n}",1,"((-2*I)*(1 + E^((2*I)*(a + b*Log[c*x^n])))*x*Hypergeometric2F1[1, 3/4 - (I/2)/(b*n), 5/4 - (I/2)/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))]*Sqrt[Sec[a + b*Log[c*x^n]]])/(-2*I + b*n)","A",0
267,1,54,54,0.1171078,"\int \frac{\sqrt{\sec \left(a+b \log \left(c x^n\right)\right)}}{x} \, dx","Integrate[Sqrt[Sec[a + b*Log[c*x^n]]]/x,x]","\frac{2 \sqrt{\sec \left(a+b \log \left(c x^n\right)\right)} \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)} F\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)\right)\right|2\right)}{b n}","\frac{2 \sqrt{\sec \left(a+b \log \left(c x^n\right)\right)} \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)} F\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)\right)\right|2\right)}{b n}",1,"(2*Sqrt[Cos[a + b*Log[c*x^n]]]*EllipticF[(a + b*Log[c*x^n])/2, 2]*Sqrt[Sec[a + b*Log[c*x^n]]])/(b*n)","A",1
268,1,415,109,5.7745811,"\int \sec ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[Sec[a + b*Log[c*x^n]]^(3/2),x]","\frac{\sqrt{2} x^{1-i b n} \left((3 b n-2 i) \left((-b n+2 i) \sqrt{\frac{e^{i a} \left(c x^n\right)^{i b}}{1+e^{2 i a} \left(c x^n\right)^{2 i b}}} \sqrt{1+e^{2 i a} \left(c x^n\right)^{2 i b}} \, _2F_1\left(\frac{1}{2},-\frac{b n+2 i}{4 b n};\frac{3}{4}-\frac{i}{2 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right)+\sqrt{2} x^{i b n} (b n \cos (b n \log (x))-2 \sin (b n \log (x))) \sqrt{\sec \left(a+b \log \left(c x^n\right)\right)}\right)-\left(b^2 n^2+4\right) x^{2 i b n} \sqrt{\frac{e^{i a} \left(c x^n\right)^{i b}}{1+e^{2 i a} \left(c x^n\right)^{2 i b}}} \sqrt{1+e^{2 i a} \left(c x^n\right)^{2 i b}} \, _2F_1\left(\frac{1}{2},\frac{3}{4}-\frac{i}{2 b n};\frac{7}{4}-\frac{i}{2 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right)\right)}{b n (3 b n-2 i) \left(b n \sin \left(a+b \log \left(c x^n\right)-b n \log (x)\right)-2 \cos \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)}","\frac{2 x \left(1+e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{3/2} \, _2F_1\left(\frac{3}{2},\frac{1}{4} \left(3-\frac{2 i}{b n}\right);\frac{1}{4} \left(7-\frac{2 i}{b n}\right);-e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sec ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}{2+3 i b n}",1,"(Sqrt[2]*x^(1 - I*b*n)*(-((4 + b^2*n^2)*x^((2*I)*b*n)*Sqrt[(E^(I*a)*(c*x^n)^(I*b))/(1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))]*Sqrt[1 + E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Hypergeometric2F1[1/2, 3/4 - (I/2)/(b*n), 7/4 - (I/2)/(b*n), -(E^((2*I)*a)*(c*x^n)^((2*I)*b))]) + (-2*I + 3*b*n)*((2*I - b*n)*Sqrt[(E^(I*a)*(c*x^n)^(I*b))/(1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))]*Sqrt[1 + E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Hypergeometric2F1[1/2, -1/4*(2*I + b*n)/(b*n), 3/4 - (I/2)/(b*n), -(E^((2*I)*a)*(c*x^n)^((2*I)*b))] + Sqrt[2]*x^(I*b*n)*Sqrt[Sec[a + b*Log[c*x^n]]]*(b*n*Cos[b*n*Log[x]] - 2*Sin[b*n*Log[x]]))))/(b*n*(-2*I + 3*b*n)*(-2*Cos[a - b*n*Log[x] + b*Log[c*x^n]] + b*n*Sin[a - b*n*Log[x] + b*Log[c*x^n]]))","B",0
269,1,68,89,0.150063,"\int \frac{\sec ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Sec[a + b*Log[c*x^n]]^(3/2)/x,x]","\frac{2 \sqrt{\sec \left(a+b \log \left(c x^n\right)\right)} \left(\sin \left(a+b \log \left(c x^n\right)\right)-\sqrt{\cos \left(a+b \log \left(c x^n\right)\right)} E\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)\right)\right|2\right)\right)}{b n}","\frac{2 \sin \left(a+b \log \left(c x^n\right)\right) \sqrt{\sec \left(a+b \log \left(c x^n\right)\right)}}{b n}-\frac{2 \sqrt{\sec \left(a+b \log \left(c x^n\right)\right)} \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)} E\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)\right)\right|2\right)}{b n}",1,"(2*Sqrt[Sec[a + b*Log[c*x^n]]]*(-(Sqrt[Cos[a + b*Log[c*x^n]]]*EllipticE[(a + b*Log[c*x^n])/2, 2]) + Sin[a + b*Log[c*x^n]]))/(b*n)","A",1
270,1,124,109,1.3836993,"\int \sec ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[Sec[a + b*Log[c*x^n]]^(5/2),x]","\frac{2 x \sqrt{\sec \left(a+b \log \left(c x^n\right)\right)} \left((2-i b n) \left(1+e^{2 i a} \left(c x^n\right)^{2 i b}\right) \, _2F_1\left(1,\frac{3}{4}-\frac{i}{2 b n};\frac{5}{4}-\frac{i}{2 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)+b n \tan \left(a+b \log \left(c x^n\right)\right)-2\right)}{3 b^2 n^2}","\frac{2 x \left(1+e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{5/2} \, _2F_1\left(\frac{5}{2},\frac{1}{4} \left(5-\frac{2 i}{b n}\right);\frac{1}{4} \left(9-\frac{2 i}{b n}\right);-e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sec ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right)}{2+5 i b n}",1,"(2*x*Sqrt[Sec[a + b*Log[c*x^n]]]*(-2 + (2 - I*b*n)*(1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))*Hypergeometric2F1[1, 3/4 - (I/2)/(b*n), 5/4 - (I/2)/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))] + b*n*Tan[a + b*Log[c*x^n]]))/(3*b^2*n^2)","A",0
271,1,69,93,0.1603599,"\int \frac{\sec ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Sec[a + b*Log[c*x^n]]^(5/2)/x,x]","\frac{2 \sec ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right) \left(\sin \left(a+b \log \left(c x^n\right)\right)+\cos ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right) F\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)\right)\right|2\right)\right)}{3 b n}","\frac{2 \sin \left(a+b \log \left(c x^n\right)\right) \sec ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}{3 b n}+\frac{2 \sqrt{\sec \left(a+b \log \left(c x^n\right)\right)} \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)} F\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)\right)\right|2\right)}{3 b n}",1,"(2*Sec[a + b*Log[c*x^n]]^(3/2)*(Cos[a + b*Log[c*x^n]]^(3/2)*EllipticF[(a + b*Log[c*x^n])/2, 2] + Sin[a + b*Log[c*x^n]]))/(3*b*n)","A",1
272,1,380,110,4.3092592,"\int \frac{1}{\sqrt{\sec \left(a+b \log \left(c x^n\right)\right)}} \, dx","Integrate[1/Sqrt[Sec[a + b*Log[c*x^n]]],x]","-\frac{2 x \cos \left(a+b \log \left(c x^n\right)-b n \log (x)\right)}{\sqrt{\sec \left(a+b \log \left(c x^n\right)\right)} \left(b n \sin \left(a+b \log \left(c x^n\right)-b n \log (x)\right)-2 \cos \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)}+\frac{2 e^{2 i a} b n x \left(c x^n\right)^{2 i b} \left((3 b n-2 i) \, _2F_1\left(\frac{1}{2},-\frac{b n+2 i}{4 b n};\frac{3}{4}-\frac{i}{2 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right)+(b n+2 i) x^{2 i b n} \, _2F_1\left(\frac{1}{2},\frac{3}{4}-\frac{i}{2 b n};\frac{7}{4}-\frac{i}{2 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right)\right)}{(b n+2 i) (3 b n-2 i) \sqrt{1+e^{2 i a} \left(c x^n\right)^{2 i b}} \sqrt{\frac{e^{i a} \left(c x^n\right)^{i b}}{2+2 e^{2 i a} \left(c x^n\right)^{2 i b}}} \left((-2+i b n) x^{2 i b n}-i e^{2 i a} (b n-2 i) \left(c x^n\right)^{2 i b}\right)}","\frac{2 x \, _2F_1\left(-\frac{1}{2},-\frac{b n+2 i}{4 b n};\frac{1}{4} \left(3-\frac{2 i}{b n}\right);-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{(2-i b n) \sqrt{1+e^{2 i a} \left(c x^n\right)^{2 i b}} \sqrt{\sec \left(a+b \log \left(c x^n\right)\right)}}",1,"(2*b*E^((2*I)*a)*n*x*(c*x^n)^((2*I)*b)*((2*I + b*n)*x^((2*I)*b*n)*Hypergeometric2F1[1/2, 3/4 - (I/2)/(b*n), 7/4 - (I/2)/(b*n), -(E^((2*I)*a)*(c*x^n)^((2*I)*b))] + (-2*I + 3*b*n)*Hypergeometric2F1[1/2, -1/4*(2*I + b*n)/(b*n), 3/4 - (I/2)/(b*n), -(E^((2*I)*a)*(c*x^n)^((2*I)*b))]))/((2*I + b*n)*(-2*I + 3*b*n)*Sqrt[1 + E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Sqrt[(E^(I*a)*(c*x^n)^(I*b))/(2 + 2*E^((2*I)*a)*(c*x^n)^((2*I)*b))]*((-2 + I*b*n)*x^((2*I)*b*n) - I*E^((2*I)*a)*(-2*I + b*n)*(c*x^n)^((2*I)*b))) - (2*x*Cos[a - b*n*Log[x] + b*Log[c*x^n]])/(Sqrt[Sec[a + b*Log[c*x^n]]]*(-2*Cos[a - b*n*Log[x] + b*Log[c*x^n]] + b*n*Sin[a - b*n*Log[x] + b*Log[c*x^n]]))","B",0
273,1,54,54,0.1138087,"\int \frac{1}{x \sqrt{\sec \left(a+b \log \left(c x^n\right)\right)}} \, dx","Integrate[1/(x*Sqrt[Sec[a + b*Log[c*x^n]]]),x]","\frac{2 E\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)\right)\right|2\right)}{b n \sqrt{\sec \left(a+b \log \left(c x^n\right)\right)} \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)}}","\frac{2 \sqrt{\sec \left(a+b \log \left(c x^n\right)\right)} \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)} E\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)\right)\right|2\right)}{b n}",1,"(2*EllipticE[(a + b*Log[c*x^n])/2, 2])/(b*n*Sqrt[Cos[a + b*Log[c*x^n]]]*Sqrt[Sec[a + b*Log[c*x^n]]])","A",1
274,1,168,109,1.6129562,"\int \frac{1}{\sec ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[Sec[a + b*Log[c*x^n]]^(-3/2),x]","\frac{2 x \left(3 b^2 n^2 \left(1+e^{2 i a} \left(c x^n\right)^{2 i b}\right) \, _2F_1\left(1,\frac{3}{4}-\frac{i}{2 b n};\frac{5}{4}-\frac{i}{2 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \sec ^2\left(a+b \log \left(c x^n\right)\right)+(2+i b n) \left(3 b n \tan \left(a+b \log \left(c x^n\right)\right)+2\right)\right)}{(2+3 i b n) (b n-2 i) (3 b n+2 i) \sec ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}","\frac{2 x \, _2F_1\left(-\frac{3}{2},\frac{1}{4} \left(-3-\frac{2 i}{b n}\right);\frac{1}{4} \left(1-\frac{2 i}{b n}\right);-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{(2-3 i b n) \left(1+e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{3/2} \sec ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}",1,"(2*x*(3*b^2*n^2*(1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))*Hypergeometric2F1[1, 3/4 - (I/2)/(b*n), 5/4 - (I/2)/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))]*Sec[a + b*Log[c*x^n]]^2 + (2 + I*b*n)*(2 + 3*b*n*Tan[a + b*Log[c*x^n]])))/((2 + (3*I)*b*n)*(-2*I + b*n)*(2*I + 3*b*n)*Sec[a + b*Log[c*x^n]]^(3/2))","A",0
275,1,72,93,0.1290782,"\int \frac{1}{x \sec ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[1/(x*Sec[a + b*Log[c*x^n]]^(3/2)),x]","\frac{\sqrt{\sec \left(a+b \log \left(c x^n\right)\right)} \left(\sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+2 \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)} F\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)\right)\right|2\right)\right)}{3 b n}","\frac{2 \sin \left(a+b \log \left(c x^n\right)\right)}{3 b n \sqrt{\sec \left(a+b \log \left(c x^n\right)\right)}}+\frac{2 \sqrt{\sec \left(a+b \log \left(c x^n\right)\right)} \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)} F\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)\right)\right|2\right)}{3 b n}",1,"(Sqrt[Sec[a + b*Log[c*x^n]]]*(2*Sqrt[Cos[a + b*Log[c*x^n]]]*EllipticF[(a + b*Log[c*x^n])/2, 2] + Sin[2*(a + b*Log[c*x^n])]))/(3*b*n)","A",1
276,1,867,110,8.6789725,"\int \frac{1}{\sec ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[Sec[a + b*Log[c*x^n]]^(-5/2),x]","\frac{30 b^3 e^{2 i \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)} x \left((b n+2 i) \, _2F_1\left(\frac{1}{2},\frac{3}{4}-\frac{i}{2 b n};\frac{7}{4}-\frac{i}{2 b n};-e^{2 i \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)} x^{2 i b n}\right) x^{2 i b n}+(3 b n-2 i) \, _2F_1\left(\frac{1}{2},-\frac{b n+2 i}{4 b n};\frac{3}{4}-\frac{i}{2 b n};-e^{2 i \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)} x^{2 i b n}\right)\right) n^3}{(2-5 i b n) (b n+2 i) (3 b n-2 i) (5 b n-2 i) \left(-b n+e^{2 i \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)} (b n-2 i)-2 i\right) \sqrt{e^{2 i \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)} x^{2 i b n}+1} \sqrt{\frac{e^{i \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)} x^{i b n}}{2 e^{2 i \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)} x^{2 i b n}+2}}}+\sqrt{\sec \left(a+b n \log (x)+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)} \left(-\frac{x \cos (b n \log (x)) \left(55 b^2 n^2+65 b^2 \cos \left(2 \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right) n^2+4 b \sin \left(2 \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right) n+12 \cos \left(2 \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right)+12\right)}{4 (5 b n-2 i) (5 b n+2 i) \left(b n \sin \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)-2 \cos \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right)}+\frac{x \sin (b n \log (x)) \left(65 b^2 \sin \left(2 \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right) n^2-16 b n-4 b \cos \left(2 \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right) n+12 \sin \left(2 \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right)\right)}{4 (5 b n-2 i) (5 b n+2 i) \left(b n \sin \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)-2 \cos \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right)}+\frac{x \sin (3 b n \log (x)) \left(5 b n \cos \left(3 \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right)-2 \sin \left(3 \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right)\right)}{2 (5 b n-2 i) (5 b n+2 i)}+\frac{x \cos (3 b n \log (x)) \left(2 \cos \left(3 \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right)+5 b n \sin \left(3 \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right)\right)}{2 (5 b n-2 i) (5 b n+2 i)}\right)","\frac{2 x \, _2F_1\left(-\frac{5}{2},\frac{1}{4} \left(-5-\frac{2 i}{b n}\right);-\frac{b n+2 i}{4 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{(2-5 i b n) \left(1+e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{5/2} \sec ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right)}",1,"(30*b^3*E^((2*I)*(a + b*(-(n*Log[x]) + Log[c*x^n])))*n^3*x*((2*I + b*n)*x^((2*I)*b*n)*Hypergeometric2F1[1/2, 3/4 - (I/2)/(b*n), 7/4 - (I/2)/(b*n), -(E^((2*I)*(a + b*(-(n*Log[x]) + Log[c*x^n])))*x^((2*I)*b*n))] + (-2*I + 3*b*n)*Hypergeometric2F1[1/2, -1/4*(2*I + b*n)/(b*n), 3/4 - (I/2)/(b*n), -(E^((2*I)*(a + b*(-(n*Log[x]) + Log[c*x^n])))*x^((2*I)*b*n))]))/((2 - (5*I)*b*n)*(2*I + b*n)*(-2*I + 3*b*n)*(-2*I + 5*b*n)*(-2*I - b*n + E^((2*I)*(a + b*(-(n*Log[x]) + Log[c*x^n])))*(-2*I + b*n))*Sqrt[1 + E^((2*I)*(a + b*(-(n*Log[x]) + Log[c*x^n])))*x^((2*I)*b*n)]*Sqrt[(E^(I*(a + b*(-(n*Log[x]) + Log[c*x^n])))*x^(I*b*n))/(2 + 2*E^((2*I)*(a + b*(-(n*Log[x]) + Log[c*x^n])))*x^((2*I)*b*n))]) + Sqrt[Sec[a + b*n*Log[x] + b*(-(n*Log[x]) + Log[c*x^n])]]*(-1/4*(x*Cos[b*n*Log[x]]*(12 + 55*b^2*n^2 + 12*Cos[2*(a + b*(-(n*Log[x]) + Log[c*x^n]))] + 65*b^2*n^2*Cos[2*(a + b*(-(n*Log[x]) + Log[c*x^n]))] + 4*b*n*Sin[2*(a + b*(-(n*Log[x]) + Log[c*x^n]))]))/((-2*I + 5*b*n)*(2*I + 5*b*n)*(-2*Cos[a + b*(-(n*Log[x]) + Log[c*x^n])] + b*n*Sin[a + b*(-(n*Log[x]) + Log[c*x^n])])) + (x*Sin[b*n*Log[x]]*(-16*b*n - 4*b*n*Cos[2*(a + b*(-(n*Log[x]) + Log[c*x^n]))] + 12*Sin[2*(a + b*(-(n*Log[x]) + Log[c*x^n]))] + 65*b^2*n^2*Sin[2*(a + b*(-(n*Log[x]) + Log[c*x^n]))]))/(4*(-2*I + 5*b*n)*(2*I + 5*b*n)*(-2*Cos[a + b*(-(n*Log[x]) + Log[c*x^n])] + b*n*Sin[a + b*(-(n*Log[x]) + Log[c*x^n])])) + (x*Sin[3*b*n*Log[x]]*(5*b*n*Cos[3*(a + b*(-(n*Log[x]) + Log[c*x^n]))] - 2*Sin[3*(a + b*(-(n*Log[x]) + Log[c*x^n]))]))/(2*(-2*I + 5*b*n)*(2*I + 5*b*n)) + (x*Cos[3*b*n*Log[x]]*(2*Cos[3*(a + b*(-(n*Log[x]) + Log[c*x^n]))] + 5*b*n*Sin[3*(a + b*(-(n*Log[x]) + Log[c*x^n]))]))/(2*(-2*I + 5*b*n)*(2*I + 5*b*n)))","B",0
277,1,83,93,0.176843,"\int \frac{1}{x \sec ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[1/(x*Sec[a + b*Log[c*x^n]]^(5/2)),x]","\frac{\sqrt{\sec \left(a+b \log \left(c x^n\right)\right)} \left(\sin \left(a+b \log \left(c x^n\right)\right)+\sin \left(3 \left(a+b \log \left(c x^n\right)\right)\right)+12 \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)} E\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)\right)\right|2\right)\right)}{10 b n}","\frac{2 \sin \left(a+b \log \left(c x^n\right)\right)}{5 b n \sec ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}+\frac{6 \sqrt{\sec \left(a+b \log \left(c x^n\right)\right)} \sqrt{\cos \left(a+b \log \left(c x^n\right)\right)} E\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)\right)\right|2\right)}{5 b n}",1,"(Sqrt[Sec[a + b*Log[c*x^n]]]*(12*Sqrt[Cos[a + b*Log[c*x^n]]]*EllipticE[(a + b*Log[c*x^n])/2, 2] + Sin[a + b*Log[c*x^n]] + Sin[3*(a + b*Log[c*x^n])]))/(10*b*n)","A",1
278,1,134,102,5.6035,"\int x^m \sec ^3\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x^m*Sec[a + b*Log[c*x^n]]^3,x]","\frac{x^{m+1} \left(-2 \sec \left(a+b \log \left(c x^n\right)\right) \left(-b n \tan \left(a+b \log \left(c x^n\right)\right)+m+1\right)+4 e^{i a} (-i b n+m+1) \left(c x^n\right)^{i b} \, _2F_1\left(1,\frac{1}{2}-\frac{i (m+1)}{2 b n};\frac{3}{2}-\frac{i (m+1)}{2 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)\right)}{4 b^2 n^2}","\frac{8 e^{3 i a} x^{m+1} \left(c x^n\right)^{3 i b} \, _2F_1\left(3,-\frac{i (m+1)-3 b n}{2 b n};-\frac{i (m+1)-5 b n}{2 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{3 i b n+m+1}",1,"(x^(1 + m)*(4*E^(I*a)*(1 + m - I*b*n)*(c*x^n)^(I*b)*Hypergeometric2F1[1, 1/2 - ((I/2)*(1 + m))/(b*n), 3/2 - ((I/2)*(1 + m))/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))] - 2*Sec[a + b*Log[c*x^n]]*(1 + m - b*n*Tan[a + b*Log[c*x^n]])))/(4*b^2*n^2)","A",0
279,1,482,102,17.1840699,"\int x^m \sec ^2\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x^m*Sec[a + b*Log[c*x^n]]^2,x]","\frac{x^{m+1} \sin (b n \log (x)) \sec \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right) \sec \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)+b n \log (x)\right)}{b n}-\frac{(m+1) \sec \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right) \left(\frac{x^{m+1} \sin (b n \log (x)) \sec \left(a+b \log \left(c x^n\right)\right)}{m+1}-\frac{i \cos \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right) \exp \left(-\frac{(2 m+1) \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)}{b n}\right) \left((2 i b n+m+1) \left(-\exp \left(\frac{2 a m+a+b (2 m+1) \left(\log \left(c x^n\right)-n \log (x)\right)+b (m+1) n \log (x)}{b n}\right)\right) \, _2F_1\left(1,-\frac{i (m+1)}{2 b n};1-\frac{i (m+1)}{2 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)+(m+1) \exp \left(\frac{a (2 i b n+2 m+1)}{b n}+\frac{(2 i b n+2 m+1) \left(\log \left(c x^n\right)-n \log (x)\right)}{n}+\log (x) (2 i b n+m+1)\right) \, _2F_1\left(1,-\frac{i (m+2 i b n+1)}{2 b n};-\frac{i (m+4 i b n+1)}{2 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)-i (2 i b n+m+1) \tan \left(a+b \log \left(c x^n\right)\right) \exp \left(\frac{2 a m+a+b (2 m+1) \left(\log \left(c x^n\right)-n \log (x)\right)+b (m+1) n \log (x)}{b n}\right)\right)}{(m+1) (2 i b n+m+1)}\right)}{b n}","\frac{4 e^{2 i a} x^{m+1} \left(c x^n\right)^{2 i b} \, _2F_1\left(2,-\frac{i (m+1)-2 b n}{2 b n};-\frac{i (m+1)-4 b n}{2 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{2 i b n+m+1}",1,"(x^(1 + m)*Sec[a + b*(-(n*Log[x]) + Log[c*x^n])]*Sec[a + b*n*Log[x] + b*(-(n*Log[x]) + Log[c*x^n])]*Sin[b*n*Log[x]])/(b*n) - ((1 + m)*Sec[a + b*(-(n*Log[x]) + Log[c*x^n])]*((x^(1 + m)*Sec[a + b*Log[c*x^n]]*Sin[b*n*Log[x]])/(1 + m) - (I*Cos[a + b*(-(n*Log[x]) + Log[c*x^n])]*(-(E^((a + 2*a*m + b*(1 + m)*n*Log[x] + b*(1 + 2*m)*(-(n*Log[x]) + Log[c*x^n]))/(b*n))*(1 + m + (2*I)*b*n)*Hypergeometric2F1[1, ((-1/2*I)*(1 + m))/(b*n), 1 - ((I/2)*(1 + m))/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))]) + E^((a*(1 + 2*m + (2*I)*b*n))/(b*n) + (1 + m + (2*I)*b*n)*Log[x] + ((1 + 2*m + (2*I)*b*n)*(-(n*Log[x]) + Log[c*x^n]))/n)*(1 + m)*Hypergeometric2F1[1, ((-1/2*I)*(1 + m + (2*I)*b*n))/(b*n), ((-1/2*I)*(1 + m + (4*I)*b*n))/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))] - I*E^((a + 2*a*m + b*(1 + m)*n*Log[x] + b*(1 + 2*m)*(-(n*Log[x]) + Log[c*x^n]))/(b*n))*(1 + m + (2*I)*b*n)*Tan[a + b*Log[c*x^n]]))/(E^(((1 + 2*m)*(a + b*(-(n*Log[x]) + Log[c*x^n])))/(b*n))*(1 + m)*(1 + m + (2*I)*b*n))))/(b*n)","B",0
280,1,94,103,0.2063148,"\int x^m \sec \left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x^m*Sec[a + b*Log[c*x^n]],x]","\frac{2 e^{i a} x^{m+1} \left(c x^n\right)^{i b} \, _2F_1\left(1,\frac{1}{2}-\frac{i (m+1)}{2 b n};\frac{3}{2}-\frac{i (m+1)}{2 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)}{i b n+m+1}","\frac{2 e^{i a} x^{m+1} \left(c x^n\right)^{i b} \, _2F_1\left(1,-\frac{i m-b n+i}{2 b n};-\frac{i (m+1)-3 b n}{2 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{i b n+m+1}",1,"(2*E^(I*a)*x^(1 + m)*(c*x^n)^(I*b)*Hypergeometric2F1[1, 1/2 - ((I/2)*(1 + m))/(b*n), 3/2 - ((I/2)*(1 + m))/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))])/(1 + m + I*b*n)","A",1
281,1,182,130,2.1447184,"\int x^m \sec ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x^m*Sec[a + b*Log[c*x^n]]^(5/2),x]","\frac{2 x^{m+1} \sqrt{\sec \left(a+b \log \left(c x^n\right)\right)} \left(\left(b^2 n^2+4 m^2+8 m+4\right) \left(1+e^{2 i a} \left(c x^n\right)^{2 i b}\right) \, _2F_1\left(1,-\frac{2 i m-3 b n+2 i}{4 b n};-\frac{2 i m-5 b n+2 i}{4 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)-(i b n+2 m+2) \left(-b n \tan \left(a+b \log \left(c x^n\right)\right)+2 m+2\right)\right)}{3 b^2 n^2 (i b n+2 m+2)}","\frac{2 x^{m+1} \left(1+e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{5/2} \, _2F_1\left(\frac{5}{2},-\frac{2 i m-5 b n+2 i}{4 b n};-\frac{2 i m-9 b n+2 i}{4 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sec ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right)}{5 i b n+2 m+2}",1,"(2*x^(1 + m)*Sqrt[Sec[a + b*Log[c*x^n]]]*((4 + 8*m + 4*m^2 + b^2*n^2)*(1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))*Hypergeometric2F1[1, -1/4*(2*I + (2*I)*m - 3*b*n)/(b*n), -1/4*(2*I + (2*I)*m - 5*b*n)/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))] - (2 + 2*m + I*b*n)*(2 + 2*m - b*n*Tan[a + b*Log[c*x^n]])))/(3*b^2*n^2*(2 + 2*m + I*b*n))","A",0
282,1,470,130,9.4861491,"\int x^m \sec ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x^m*Sec[a + b*Log[c*x^n]]^(3/2),x]","\frac{\sqrt{2} x^{-i b n+m+1} \left((3 i b n+2 m+2) \left((i b n+2 m+2) \sqrt{\frac{e^{i a} \left(c x^n\right)^{i b}}{1+e^{2 i a} \left(c x^n\right)^{2 i b}}} \sqrt{1+e^{2 i a} \left(c x^n\right)^{2 i b}} \, _2F_1\left(\frac{1}{2},-\frac{2 i m+b n+2 i}{4 b n};-\frac{2 i m-3 b n+2 i}{4 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right)-i \sqrt{2} x^{i b n} \sqrt{\sec \left(a+b \log \left(c x^n\right)\right)} (b n \cos (b n \log (x))-2 (m+1) \sin (b n \log (x)))\right)-\left(b^2 n^2+4 m^2+8 m+4\right) x^{2 i b n} \sqrt{\frac{e^{i a} \left(c x^n\right)^{i b}}{1+e^{2 i a} \left(c x^n\right)^{2 i b}}} \sqrt{1+e^{2 i a} \left(c x^n\right)^{2 i b}} \, _2F_1\left(\frac{1}{2},-\frac{i \left(m+\frac{3 i b n}{2}+1\right)}{2 b n};-\frac{2 i m-7 b n+2 i}{4 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right)\right)}{b n (3 b n-2 i m-2 i) \left(b n \sin \left(a+b \log \left(c x^n\right)-b n \log (x)\right)-2 (m+1) \cos \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)}","\frac{2 x^{m+1} \left(1+e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{3/2} \, _2F_1\left(\frac{3}{2},-\frac{2 i m-3 b n+2 i}{4 b n};-\frac{2 i m-7 b n+2 i}{4 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sec ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}{3 i b n+2 m+2}",1,"(Sqrt[2]*x^(1 + m - I*b*n)*(-((4 + 8*m + 4*m^2 + b^2*n^2)*x^((2*I)*b*n)*Sqrt[(E^(I*a)*(c*x^n)^(I*b))/(1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))]*Sqrt[1 + E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Hypergeometric2F1[1/2, ((-1/2*I)*(1 + m + ((3*I)/2)*b*n))/(b*n), -1/4*(2*I + (2*I)*m - 7*b*n)/(b*n), -(E^((2*I)*a)*(c*x^n)^((2*I)*b))]) + (2 + 2*m + (3*I)*b*n)*((2 + 2*m + I*b*n)*Sqrt[(E^(I*a)*(c*x^n)^(I*b))/(1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))]*Sqrt[1 + E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Hypergeometric2F1[1/2, -1/4*(2*I + (2*I)*m + b*n)/(b*n), -1/4*(2*I + (2*I)*m - 3*b*n)/(b*n), -(E^((2*I)*a)*(c*x^n)^((2*I)*b))] - I*Sqrt[2]*x^(I*b*n)*Sqrt[Sec[a + b*Log[c*x^n]]]*(b*n*Cos[b*n*Log[x]] - 2*(1 + m)*Sin[b*n*Log[x]]))))/(b*n*(-2*I - (2*I)*m + 3*b*n)*(-2*(1 + m)*Cos[a - b*n*Log[x] + b*Log[c*x^n]] + b*n*Sin[a - b*n*Log[x] + b*Log[c*x^n]]))","B",0
283,1,119,130,0.7846195,"\int x^m \sqrt{\sec \left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[x^m*Sqrt[Sec[a + b*Log[c*x^n]]],x]","\frac{2 x^{m+1} \left(1+e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \sqrt{\sec \left(a+b \log \left(c x^n\right)\right)} \, _2F_1\left(1,-\frac{2 i m-3 b n+2 i}{4 b n};-\frac{2 i m-5 b n+2 i}{4 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)}{i b n+2 m+2}","\frac{2 x^{m+1} \sqrt{1+e^{2 i a} \left(c x^n\right)^{2 i b}} \, _2F_1\left(\frac{1}{2},-\frac{2 i m-b n+2 i}{4 b n};-\frac{2 i m-5 b n+2 i}{4 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sqrt{\sec \left(a+b \log \left(c x^n\right)\right)}}{i b n+2 m+2}",1,"(2*(1 + E^((2*I)*(a + b*Log[c*x^n])))*x^(1 + m)*Hypergeometric2F1[1, -1/4*(2*I + (2*I)*m - 3*b*n)/(b*n), -1/4*(2*I + (2*I)*m - 5*b*n)/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))]*Sqrt[Sec[a + b*Log[c*x^n]]])/(2 + 2*m + I*b*n)","A",0
284,1,437,129,6.9292407,"\int \frac{x^m}{\sqrt{\sec \left(a+b \log \left(c x^n\right)\right)}} \, dx","Integrate[x^m/Sqrt[Sec[a + b*Log[c*x^n]]],x]","\frac{2 x^{m+1} \cos \left(a+b \log \left(c x^n\right)-b n \log (x)\right)}{\sqrt{\sec \left(a+b \log \left(c x^n\right)\right)} \left(2 (m+1) \cos \left(a+b \log \left(c x^n\right)-b n \log (x)\right)-b n \sin \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)}-\frac{2 b n x^{m+1} e^{2 i \left(a+b \log \left(c x^n\right)-b n \log (x)\right)} \left((3 b n-2 i m-2 i) \, _2F_1\left(\frac{1}{2},-\frac{2 i m+b n+2 i}{4 b n};-\frac{2 i m-3 b n+2 i}{4 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right)+(b n+2 i m+2 i) x^{2 i b n} \, _2F_1\left(\frac{1}{2},-\frac{i \left(m+\frac{3 i b n}{2}+1\right)}{2 b n};-\frac{2 i m-7 b n+2 i}{4 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right)\right)}{(-i b n+2 m+2) (3 i b n+2 m+2) \sqrt{1+e^{2 i a} \left(c x^n\right)^{2 i b}} \sqrt{\frac{e^{i a} \left(c x^n\right)^{i b}}{2+2 e^{2 i a} \left(c x^n\right)^{2 i b}}} \left((i b n+2 m+2) e^{2 i \left(a+b \log \left(c x^n\right)-b n \log (x)\right)}-i b n+2 m+2\right)}","\frac{2 x^{m+1} \, _2F_1\left(-\frac{1}{2},-\frac{2 i m+b n+2 i}{4 b n};-\frac{2 i m-3 b n+2 i}{4 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{(-i b n+2 m+2) \sqrt{1+e^{2 i a} \left(c x^n\right)^{2 i b}} \sqrt{\sec \left(a+b \log \left(c x^n\right)\right)}}",1,"(-2*b*E^((2*I)*(a - b*n*Log[x] + b*Log[c*x^n]))*n*x^(1 + m)*((2*I + (2*I)*m + b*n)*x^((2*I)*b*n)*Hypergeometric2F1[1/2, ((-1/2*I)*(1 + m + ((3*I)/2)*b*n))/(b*n), -1/4*(2*I + (2*I)*m - 7*b*n)/(b*n), -(E^((2*I)*a)*(c*x^n)^((2*I)*b))] + (-2*I - (2*I)*m + 3*b*n)*Hypergeometric2F1[1/2, -1/4*(2*I + (2*I)*m + b*n)/(b*n), -1/4*(2*I + (2*I)*m - 3*b*n)/(b*n), -(E^((2*I)*a)*(c*x^n)^((2*I)*b))]))/((2 + 2*m - I*b*n)*(2 + 2*m + (3*I)*b*n)*(2 + 2*m - I*b*n + E^((2*I)*(a - b*n*Log[x] + b*Log[c*x^n]))*(2 + 2*m + I*b*n))*Sqrt[1 + E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Sqrt[(E^(I*a)*(c*x^n)^(I*b))/(2 + 2*E^((2*I)*a)*(c*x^n)^((2*I)*b))]) + (2*x^(1 + m)*Cos[a - b*n*Log[x] + b*Log[c*x^n]])/(Sqrt[Sec[a + b*Log[c*x^n]]]*(2*(1 + m)*Cos[a - b*n*Log[x] + b*Log[c*x^n]] - b*n*Sin[a - b*n*Log[x] + b*Log[c*x^n]]))","B",0
285,1,202,130,2.497311,"\int \frac{x^m}{\sec ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[x^m/Sec[a + b*Log[c*x^n]]^(3/2),x]","\frac{2 x^{m+1} \left(3 b^2 n^2 \left(1+e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sec ^2\left(a+b \log \left(c x^n\right)\right) \, _2F_1\left(1,-\frac{2 i m-3 b n+2 i}{4 b n};-\frac{2 i m-5 b n+2 i}{4 b n};-e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)+(i b n+2 m+2) \left(3 b n \tan \left(a+b \log \left(c x^n\right)\right)+2 m+2\right)\right)}{(i b n+2 m+2) (-3 i b n+2 m+2) (3 i b n+2 m+2) \sec ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}","\frac{2 x^{m+1} \, _2F_1\left(-\frac{3}{2},-\frac{2 i m+3 b n+2 i}{4 b n};-\frac{2 i m-b n+2 i}{4 b n};-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{(-3 i b n+2 m+2) \left(1+e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{3/2} \sec ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}",1,"(2*x^(1 + m)*(3*b^2*n^2*(1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))*Hypergeometric2F1[1, -1/4*(2*I + (2*I)*m - 3*b*n)/(b*n), -1/4*(2*I + (2*I)*m - 5*b*n)/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))]*Sec[a + b*Log[c*x^n]]^2 + (2 + 2*m + I*b*n)*(2 + 2*m + 3*b*n*Tan[a + b*Log[c*x^n]])))/((2 + 2*m + I*b*n)*(2 + 2*m - (3*I)*b*n)*(2 + 2*m + (3*I)*b*n)*Sec[a + b*Log[c*x^n]]^(3/2))","A",0
286,1,169,139,1.634466,"\int (e x)^m \sec ^p\left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[(e*x)^m*Sec[d*(a + b*Log[c*x^n])]^p,x]","\frac{2^p x (e x)^m \left(\frac{e^{i a d} \left(c x^n\right)^{i b d}}{1+e^{2 i a d} \left(c x^n\right)^{2 i b d}}\right)^p \left(1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)^p \, _2F_1\left(p,-\frac{i (m+i b d n p+1)}{2 b d n};\frac{1}{2} \left(-\frac{i (m+1)}{b d n}+p+2\right);-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{i b d n p+m+1}","\frac{(e x)^{m+1} \left(1+e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)^p \, _2F_1\left(p,-\frac{i m-b d n p+i}{2 b d n};\frac{1}{2} \left(-\frac{i (m+1)}{b d n}+p+2\right);-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right) \sec ^p\left(d \left(a+b \log \left(c x^n\right)\right)\right)}{e (i b d n p+m+1)}",1,"(2^p*x*(e*x)^m*((E^(I*a*d)*(c*x^n)^(I*b*d))/(1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)))^p*(1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))^p*Hypergeometric2F1[p, ((-1/2*I)*(1 + m + I*b*d*n*p))/(b*d*n), (2 - (I*(1 + m))/(b*d*n) + p)/2, -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))])/(1 + m + I*b*d*n*p)","A",0
287,1,142,106,1.0155961,"\int x \sec ^p\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x*Sec[a + b*Log[c*x^n]]^p,x]","-\frac{i 2^p x^2 \left(\frac{e^{i a} \left(c x^n\right)^{i b}}{1+e^{2 i a} \left(c x^n\right)^{2 i b}}\right)^p \left(1+e^{2 i a} \left(c x^n\right)^{2 i b}\right)^p \, _2F_1\left(\frac{p}{2}-\frac{i}{b n},p;\frac{p}{2}-\frac{i}{b n}+1;-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{b n p-2 i}","\frac{x^2 \left(1+e^{2 i a} \left(c x^n\right)^{2 i b}\right)^p \, _2F_1\left(p,\frac{1}{2} \left(p-\frac{2 i}{b n}\right);\frac{1}{2} \left(p-\frac{2 i}{b n}+2\right);-e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sec ^p\left(a+b \log \left(c x^n\right)\right)}{2+i b n p}",1,"((-I)*2^p*x^2*((E^(I*a)*(c*x^n)^(I*b))/(1 + E^((2*I)*a)*(c*x^n)^((2*I)*b)))^p*(1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))^p*Hypergeometric2F1[(-I)/(b*n) + p/2, p, 1 - I/(b*n) + p/2, -(E^((2*I)*a)*(c*x^n)^((2*I)*b))])/(-2*I + b*n*p)","A",0
288,1,142,107,0.8090816,"\int \sec ^p\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[Sec[a + b*Log[c*x^n]]^p,x]","-\frac{i 2^p x \left(\frac{e^{i a} \left(c x^n\right)^{i b}}{1+e^{2 i a} \left(c x^n\right)^{2 i b}}\right)^p \left(1+e^{2 i a} \left(c x^n\right)^{2 i b}\right)^p \, _2F_1\left(p,\frac{b n p-i}{2 b n};\frac{1}{2} \left(p-\frac{i}{b n}+2\right);-e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{b n p-i}","\frac{x \left(1+e^{2 i a} \left(c x^n\right)^{2 i b}\right)^p \, _2F_1\left(p,-\frac{i-b n p}{2 b n};\frac{1}{2} \left(p-\frac{i}{b n}+2\right);-e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sec ^p\left(a+b \log \left(c x^n\right)\right)}{1+i b n p}",1,"((-I)*2^p*x*((E^(I*a)*(c*x^n)^(I*b))/(1 + E^((2*I)*a)*(c*x^n)^((2*I)*b)))^p*(1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))^p*Hypergeometric2F1[p, (-I + b*n*p)/(2*b*n), (2 - I/(b*n) + p)/2, -(E^((2*I)*a)*(c*x^n)^((2*I)*b))])/(-I + b*n*p)","A",0
289,1,82,86,1.5666241,"\int x^2 \csc \left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x^2*Csc[a + b*Log[c*x^n]],x]","-\frac{2 e^{i a} x^3 \left(c x^n\right)^{i b} \, _2F_1\left(1,\frac{1}{2}-\frac{3 i}{2 b n};\frac{3}{2}-\frac{3 i}{2 b n};e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)}{b n-3 i}","\frac{2 e^{i a} x^3 \left(c x^n\right)^{i b} \, _2F_1\left(1,\frac{1}{2} \left(1-\frac{3 i}{b n}\right);\frac{3}{2} \left(1-\frac{i}{b n}\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{-b n+3 i}",1,"(-2*E^(I*a)*x^3*(c*x^n)^(I*b)*Hypergeometric2F1[1, 1/2 - ((3*I)/2)/(b*n), 3/2 - ((3*I)/2)/(b*n), E^((2*I)*(a + b*Log[c*x^n]))])/(-3*I + b*n)","A",1
290,1,78,86,1.5069713,"\int x \csc \left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x*Csc[a + b*Log[c*x^n]],x]","-\frac{2 e^{i a} x^2 \left(c x^n\right)^{i b} \, _2F_1\left(1,\frac{1}{2}-\frac{i}{b n};\frac{3}{2}-\frac{i}{b n};e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)}{b n-2 i}","\frac{2 e^{i a} x^2 \left(c x^n\right)^{i b} \, _2F_1\left(1,\frac{1}{2} \left(1-\frac{2 i}{b n}\right);\frac{1}{2} \left(3-\frac{2 i}{b n}\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{-b n+2 i}",1,"(-2*E^(I*a)*x^2*(c*x^n)^(I*b)*Hypergeometric2F1[1, 1/2 - I/(b*n), 3/2 - I/(b*n), E^((2*I)*(a + b*Log[c*x^n]))])/(-2*I + b*n)","A",1
291,1,80,84,1.3178323,"\int \csc \left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[Csc[a + b*Log[c*x^n]],x]","-\frac{2 e^{i a} x \left(c x^n\right)^{i b} \, _2F_1\left(1,\frac{1}{2}-\frac{i}{2 b n};\frac{3}{2}-\frac{i}{2 b n};e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)}{b n-i}","\frac{2 e^{i a} x \left(c x^n\right)^{i b} \, _2F_1\left(1,\frac{1}{2} \left(1-\frac{i}{b n}\right);\frac{1}{2} \left(3-\frac{i}{b n}\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{-b n+i}",1,"(-2*E^(I*a)*x*(c*x^n)^(I*b)*Hypergeometric2F1[1, 1/2 - (I/2)/(b*n), 3/2 - (I/2)/(b*n), E^((2*I)*(a + b*Log[c*x^n]))])/(-I + b*n)","A",1
292,1,54,20,0.0614606,"\int \frac{\csc \left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Csc[a + b*Log[c*x^n]]/x,x]","\frac{\log \left(\sin \left(\frac{a}{2}+\frac{1}{2} b \log \left(c x^n\right)\right)\right)}{b n}-\frac{\log \left(\cos \left(\frac{a}{2}+\frac{1}{2} b \log \left(c x^n\right)\right)\right)}{b n}","-\frac{\tanh ^{-1}\left(\cos \left(a+b \log \left(c x^n\right)\right)\right)}{b n}",1,"-(Log[Cos[a/2 + (b*Log[c*x^n])/2]]/(b*n)) + Log[Sin[a/2 + (b*Log[c*x^n])/2]]/(b*n)","B",1
293,1,82,85,1.1658907,"\int \frac{\csc \left(a+b \log \left(c x^n\right)\right)}{x^2} \, dx","Integrate[Csc[a + b*Log[c*x^n]]/x^2,x]","-\frac{2 e^{i a} \left(c x^n\right)^{i b} \, _2F_1\left(1,\frac{1}{2}+\frac{i}{2 b n};\frac{3}{2}+\frac{i}{2 b n};e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)}{x (b n+i)}","-\frac{2 e^{i a} \left(c x^n\right)^{i b} \, _2F_1\left(1,\frac{1}{2} \left(1+\frac{i}{b n}\right);\frac{1}{2} \left(3+\frac{i}{b n}\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{x (b n+i)}",1,"(-2*E^(I*a)*(c*x^n)^(I*b)*Hypergeometric2F1[1, 1/2 + (I/2)/(b*n), 3/2 + (I/2)/(b*n), E^((2*I)*(a + b*Log[c*x^n]))])/((I + b*n)*x)","A",1
294,1,78,85,1.1261814,"\int \frac{\csc \left(a+b \log \left(c x^n\right)\right)}{x^3} \, dx","Integrate[Csc[a + b*Log[c*x^n]]/x^3,x]","-\frac{2 e^{i a} \left(c x^n\right)^{i b} \, _2F_1\left(1,\frac{1}{2}+\frac{i}{b n};\frac{3}{2}+\frac{i}{b n};e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)}{x^2 (b n+2 i)}","-\frac{2 e^{i a} \left(c x^n\right)^{i b} \, _2F_1\left(1,\frac{1}{2} \left(1+\frac{2 i}{b n}\right);\frac{1}{2} \left(3+\frac{2 i}{b n}\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{x^2 (b n+2 i)}",1,"(-2*E^(I*a)*(c*x^n)^(I*b)*Hypergeometric2F1[1, 1/2 + I/(b*n), 3/2 + I/(b*n), E^((2*I)*(a + b*Log[c*x^n]))])/((2*I + b*n)*x^2)","A",1
295,1,146,84,5.3200562,"\int \csc ^2\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[Csc[a + b*Log[c*x^n]]^2,x]","\frac{x \left(-\frac{e^{2 i a} \left(c x^n\right)^{2 i b} \, _2F_1\left(1,1-\frac{i}{2 b n};2-\frac{i}{2 b n};e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)}{2 b n-i}-i \, _2F_1\left(1,-\frac{i}{2 b n};1-\frac{i}{2 b n};e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)-\cot \left(a+b \log \left(c x^n\right)\right)\right)}{b n}","-\frac{4 e^{2 i a} x \left(c x^n\right)^{2 i b} \, _2F_1\left(2,\frac{1}{2} \left(2-\frac{i}{b n}\right);\frac{1}{2} \left(4-\frac{i}{b n}\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{1+2 i b n}",1,"(x*(-Cot[a + b*Log[c*x^n]] - (E^((2*I)*a)*(c*x^n)^((2*I)*b)*Hypergeometric2F1[1, 1 - (I/2)/(b*n), 2 - (I/2)/(b*n), E^((2*I)*(a + b*Log[c*x^n]))])/(-I + 2*b*n) - I*Hypergeometric2F1[1, (-1/2*I)/(b*n), 1 - (I/2)/(b*n), E^((2*I)*(a + b*Log[c*x^n]))]))/(b*n)","A",0
296,1,19,19,0.0895719,"\int \frac{\csc ^2\left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Csc[a + b*Log[c*x^n]]^2/x,x]","-\frac{\cot \left(a+b \log \left(c x^n\right)\right)}{b n}","-\frac{\cot \left(a+b \log \left(c x^n\right)\right)}{b n}",1,"-(Cot[a + b*Log[c*x^n]]/(b*n))","A",1
297,1,117,84,5.6437705,"\int \csc ^3\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[Csc[a + b*Log[c*x^n]]^3,x]","-\frac{x \left(\left(b n \cot \left(a+b \log \left(c x^n\right)\right)+1\right) \csc \left(a+b \log \left(c x^n\right)\right)+2 e^{i a} (b n+i) \left(c x^n\right)^{i b} \, _2F_1\left(1,\frac{1}{2}-\frac{i}{2 b n};\frac{3}{2}-\frac{i}{2 b n};e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)\right)}{2 b^2 n^2}","-\frac{8 e^{3 i a} x \left(c x^n\right)^{3 i b} \, _2F_1\left(3,\frac{1}{2} \left(3-\frac{i}{b n}\right);\frac{1}{2} \left(5-\frac{i}{b n}\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{-3 b n+i}",1,"-1/2*(x*((1 + b*n*Cot[a + b*Log[c*x^n]])*Csc[a + b*Log[c*x^n]] + 2*E^(I*a)*(I + b*n)*(c*x^n)^(I*b)*Hypergeometric2F1[1, 1/2 - (I/2)/(b*n), 3/2 - (I/2)/(b*n), E^((2*I)*(a + b*Log[c*x^n]))]))/(b^2*n^2)","A",0
298,1,107,55,0.0787145,"\int \frac{\csc ^3\left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Csc[a + b*Log[c*x^n]]^3/x,x]","\frac{\log \left(\sin \left(\frac{1}{2} \left(a+b \log \left(c x^n\right)\right)\right)\right)}{2 b n}+\frac{\sec ^2\left(\frac{1}{2} \left(a+b \log \left(c x^n\right)\right)\right)}{8 b n}-\frac{\log \left(\cos \left(\frac{1}{2} \left(a+b \log \left(c x^n\right)\right)\right)\right)}{2 b n}-\frac{\csc ^2\left(\frac{1}{2} \left(a+b \log \left(c x^n\right)\right)\right)}{8 b n}","-\frac{\tanh ^{-1}\left(\cos \left(a+b \log \left(c x^n\right)\right)\right)}{2 b n}-\frac{\cot \left(a+b \log \left(c x^n\right)\right) \csc \left(a+b \log \left(c x^n\right)\right)}{2 b n}",1,"-1/8*Csc[(a + b*Log[c*x^n])/2]^2/(b*n) - Log[Cos[(a + b*Log[c*x^n])/2]]/(2*b*n) + Log[Sin[(a + b*Log[c*x^n])/2]]/(2*b*n) + Sec[(a + b*Log[c*x^n])/2]^2/(8*b*n)","A",1
299,1,221,84,13.2352387,"\int \csc ^4\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[Csc[a + b*Log[c*x^n]]^4,x]","\frac{x \left(-4 i \left(4 b^2 n^2+1\right) \, _2F_1\left(1,-\frac{i}{2 b n};1-\frac{i}{2 b n};e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)+\csc ^3\left(a+b \log \left(c x^n\right)\right) \left(-\left(\left(12 b^2 n^2+1\right) \cos \left(a+b \log \left(c x^n\right)\right)\right)+\left(4 b^2 n^2+1\right) \cos \left(3 \left(a+b \log \left(c x^n\right)\right)\right)-4 b n \sin \left(a+b \log \left(c x^n\right)\right)\right)-4 e^{2 i a} (2 b n+i) \left(c x^n\right)^{2 i b} \, _2F_1\left(1,1-\frac{i}{2 b n};2-\frac{i}{2 b n};e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right)\right)}{24 b^3 n^3}","\frac{16 e^{4 i a} x \left(c x^n\right)^{4 i b} \, _2F_1\left(4,\frac{1}{2} \left(4-\frac{i}{b n}\right);\frac{1}{2} \left(6-\frac{i}{b n}\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{1+4 i b n}",1,"(x*(-4*E^((2*I)*a)*(I + 2*b*n)*(c*x^n)^((2*I)*b)*Hypergeometric2F1[1, 1 - (I/2)/(b*n), 2 - (I/2)/(b*n), E^((2*I)*(a + b*Log[c*x^n]))] - (4*I)*(1 + 4*b^2*n^2)*Hypergeometric2F1[1, (-1/2*I)/(b*n), 1 - (I/2)/(b*n), E^((2*I)*(a + b*Log[c*x^n]))] + Csc[a + b*Log[c*x^n]]^3*(-((1 + 12*b^2*n^2)*Cos[a + b*Log[c*x^n]]) + (1 + 4*b^2*n^2)*Cos[3*(a + b*Log[c*x^n])] - 4*b*n*Sin[a + b*Log[c*x^n]])))/(24*b^3*n^3)","B",0
300,1,56,43,0.076018,"\int \frac{\csc ^4\left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Csc[a + b*Log[c*x^n]]^4/x,x]","-\frac{2 \cot \left(a+b \log \left(c x^n\right)\right)}{3 b n}-\frac{\cot \left(a+b \log \left(c x^n\right)\right) \csc ^2\left(a+b \log \left(c x^n\right)\right)}{3 b n}","-\frac{\cot ^3\left(a+b \log \left(c x^n\right)\right)}{3 b n}-\frac{\cot \left(a+b \log \left(c x^n\right)\right)}{b n}",1,"(-2*Cot[a + b*Log[c*x^n]])/(3*b*n) - (Cot[a + b*Log[c*x^n]]*Csc[a + b*Log[c*x^n]]^2)/(3*b*n)","A",1
301,1,30,42,0.4362401,"\int \left(-\left(\left(1+b^2 n^2\right) \csc \left(a+b \log \left(c x^n\right)\right)\right)+2 b^2 n^2 \csc ^3\left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[-((1 + b^2*n^2)*Csc[a + b*Log[c*x^n]]) + 2*b^2*n^2*Csc[a + b*Log[c*x^n]]^3,x]","-x \left(b n \cot \left(a+b \log \left(c x^n\right)\right)+1\right) \csc \left(a+b \log \left(c x^n\right)\right)","-x \csc \left(a+b \log \left(c x^n\right)\right)-b n x \cot \left(a+b \log \left(c x^n\right)\right) \csc \left(a+b \log \left(c x^n\right)\right)",1,"-(x*(1 + b*n*Cot[a + b*Log[c*x^n]])*Csc[a + b*Log[c*x^n]])","A",1
302,1,79,110,2.0486726,"\int x^m \csc ^3\left(a+2 \log \left(c x^{\frac{1}{2} \sqrt{-(1+m)^2}}\right)\right) \, dx","Integrate[x^m*Csc[a + 2*Log[c*x^(Sqrt[-(1 + m)^2]/2)]]^3,x]","\frac{x^{m+1} \left(\sqrt{-(m+1)^2} \cot \left(a+2 \log \left(c x^{\frac{1}{2} \sqrt{-(m+1)^2}}\right)\right)+m+1\right) \csc \left(a+2 \log \left(c x^{\frac{1}{2} \sqrt{-(m+1)^2}}\right)\right)}{2 (m+1)^2}","\frac{x^{m+1} \csc \left(a+2 \log \left(c x^{\frac{1}{2} \sqrt{-(m+1)^2}}\right)\right)}{2 (m+1)}-\frac{x^{m+1} \cot \left(a+2 \log \left(c x^{\frac{1}{2} \sqrt{-(m+1)^2}}\right)\right) \csc \left(a+2 \log \left(c x^{\frac{1}{2} \sqrt{-(m+1)^2}}\right)\right)}{2 \sqrt{-(m+1)^2}}",1,"(x^(1 + m)*(1 + m + Sqrt[-(1 + m)^2]*Cot[a + 2*Log[c*x^(Sqrt[-(1 + m)^2]/2)]])*Csc[a + 2*Log[c*x^(Sqrt[-(1 + m)^2]/2)]])/(2*(1 + m)^2)","A",1
303,1,127,49,0.2061421,"\int x \csc ^3\left(a+2 \log \left(c x^i\right)\right) \, dx","Integrate[x*Csc[a + 2*Log[c*x^I]]^3,x]","\frac{\csc ^2\left(a+2 \log \left(c x^i\right)\right) \left(\left(2 x^4+1\right) \sin \left(a+2 \log \left(c x^i\right)-2 i \log (x)\right)+i \left(2 x^4-1\right) \cos \left(a+2 \log \left(c x^i\right)-2 i \log (x)\right)\right) \left(i \sin \left(2 \left(a+2 \log \left(c x^i\right)-2 i \log (x)\right)\right)+\cos \left(2 \left(a+2 \log \left(c x^i\right)-2 i \log (x)\right)\right)\right)}{4 x^4}","-\frac{i e^{i a} x^2 \left(c x^i\right)^{2 i}}{\left(1-e^{2 i a} \left(c x^i\right)^{4 i}\right)^2}",1,"(Csc[a + 2*Log[c*x^I]]^2*(I*(-1 + 2*x^4)*Cos[a + 2*Log[c*x^I] - (2*I)*Log[x]] + (1 + 2*x^4)*Sin[a + 2*Log[c*x^I] - (2*I)*Log[x]])*(Cos[2*(a + 2*Log[c*x^I] - (2*I)*Log[x])] + I*Sin[2*(a + 2*Log[c*x^I] - (2*I)*Log[x])]))/(4*x^4)","B",1
304,1,137,58,0.1657805,"\int \csc ^3\left(a+2 \log \left(c x^{\frac{i}{2}}\right)\right) \, dx","Integrate[Csc[a + 2*Log[c*x^(I/2)]]^3,x]","\frac{\csc ^2\left(a+2 \log \left(c x^{\frac{i}{2}}\right)\right) \left(\left(2 x^2+1\right) \sin \left(a+2 \log \left(c x^{\frac{i}{2}}\right)-i \log (x)\right)+i \left(2 x^2-1\right) \cos \left(a+2 \log \left(c x^{\frac{i}{2}}\right)-i \log (x)\right)\right) \left(i \sin \left(2 \left(a+2 \log \left(c x^{\frac{i}{2}}\right)-i \log (x)\right)\right)+\cos \left(2 \left(a+2 \log \left(c x^{\frac{i}{2}}\right)-i \log (x)\right)\right)\right)}{2 x^2}","\frac{1}{2} x \csc \left(a+2 \log \left(c x^{\frac{i}{2}}\right)\right)+\frac{1}{2} i x \cot \left(a+2 \log \left(c x^{\frac{i}{2}}\right)\right) \csc \left(a+2 \log \left(c x^{\frac{i}{2}}\right)\right)",1,"(Csc[a + 2*Log[c*x^(I/2)]]^2*(I*(-1 + 2*x^2)*Cos[a + 2*Log[c*x^(I/2)] - I*Log[x]] + (1 + 2*x^2)*Sin[a + 2*Log[c*x^(I/2)] - I*Log[x]])*(Cos[2*(a + 2*Log[c*x^(I/2)] - I*Log[x])] + I*Sin[2*(a + 2*Log[c*x^(I/2)] - I*Log[x])]))/(2*x^2)","B",1
305,1,137,51,0.1730079,"\int \csc ^3\left(a+2 \log \left(c x^{-\frac{i}{2}}\right)\right) \, dx","Integrate[Csc[a + 2*Log[c/x^(I/2)]]^3,x]","-\frac{\csc ^2\left(a+2 \log \left(c x^{-\frac{i}{2}}\right)\right) \left(i \left(2 x^2+1\right) \sin \left(a+2 \log \left(c x^{-\frac{i}{2}}\right)+i \log (x)\right)+\left(2 x^2-1\right) \cos \left(a+2 \log \left(c x^{-\frac{i}{2}}\right)+i \log (x)\right)\right) \left(\sin \left(2 \left(a+2 \log \left(c x^{-\frac{i}{2}}\right)+i \log (x)\right)\right)+i \cos \left(2 \left(a+2 \log \left(c x^{-\frac{i}{2}}\right)+i \log (x)\right)\right)\right)}{2 x^2}","\frac{2 i e^{3 i a} x \left(c x^{-\frac{i}{2}}\right)^{6 i}}{\left(1-e^{2 i a} \left(c x^{-\frac{i}{2}}\right)^{4 i}\right)^2}",1,"-1/2*(Csc[a + 2*Log[c/x^(I/2)]]^2*((-1 + 2*x^2)*Cos[a + 2*Log[c/x^(I/2)] + I*Log[x]] + I*(1 + 2*x^2)*Sin[a + 2*Log[c/x^(I/2)] + I*Log[x]])*(I*Cos[2*(a + 2*Log[c/x^(I/2)] + I*Log[x])] + Sin[2*(a + 2*Log[c/x^(I/2)] + I*Log[x])]))/x^2","B",1
306,1,155,96,2.1099236,"\int \csc ^p\left(a+\frac{i \log \left(c x^n\right)}{n (-2+p)}\right) \, dx","Integrate[Csc[a + (I*Log[c*x^n])/(n*(-2 + p))]^p,x]","\frac{2^{p-1} (p-2) x e^{-\frac{2 i a p}{p-2}} \left(e^{\frac{2 i a p}{p-2}}-e^{\frac{4 i a}{p-2}} \left(c x^n\right)^{\frac{2}{n (p-2)}}\right) \left(-\frac{i e^{\frac{i a (p+2)}{p-2}} \left(c x^n\right)^{\frac{1}{n (p-2)}}}{e^{\frac{4 i a}{p-2}} \left(c x^n\right)^{\frac{2}{n (p-2)}}-e^{\frac{2 i a p}{p-2}}}\right)^p}{p-1}","-\frac{e^{-2 i a} (2-p) x \left(c x^n\right)^{-\frac{2}{n (2-p)}} \left(1-e^{2 i a} \left(c x^n\right)^{\frac{2}{n (2-p)}}\right) \csc ^p\left(a-\frac{i \log \left(c x^n\right)}{n (2-p)}\right)}{2 (1-p)}",1,"(2^(-1 + p)*(-2 + p)*x*(E^(((2*I)*a*p)/(-2 + p)) - E^(((4*I)*a)/(-2 + p))*(c*x^n)^(2/(n*(-2 + p))))*(((-I)*E^((I*a*(2 + p))/(-2 + p))*(c*x^n)^(1/(n*(-2 + p))))/(-E^(((2*I)*a*p)/(-2 + p)) + E^(((4*I)*a)/(-2 + p))*(c*x^n)^(2/(n*(-2 + p)))))^p)/(E^(((2*I)*a*p)/(-2 + p))*(-1 + p))","A",0
307,1,128,71,3.1037924,"\int \csc ^p\left(a-\frac{i \log \left(c x^n\right)}{n (-2+p)}\right) \, dx","Integrate[Csc[a - (I*Log[c*x^n])/(n*(-2 + p))]^p,x]","\frac{2^{p-1} (p-2) x \left(\frac{i e^{i a} \left(c x^n\right)^{\frac{1}{n (p-2)}}}{-1+e^{2 i a} \left(c x^n\right)^{\frac{2}{n (p-2)}}}\right)^p \left(1+e^{2 i a} \left(c x^n\right)^{\frac{2}{n (p-2)}} \left(-1+\left(1-e^{-2 i a} \left(c x^n\right)^{-\frac{2}{n (p-2)}}\right)^p\right)\right)}{p-1}","\frac{(2-p) x \left(1-e^{2 i a} \left(c x^n\right)^{-\frac{2}{n (2-p)}}\right) \csc ^p\left(a+\frac{i \log \left(c x^n\right)}{n (2-p)}\right)}{2 (1-p)}",1,"(2^(-1 + p)*(-2 + p)*x*((I*E^(I*a)*(c*x^n)^(1/(n*(-2 + p))))/(-1 + E^((2*I)*a)*(c*x^n)^(2/(n*(-2 + p)))))^p*(1 + E^((2*I)*a)*(c*x^n)^(2/(n*(-2 + p)))*(-1 + (1 - 1/(E^((2*I)*a)*(c*x^n)^(2/(n*(-2 + p)))))^p)))/(-1 + p)","A",0
308,1,115,109,0.6267575,"\int \sqrt{\csc \left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[Sqrt[Csc[a + b*Log[c*x^n]]],x]","\frac{2 i e^{-2 i a} x \left(c x^n\right)^{-2 i b} \left(-1+e^{2 i \left(a+b \log \left(c x^n\right)\right)}\right) \, _2F_1\left(1,\frac{3}{4}+\frac{i}{2 b n};\frac{5}{4}+\frac{i}{2 b n};e^{-2 i \left(a+b \log \left(c x^n\right)\right)}\right) \sqrt{\csc \left(a+b \log \left(c x^n\right)\right)}}{b n+2 i}","\frac{2 x \sqrt{1-e^{2 i a} \left(c x^n\right)^{2 i b}} \, _2F_1\left(\frac{1}{2},\frac{1}{4} \left(1-\frac{2 i}{b n}\right);\frac{1}{4} \left(5-\frac{2 i}{b n}\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sqrt{\csc \left(a+b \log \left(c x^n\right)\right)}}{2+i b n}",1,"((2*I)*(-1 + E^((2*I)*(a + b*Log[c*x^n])))*x*Sqrt[Csc[a + b*Log[c*x^n]]]*Hypergeometric2F1[1, 3/4 + (I/2)/(b*n), 5/4 + (I/2)/(b*n), E^((-2*I)*(a + b*Log[c*x^n]))])/(E^((2*I)*a)*(2*I + b*n)*(c*x^n)^((2*I)*b))","A",0
309,1,58,59,0.1127148,"\int \frac{\sqrt{\csc \left(a+b \log \left(c x^n\right)\right)}}{x} \, dx","Integrate[Sqrt[Csc[a + b*Log[c*x^n]]]/x,x]","-\frac{2 \sqrt{\sin \left(a+b \log \left(c x^n\right)\right)} \sqrt{\csc \left(a+b \log \left(c x^n\right)\right)} F\left(\left.\frac{1}{4} \left(-2 a-2 b \log \left(c x^n\right)+\pi \right)\right|2\right)}{b n}","\frac{2 \sqrt{\sin \left(a+b \log \left(c x^n\right)\right)} \sqrt{\csc \left(a+b \log \left(c x^n\right)\right)} F\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)-\frac{\pi }{2}\right)\right|2\right)}{b n}",1,"(-2*Sqrt[Csc[a + b*Log[c*x^n]]]*EllipticF[(-2*a + Pi - 2*b*Log[c*x^n])/4, 2]*Sqrt[Sin[a + b*Log[c*x^n]]])/(b*n)","A",1
310,1,411,109,6.0354454,"\int \csc ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[Csc[a + b*Log[c*x^n]]^(3/2),x]","\frac{x \left(\left(b^2 n^2+4\right) x^{i b n} \sqrt{2-2 e^{2 i a} \left(c x^n\right)^{2 i b}} \sqrt{\frac{i e^{i a} \left(c x^n\right)^{i b}}{-1+e^{2 i a} \left(c x^n\right)^{2 i b}}} \, _2F_1\left(\frac{1}{2},\frac{3}{4}-\frac{i}{2 b n};\frac{7}{4}-\frac{i}{2 b n};e^{2 i a} \left(c x^n\right)^{2 i b}\right)-(3 b n-2 i) x^{-i b n} \left((-b n+2 i) \sqrt{2-2 e^{2 i a} \left(c x^n\right)^{2 i b}} \sqrt{\frac{i e^{i a} \left(c x^n\right)^{i b}}{-1+e^{2 i a} \left(c x^n\right)^{2 i b}}} \, _2F_1\left(\frac{1}{2},-\frac{b n+2 i}{4 b n};\frac{3}{4}-\frac{i}{2 b n};e^{2 i a} \left(c x^n\right)^{2 i b}\right)+2 x^{i b n} (b n \cos (b n \log (x))-2 \sin (b n \log (x))) \sqrt{\csc \left(a+b \log \left(c x^n\right)\right)}\right)\right)}{b n (3 b n-2 i) \left(2 \sin \left(a+b \log \left(c x^n\right)-b n \log (x)\right)+b n \cos \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)}","\frac{2 x \left(1-e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{3/2} \, _2F_1\left(\frac{3}{2},\frac{1}{4} \left(3-\frac{2 i}{b n}\right);\frac{1}{4} \left(7-\frac{2 i}{b n}\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right) \csc ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}{2+3 i b n}",1,"(x*((4 + b^2*n^2)*x^(I*b*n)*Sqrt[2 - 2*E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Sqrt[(I*E^(I*a)*(c*x^n)^(I*b))/(-1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))]*Hypergeometric2F1[1/2, 3/4 - (I/2)/(b*n), 7/4 - (I/2)/(b*n), E^((2*I)*a)*(c*x^n)^((2*I)*b)] - ((-2*I + 3*b*n)*((2*I - b*n)*Sqrt[2 - 2*E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Sqrt[(I*E^(I*a)*(c*x^n)^(I*b))/(-1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))]*Hypergeometric2F1[1/2, -1/4*(2*I + b*n)/(b*n), 3/4 - (I/2)/(b*n), E^((2*I)*a)*(c*x^n)^((2*I)*b)] + 2*x^(I*b*n)*Sqrt[Csc[a + b*Log[c*x^n]]]*(b*n*Cos[b*n*Log[x]] - 2*Sin[b*n*Log[x]])))/x^(I*b*n)))/(b*n*(-2*I + 3*b*n)*(b*n*Cos[a - b*n*Log[x] + b*Log[c*x^n]] + 2*Sin[a - b*n*Log[x] + b*Log[c*x^n]]))","B",0
311,1,72,94,0.1364633,"\int \frac{\csc ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Csc[a + b*Log[c*x^n]]^(3/2)/x,x]","-\frac{2 \sqrt{\csc \left(a+b \log \left(c x^n\right)\right)} \left(\cos \left(a+b \log \left(c x^n\right)\right)-\sqrt{\sin \left(a+b \log \left(c x^n\right)\right)} E\left(\left.\frac{1}{4} \left(-2 a-2 b \log \left(c x^n\right)+\pi \right)\right|2\right)\right)}{b n}","-\frac{2 \cos \left(a+b \log \left(c x^n\right)\right) \sqrt{\csc \left(a+b \log \left(c x^n\right)\right)}}{b n}-\frac{2 \sqrt{\sin \left(a+b \log \left(c x^n\right)\right)} \sqrt{\csc \left(a+b \log \left(c x^n\right)\right)} E\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)-\frac{\pi }{2}\right)\right|2\right)}{b n}",1,"(-2*Sqrt[Csc[a + b*Log[c*x^n]]]*(Cos[a + b*Log[c*x^n]] - EllipticE[(-2*a + Pi - 2*b*Log[c*x^n])/4, 2]*Sqrt[Sin[a + b*Log[c*x^n]]]))/(b*n)","A",1
312,1,174,109,1.7267346,"\int \csc ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[Csc[a + b*Log[c*x^n]]^(5/2),x]","\frac{2 x^{1-2 i b n} e^{-2 i \left(a+b \log \left(c x^n\right)-b n \log (x)\right)} \sqrt{\csc \left(a+b \log \left(c x^n\right)\right)} \left((2+i b n) \left(-1+e^{2 i a} \left(c x^n\right)^{2 i b}\right) \, _2F_1\left(1,\frac{3}{4}+\frac{i}{2 b n};\frac{5}{4}+\frac{i}{2 b n};e^{-2 i \left(a+b \log \left(c x^n\right)\right)}\right)-e^{2 i a} \left(c x^n\right)^{2 i b} \left(b n \cot \left(a+b \log \left(c x^n\right)\right)+2\right)\right)}{3 b^2 n^2}","\frac{2 x \left(1-e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{5/2} \, _2F_1\left(\frac{5}{2},\frac{1}{4} \left(5-\frac{2 i}{b n}\right);\frac{1}{4} \left(9-\frac{2 i}{b n}\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right) \csc ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right)}{2+5 i b n}",1,"(2*x^(1 - (2*I)*b*n)*Sqrt[Csc[a + b*Log[c*x^n]]]*(-(E^((2*I)*a)*(c*x^n)^((2*I)*b)*(2 + b*n*Cot[a + b*Log[c*x^n]])) + (2 + I*b*n)*(-1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))*Hypergeometric2F1[1, 3/4 + (I/2)/(b*n), 5/4 + (I/2)/(b*n), E^((-2*I)*(a + b*Log[c*x^n]))]))/(3*b^2*E^((2*I)*(a - b*n*Log[x] + b*Log[c*x^n]))*n^2)","A",0
313,1,73,98,0.1778012,"\int \frac{\csc ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right)}{x} \, dx","Integrate[Csc[a + b*Log[c*x^n]]^(5/2)/x,x]","-\frac{2 \csc ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right) \left(\cos \left(a+b \log \left(c x^n\right)\right)+\sin ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right) F\left(\left.\frac{1}{4} \left(-2 a-2 b \log \left(c x^n\right)+\pi \right)\right|2\right)\right)}{3 b n}","\frac{2 \sqrt{\sin \left(a+b \log \left(c x^n\right)\right)} \sqrt{\csc \left(a+b \log \left(c x^n\right)\right)} F\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)-\frac{\pi }{2}\right)\right|2\right)}{3 b n}-\frac{2 \cos \left(a+b \log \left(c x^n\right)\right) \csc ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}{3 b n}",1,"(-2*Csc[a + b*Log[c*x^n]]^(3/2)*(Cos[a + b*Log[c*x^n]] + EllipticF[(-2*a + Pi - 2*b*Log[c*x^n])/4, 2]*Sin[a + b*Log[c*x^n]]^(3/2)))/(3*b*n)","A",1
314,1,377,110,3.9607714,"\int \frac{1}{\sqrt{\csc \left(a+b \log \left(c x^n\right)\right)}} \, dx","Integrate[1/Sqrt[Csc[a + b*Log[c*x^n]]],x]","\frac{2 x \sin \left(a+b \log \left(c x^n\right)-b n \log (x)\right)}{\sqrt{\csc \left(a+b \log \left(c x^n\right)\right)} \left(2 \sin \left(a+b \log \left(c x^n\right)-b n \log (x)\right)+b n \cos \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)}-\frac{2 e^{i a} b n x \left(c x^n\right)^{i b} \sqrt{2-2 e^{2 i a} \left(c x^n\right)^{2 i b}} \sqrt{\frac{i e^{i a} \left(c x^n\right)^{i b}}{-1+e^{2 i a} \left(c x^n\right)^{2 i b}}} \left((3 b n-2 i) \, _2F_1\left(\frac{1}{2},-\frac{b n+2 i}{4 b n};\frac{3}{4}-\frac{i}{2 b n};e^{2 i a} \left(c x^n\right)^{2 i b}\right)+(b n+2 i) x^{2 i b n} \, _2F_1\left(\frac{1}{2},\frac{3}{4}-\frac{i}{2 b n};\frac{7}{4}-\frac{i}{2 b n};e^{2 i a} \left(c x^n\right)^{2 i b}\right)\right)}{(b n+2 i) (3 b n-2 i) \left(e^{2 i a} (b n-2 i) \left(c x^n\right)^{2 i b}+(b n+2 i) x^{2 i b n}\right)}","\frac{2 x \, _2F_1\left(-\frac{1}{2},-\frac{b n+2 i}{4 b n};\frac{1}{4} \left(3-\frac{2 i}{b n}\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{(2-i b n) \sqrt{1-e^{2 i a} \left(c x^n\right)^{2 i b}} \sqrt{\csc \left(a+b \log \left(c x^n\right)\right)}}",1,"(-2*b*E^(I*a)*n*x*(c*x^n)^(I*b)*Sqrt[2 - 2*E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Sqrt[(I*E^(I*a)*(c*x^n)^(I*b))/(-1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))]*((2*I + b*n)*x^((2*I)*b*n)*Hypergeometric2F1[1/2, 3/4 - (I/2)/(b*n), 7/4 - (I/2)/(b*n), E^((2*I)*a)*(c*x^n)^((2*I)*b)] + (-2*I + 3*b*n)*Hypergeometric2F1[1/2, -1/4*(2*I + b*n)/(b*n), 3/4 - (I/2)/(b*n), E^((2*I)*a)*(c*x^n)^((2*I)*b)]))/((2*I + b*n)*(-2*I + 3*b*n)*((2*I + b*n)*x^((2*I)*b*n) + E^((2*I)*a)*(-2*I + b*n)*(c*x^n)^((2*I)*b))) + (2*x*Sin[a - b*n*Log[x] + b*Log[c*x^n]])/(Sqrt[Csc[a + b*Log[c*x^n]]]*(b*n*Cos[a - b*n*Log[x] + b*Log[c*x^n]] + 2*Sin[a - b*n*Log[x] + b*Log[c*x^n]]))","B",0
315,1,58,59,0.1006822,"\int \frac{1}{x \sqrt{\csc \left(a+b \log \left(c x^n\right)\right)}} \, dx","Integrate[1/(x*Sqrt[Csc[a + b*Log[c*x^n]]]),x]","-\frac{2 \sqrt{\sin \left(a+b \log \left(c x^n\right)\right)} \sqrt{\csc \left(a+b \log \left(c x^n\right)\right)} E\left(\left.\frac{1}{4} \left(-2 a-2 b \log \left(c x^n\right)+\pi \right)\right|2\right)}{b n}","\frac{2 \sqrt{\sin \left(a+b \log \left(c x^n\right)\right)} \sqrt{\csc \left(a+b \log \left(c x^n\right)\right)} E\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)-\frac{\pi }{2}\right)\right|2\right)}{b n}",1,"(-2*Sqrt[Csc[a + b*Log[c*x^n]]]*EllipticE[(-2*a + Pi - 2*b*Log[c*x^n])/4, 2]*Sqrt[Sin[a + b*Log[c*x^n]]])/(b*n)","A",1
316,1,186,109,2.3117636,"\int \frac{1}{\csc ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[Csc[a + b*Log[c*x^n]]^(-3/2),x]","\frac{2 i x \left((2-i b n) \left(3 b n \cot \left(a+b \log \left(c x^n\right)\right)-2\right)-3 e^{-2 i a} b^2 n^2 \left(c x^n\right)^{-2 i b} \left(-1+e^{2 i a} \left(c x^n\right)^{2 i b}\right) \, _2F_1\left(1,\frac{3}{4}+\frac{i}{2 b n};\frac{5}{4}+\frac{i}{2 b n};e^{-2 i \left(a+b \log \left(c x^n\right)\right)}\right) \csc ^2\left(a+b \log \left(c x^n\right)\right)\right)}{(-3 b n+2 i) (b n+2 i) (3 b n+2 i) \csc ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}","\frac{2 x \, _2F_1\left(-\frac{3}{2},\frac{1}{4} \left(-3-\frac{2 i}{b n}\right);\frac{1}{4} \left(1-\frac{2 i}{b n}\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{(2-3 i b n) \left(1-e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{3/2} \csc ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}",1,"((2*I)*x*((2 - I*b*n)*(-2 + 3*b*n*Cot[a + b*Log[c*x^n]]) - (3*b^2*n^2*(-1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))*Csc[a + b*Log[c*x^n]]^2*Hypergeometric2F1[1, 3/4 + (I/2)/(b*n), 5/4 + (I/2)/(b*n), E^((-2*I)*(a + b*Log[c*x^n]))])/(E^((2*I)*a)*(c*x^n)^((2*I)*b))))/((2*I - 3*b*n)*(2*I + b*n)*(2*I + 3*b*n)*Csc[a + b*Log[c*x^n]]^(3/2))","A",0
317,1,76,98,0.1606533,"\int \frac{1}{x \csc ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[1/(x*Csc[a + b*Log[c*x^n]]^(3/2)),x]","-\frac{\sqrt{\csc \left(a+b \log \left(c x^n\right)\right)} \left(\sin \left(2 \left(a+b \log \left(c x^n\right)\right)\right)+2 \sqrt{\sin \left(a+b \log \left(c x^n\right)\right)} F\left(\left.\frac{1}{4} \left(-2 a-2 b \log \left(c x^n\right)+\pi \right)\right|2\right)\right)}{3 b n}","\frac{2 \sqrt{\sin \left(a+b \log \left(c x^n\right)\right)} \sqrt{\csc \left(a+b \log \left(c x^n\right)\right)} F\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)-\frac{\pi }{2}\right)\right|2\right)}{3 b n}-\frac{2 \cos \left(a+b \log \left(c x^n\right)\right)}{3 b n \sqrt{\csc \left(a+b \log \left(c x^n\right)\right)}}",1,"-1/3*(Sqrt[Csc[a + b*Log[c*x^n]]]*(2*EllipticF[(-2*a + Pi - 2*b*Log[c*x^n])/4, 2]*Sqrt[Sin[a + b*Log[c*x^n]]] + Sin[2*(a + b*Log[c*x^n])]))/(b*n)","A",1
318,1,876,110,8.6625576,"\int \frac{1}{\csc ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[Csc[a + b*Log[c*x^n]]^(-5/2),x]","\sqrt{\csc \left(a+b n \log (x)+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)} \left(-\frac{x \cos (b n \log (x)) \left(-55 b^2 n^2+65 b^2 \cos \left(2 \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right) n^2+4 b \sin \left(2 \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right) n+12 \cos \left(2 \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right)-12\right)}{4 (5 b n-2 i) (5 b n+2 i) \left(b n \cos \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)+2 \sin \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right)}+\frac{x \sin (b n \log (x)) \left(65 b^2 \sin \left(2 \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right) n^2+16 b n-4 b \cos \left(2 \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right) n+12 \sin \left(2 \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right)\right)}{4 (5 b n-2 i) (5 b n+2 i) \left(b n \cos \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)+2 \sin \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right)}+\frac{x \cos (3 b n \log (x)) \left(5 b n \cos \left(3 \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right)-2 \sin \left(3 \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right)\right)}{2 (5 b n-2 i) (5 b n+2 i)}-\frac{x \sin (3 b n \log (x)) \left(2 \cos \left(3 \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right)+5 b n \sin \left(3 \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right)\right)}{2 (5 b n-2 i) (5 b n+2 i)}\right)-\frac{30 b^3 e^{i \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)} n^3 x^{1-i b n} \sqrt{2-2 e^{2 i \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)} x^{2 i b n}} \sqrt{\frac{i e^{i \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)} x^{i b n}}{e^{2 i \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)} x^{2 i b n}-1}} \left((b n+2 i) \, _2F_1\left(\frac{1}{2},\frac{3}{4}-\frac{i}{2 b n};\frac{7}{4}-\frac{i}{2 b n};e^{2 i \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)} x^{2 i b n}\right) x^{2 i b n}+(3 b n-2 i) \, _2F_1\left(\frac{1}{2},-\frac{b n+2 i}{4 b n};\frac{3}{4}-\frac{i}{2 b n};e^{2 i \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)} x^{2 i b n}\right)\right)}{(b n+2 i) (3 b n-2 i) (5 b n-2 i) (5 b n+2 i) \left(b n+e^{2 i \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)} (b n-2 i)+2 i\right)}","\frac{2 x \, _2F_1\left(-\frac{5}{2},\frac{1}{4} \left(-5-\frac{2 i}{b n}\right);-\frac{b n+2 i}{4 b n};e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{(2-5 i b n) \left(1-e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{5/2} \csc ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right)}",1,"(-30*b^3*E^(I*(a + b*(-(n*Log[x]) + Log[c*x^n])))*n^3*x^(1 - I*b*n)*Sqrt[2 - 2*E^((2*I)*(a + b*(-(n*Log[x]) + Log[c*x^n])))*x^((2*I)*b*n)]*Sqrt[(I*E^(I*(a + b*(-(n*Log[x]) + Log[c*x^n])))*x^(I*b*n))/(-1 + E^((2*I)*(a + b*(-(n*Log[x]) + Log[c*x^n])))*x^((2*I)*b*n))]*((2*I + b*n)*x^((2*I)*b*n)*Hypergeometric2F1[1/2, 3/4 - (I/2)/(b*n), 7/4 - (I/2)/(b*n), E^((2*I)*(a + b*(-(n*Log[x]) + Log[c*x^n])))*x^((2*I)*b*n)] + (-2*I + 3*b*n)*Hypergeometric2F1[1/2, -1/4*(2*I + b*n)/(b*n), 3/4 - (I/2)/(b*n), E^((2*I)*(a + b*(-(n*Log[x]) + Log[c*x^n])))*x^((2*I)*b*n)]))/((2*I + b*n)*(-2*I + 3*b*n)*(-2*I + 5*b*n)*(2*I + 5*b*n)*(2*I + b*n + E^((2*I)*(a + b*(-(n*Log[x]) + Log[c*x^n])))*(-2*I + b*n))) + Sqrt[Csc[a + b*n*Log[x] + b*(-(n*Log[x]) + Log[c*x^n])]]*(-1/4*(x*Cos[b*n*Log[x]]*(-12 - 55*b^2*n^2 + 12*Cos[2*(a + b*(-(n*Log[x]) + Log[c*x^n]))] + 65*b^2*n^2*Cos[2*(a + b*(-(n*Log[x]) + Log[c*x^n]))] + 4*b*n*Sin[2*(a + b*(-(n*Log[x]) + Log[c*x^n]))]))/((-2*I + 5*b*n)*(2*I + 5*b*n)*(b*n*Cos[a + b*(-(n*Log[x]) + Log[c*x^n])] + 2*Sin[a + b*(-(n*Log[x]) + Log[c*x^n])])) + (x*Sin[b*n*Log[x]]*(16*b*n - 4*b*n*Cos[2*(a + b*(-(n*Log[x]) + Log[c*x^n]))] + 12*Sin[2*(a + b*(-(n*Log[x]) + Log[c*x^n]))] + 65*b^2*n^2*Sin[2*(a + b*(-(n*Log[x]) + Log[c*x^n]))]))/(4*(-2*I + 5*b*n)*(2*I + 5*b*n)*(b*n*Cos[a + b*(-(n*Log[x]) + Log[c*x^n])] + 2*Sin[a + b*(-(n*Log[x]) + Log[c*x^n])])) + (x*Cos[3*b*n*Log[x]]*(5*b*n*Cos[3*(a + b*(-(n*Log[x]) + Log[c*x^n]))] - 2*Sin[3*(a + b*(-(n*Log[x]) + Log[c*x^n]))]))/(2*(-2*I + 5*b*n)*(2*I + 5*b*n)) - (x*Sin[3*b*n*Log[x]]*(2*Cos[3*(a + b*(-(n*Log[x]) + Log[c*x^n]))] + 5*b*n*Sin[3*(a + b*(-(n*Log[x]) + Log[c*x^n]))]))/(2*(-2*I + 5*b*n)*(2*I + 5*b*n)))","B",0
319,1,88,98,0.2006227,"\int \frac{1}{x \csc ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[1/(x*Csc[a + b*Log[c*x^n]]^(5/2)),x]","-\frac{2 \sqrt{\csc \left(a+b \log \left(c x^n\right)\right)} \left(\sin ^2\left(a+b \log \left(c x^n\right)\right) \cos \left(a+b \log \left(c x^n\right)\right)+3 \sqrt{\sin \left(a+b \log \left(c x^n\right)\right)} E\left(\left.\frac{1}{4} \left(-2 a-2 b \log \left(c x^n\right)+\pi \right)\right|2\right)\right)}{5 b n}","\frac{6 \sqrt{\sin \left(a+b \log \left(c x^n\right)\right)} \sqrt{\csc \left(a+b \log \left(c x^n\right)\right)} E\left(\left.\frac{1}{2} \left(a+b \log \left(c x^n\right)-\frac{\pi }{2}\right)\right|2\right)}{5 b n}-\frac{2 \cos \left(a+b \log \left(c x^n\right)\right)}{5 b n \csc ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}",1,"(-2*Sqrt[Csc[a + b*Log[c*x^n]]]*(3*EllipticE[(-2*a + Pi - 2*b*Log[c*x^n])/4, 2]*Sqrt[Sin[a + b*Log[c*x^n]]] + Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^2))/(5*b*n)","A",1
320,1,367,122,2.2938161,"\int (e x)^m \csc ^3\left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[(e*x)^m*Csc[d*(a + b*Log[c*x^n])]^3,x]","\frac{x (e x)^m \left(8 (-i b d n+m+1) x^{i b d n} \left(\sin \left(d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)-i \cos \left(d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)\right) \, _2F_1\left(1,\frac{-i m+b d n-i}{2 b d n};-\frac{i (m+3 i b d n+1)}{2 b d n};x^{2 i b d n} \left(\cos \left(2 d \left(a-b n \log (x)+b \log \left(c x^n\right)\right)\right)+i \sin \left(2 d \left(a-b n \log (x)+b \log \left(c x^n\right)\right)\right)\right)\right)-4 (m+1) \csc \left(d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)-2 (m+1) \sin \left(\frac{1}{2} b d n \log (x)\right) \sec \left(\frac{1}{2} d \left(a+b \log \left(c x^n\right)\right)\right) \sec \left(\frac{1}{2} d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)+2 (m+1) \sin \left(\frac{1}{2} b d n \log (x)\right) \csc \left(\frac{1}{2} d \left(a+b \log \left(c x^n\right)\right)\right) \csc \left(\frac{1}{2} d \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)+b d n \sec ^2\left(\frac{1}{2} d \left(a+b \log \left(c x^n\right)\right)\right)-b d n \csc ^2\left(\frac{1}{2} d \left(a+b \log \left(c x^n\right)\right)\right)\right)}{8 b^2 d^2 n^2}","-\frac{8 e^{3 i a d} (e x)^{m+1} \left(c x^n\right)^{3 i b d} \, _2F_1\left(3,-\frac{i (m+1)-3 b d n}{2 b d n};-\frac{i (m+1)-5 b d n}{2 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{e (-3 b d n+i (m+1))}",1,"(x*(e*x)^m*(-(b*d*n*Csc[(d*(a + b*Log[c*x^n]))/2]^2) - 4*(1 + m)*Csc[d*(a - b*n*Log[x] + b*Log[c*x^n])] + b*d*n*Sec[(d*(a + b*Log[c*x^n]))/2]^2 + 2*(1 + m)*Csc[(d*(a + b*Log[c*x^n]))/2]*Csc[(d*(a - b*n*Log[x] + b*Log[c*x^n]))/2]*Sin[(b*d*n*Log[x])/2] - 2*(1 + m)*Sec[(d*(a + b*Log[c*x^n]))/2]*Sec[(d*(a - b*n*Log[x] + b*Log[c*x^n]))/2]*Sin[(b*d*n*Log[x])/2] + 8*(1 + m - I*b*d*n)*x^(I*b*d*n)*Hypergeometric2F1[1, (-I - I*m + b*d*n)/(2*b*d*n), ((-1/2*I)*(1 + m + (3*I)*b*d*n))/(b*d*n), x^((2*I)*b*d*n)*(Cos[2*d*(a - b*n*Log[x] + b*Log[c*x^n])] + I*Sin[2*d*(a - b*n*Log[x] + b*Log[c*x^n])])]*((-I)*Cos[d*(a - b*n*Log[x] + b*Log[c*x^n])] + Sin[d*(a - b*n*Log[x] + b*Log[c*x^n])])))/(8*b^2*d^2*n^2)","B",0
321,1,534,119,6.5305494,"\int (e x)^m \csc ^2\left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[(e*x)^m*Csc[d*(a + b*Log[c*x^n])]^2,x]","\frac{x (e x)^m \sin (b d n \log (x)) \csc \left(d \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right) \csc \left(d \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)+b d n \log (x)\right)}{b d n}-\frac{(m+1) x^{-m} (e x)^m \csc \left(d \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right) \left(\frac{x^{m+1} \sin (b d n \log (x)) \csc \left(d \left(a+b \log \left(c x^n\right)\right)\right)}{m+1}-\frac{i \sin \left(d \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right) \exp \left(-\frac{(2 m+1) \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)}{b n}\right) \left(-(2 i b d n+m+1) \exp \left(\frac{2 a m+a+b (2 m+1) \left(\log \left(c x^n\right)-n \log (x)\right)+b (m+1) n \log (x)}{b n}\right) \, _2F_1\left(1,-\frac{i (m+1)}{2 b d n};1-\frac{i (m+1)}{2 b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)-(m+1) \exp \left(\frac{a (2 i b d n+2 m+1)}{b n}+\frac{(2 i b d n+2 m+1) \left(\log \left(c x^n\right)-n \log (x)\right)}{n}+\log (x) (2 i b d n+m+1)\right) \, _2F_1\left(1,-\frac{i (m+2 i b d n+1)}{2 b d n};-\frac{i (m+4 i b d n+1)}{2 b d n};e^{2 i d \left(a+b \log \left(c x^n\right)\right)}\right)+i (2 i b d n+m+1) \cot \left(d \left(a+b \log \left(c x^n\right)\right)\right) \exp \left(\frac{2 a m+a+b (2 m+1) \left(\log \left(c x^n\right)-n \log (x)\right)+b (m+1) n \log (x)}{b n}\right)\right)}{(m+1) (2 i b d n+m+1)}\right)}{b d n}","-\frac{4 e^{2 i a d} (e x)^{m+1} \left(c x^n\right)^{2 i b d} \, _2F_1\left(2,-\frac{i (m+1)-2 b d n}{2 b d n};-\frac{i (m+1)-4 b d n}{2 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{e (2 i b d n+m+1)}",1,"(x*(e*x)^m*Csc[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]*Csc[b*d*n*Log[x] + d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]*Sin[b*d*n*Log[x]])/(b*d*n) - ((1 + m)*(e*x)^m*Csc[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]*((x^(1 + m)*Csc[d*(a + b*Log[c*x^n])]*Sin[b*d*n*Log[x]])/(1 + m) - (I*(I*E^((a + 2*a*m + b*(1 + m)*n*Log[x] + b*(1 + 2*m)*(-(n*Log[x]) + Log[c*x^n]))/(b*n))*(1 + m + (2*I)*b*d*n)*Cot[d*(a + b*Log[c*x^n])] - E^((a + 2*a*m + b*(1 + m)*n*Log[x] + b*(1 + 2*m)*(-(n*Log[x]) + Log[c*x^n]))/(b*n))*(1 + m + (2*I)*b*d*n)*Hypergeometric2F1[1, ((-1/2*I)*(1 + m))/(b*d*n), 1 - ((I/2)*(1 + m))/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] - E^((a*(1 + 2*m + (2*I)*b*d*n))/(b*n) + (1 + m + (2*I)*b*d*n)*Log[x] + ((1 + 2*m + (2*I)*b*d*n)*(-(n*Log[x]) + Log[c*x^n]))/n)*(1 + m)*Hypergeometric2F1[1, ((-1/2*I)*(1 + m + (2*I)*b*d*n))/(b*d*n), ((-1/2*I)*(1 + m + (4*I)*b*d*n))/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))])*Sin[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))])/(E^(((1 + 2*m)*(a + b*(-(n*Log[x]) + Log[c*x^n])))/(b*n))*(1 + m)*(1 + m + (2*I)*b*d*n))))/(b*d*n*x^m)","B",1
322,1,181,123,0.4267975,"\int (e x)^m \csc \left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[(e*x)^m*Csc[d*(a + b*Log[c*x^n])],x]","\frac{2 (e x)^m x^{1+i b d n} \left(\sin \left(d \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right)-i \cos \left(d \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right)\right) \, _2F_1\left(1,\frac{-i m+b d n-i}{2 b d n};-\frac{i (m+3 i b d n+1)}{2 b d n};x^{2 i b d n} \left(\cos \left(2 d \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right)+i \sin \left(2 d \left(a+b \left(\log \left(c x^n\right)-n \log (x)\right)\right)\right)\right)\right)}{i b d n+m+1}","\frac{2 e^{i a d} (e x)^{m+1} \left(c x^n\right)^{i b d} \, _2F_1\left(1,-\frac{i m-b d n+i}{2 b d n};-\frac{i (m+1)-3 b d n}{2 b d n};e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{e (-b d n+i (m+1))}",1,"(2*x^(1 + I*b*d*n)*(e*x)^m*Hypergeometric2F1[1, (-I - I*m + b*d*n)/(2*b*d*n), ((-1/2*I)*(1 + m + (3*I)*b*d*n))/(b*d*n), x^((2*I)*b*d*n)*(Cos[2*d*(a + b*(-(n*Log[x]) + Log[c*x^n]))] + I*Sin[2*d*(a + b*(-(n*Log[x]) + Log[c*x^n]))])]*((-I)*Cos[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))] + Sin[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]))/(1 + m + I*b*d*n)","A",0
323,1,165,130,2.9867145,"\int x^m \csc ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x^m*Csc[a + b*Log[c*x^n]]^(5/2),x]","\frac{2 x^{m+1} \sqrt{\csc \left(a+b \log \left(c x^n\right)\right)} \left(e^{-2 i a} (i b n+2 m+2) \left(c x^n\right)^{-2 i b} \left(-1+e^{2 i a} \left(c x^n\right)^{2 i b}\right) \, _2F_1\left(1,\frac{2 i m+3 b n+2 i}{4 b n};\frac{2 i m+5 b n+2 i}{4 b n};e^{-2 i \left(a+b \log \left(c x^n\right)\right)}\right)-b n \cot \left(a+b \log \left(c x^n\right)\right)-2 m-2\right)}{3 b^2 n^2}","\frac{2 x^{m+1} \left(1-e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{5/2} \, _2F_1\left(\frac{5}{2},-\frac{2 i m-5 b n+2 i}{4 b n};-\frac{2 i m-9 b n+2 i}{4 b n};e^{2 i a} \left(c x^n\right)^{2 i b}\right) \csc ^{\frac{5}{2}}\left(a+b \log \left(c x^n\right)\right)}{5 i b n+2 m+2}",1,"(2*x^(1 + m)*Sqrt[Csc[a + b*Log[c*x^n]]]*(-2 - 2*m - b*n*Cot[a + b*Log[c*x^n]] + ((2 + 2*m + I*b*n)*(-1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))*Hypergeometric2F1[1, (2*I + (2*I)*m + 3*b*n)/(4*b*n), (2*I + (2*I)*m + 5*b*n)/(4*b*n), E^((-2*I)*(a + b*Log[c*x^n]))])/(E^((2*I)*a)*(c*x^n)^((2*I)*b))))/(3*b^2*n^2)","A",0
324,1,466,130,9.4198876,"\int x^m \csc ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x^m*Csc[a + b*Log[c*x^n]]^(3/2),x]","\frac{x^{-i b n+m+1} \left(\left(b^2 n^2+4 m^2+8 m+4\right) x^{2 i b n} \sqrt{2-2 e^{2 i a} \left(c x^n\right)^{2 i b}} \sqrt{\frac{i e^{i a} \left(c x^n\right)^{i b}}{-1+e^{2 i a} \left(c x^n\right)^{2 i b}}} \, _2F_1\left(\frac{1}{2},-\frac{i \left(m+\frac{3 i b n}{2}+1\right)}{2 b n};-\frac{2 i m-7 b n+2 i}{4 b n};e^{2 i a} \left(c x^n\right)^{2 i b}\right)+(3 b n-2 i m-2 i) \left((b n-2 i m-2 i) \sqrt{2-2 e^{2 i a} \left(c x^n\right)^{2 i b}} \sqrt{\frac{i e^{i a} \left(c x^n\right)^{i b}}{-1+e^{2 i a} \left(c x^n\right)^{2 i b}}} \, _2F_1\left(\frac{1}{2},-\frac{2 i m+b n+2 i}{4 b n};-\frac{2 i m-3 b n+2 i}{4 b n};e^{2 i a} \left(c x^n\right)^{2 i b}\right)-2 x^{i b n} \sqrt{\csc \left(a+b \log \left(c x^n\right)\right)} (b n \cos (b n \log (x))-2 (m+1) \sin (b n \log (x)))\right)\right)}{b n (3 b n-2 i m-2 i) \left(2 (m+1) \sin \left(a+b \log \left(c x^n\right)-b n \log (x)\right)+b n \cos \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)}","\frac{2 x^{m+1} \left(1-e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{3/2} \, _2F_1\left(\frac{3}{2},-\frac{2 i m-3 b n+2 i}{4 b n};-\frac{2 i m-7 b n+2 i}{4 b n};e^{2 i a} \left(c x^n\right)^{2 i b}\right) \csc ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}{3 i b n+2 m+2}",1,"(x^(1 + m - I*b*n)*((4 + 8*m + 4*m^2 + b^2*n^2)*x^((2*I)*b*n)*Sqrt[2 - 2*E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Sqrt[(I*E^(I*a)*(c*x^n)^(I*b))/(-1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))]*Hypergeometric2F1[1/2, ((-1/2*I)*(1 + m + ((3*I)/2)*b*n))/(b*n), -1/4*(2*I + (2*I)*m - 7*b*n)/(b*n), E^((2*I)*a)*(c*x^n)^((2*I)*b)] + (-2*I - (2*I)*m + 3*b*n)*((-2*I - (2*I)*m + b*n)*Sqrt[2 - 2*E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Sqrt[(I*E^(I*a)*(c*x^n)^(I*b))/(-1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))]*Hypergeometric2F1[1/2, -1/4*(2*I + (2*I)*m + b*n)/(b*n), -1/4*(2*I + (2*I)*m - 3*b*n)/(b*n), E^((2*I)*a)*(c*x^n)^((2*I)*b)] - 2*x^(I*b*n)*Sqrt[Csc[a + b*Log[c*x^n]]]*(b*n*Cos[b*n*Log[x]] - 2*(1 + m)*Sin[b*n*Log[x]]))))/(b*n*(-2*I - (2*I)*m + 3*b*n)*(b*n*Cos[a - b*n*Log[x] + b*Log[c*x^n]] + 2*(1 + m)*Sin[a - b*n*Log[x] + b*Log[c*x^n]]))","B",0
325,1,138,130,0.9279247,"\int x^m \sqrt{\csc \left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[x^m*Sqrt[Csc[a + b*Log[c*x^n]]],x]","\frac{2 e^{-2 i a} x^{m+1} \left(c x^n\right)^{-2 i b} \left(-1+e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sqrt{\csc \left(a+b \log \left(c x^n\right)\right)} \, _2F_1\left(1,\frac{2 i m+3 b n+2 i}{4 b n};\frac{2 i m+5 b n+2 i}{4 b n};e^{-2 i \left(a+b \log \left(c x^n\right)\right)}\right)}{-i b n+2 m+2}","\frac{2 x^{m+1} \sqrt{1-e^{2 i a} \left(c x^n\right)^{2 i b}} \, _2F_1\left(\frac{1}{2},-\frac{2 i m-b n+2 i}{4 b n};-\frac{2 i m-5 b n+2 i}{4 b n};e^{2 i a} \left(c x^n\right)^{2 i b}\right) \sqrt{\csc \left(a+b \log \left(c x^n\right)\right)}}{i b n+2 m+2}",1,"(2*x^(1 + m)*(-1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))*Sqrt[Csc[a + b*Log[c*x^n]]]*Hypergeometric2F1[1, (2*I + (2*I)*m + 3*b*n)/(4*b*n), (2*I + (2*I)*m + 5*b*n)/(4*b*n), E^((-2*I)*(a + b*Log[c*x^n]))])/(E^((2*I)*a)*(2 + 2*m - I*b*n)*(c*x^n)^((2*I)*b))","A",0
326,1,441,129,7.2274852,"\int \frac{x^m}{\sqrt{\csc \left(a+b \log \left(c x^n\right)\right)}} \, dx","Integrate[x^m/Sqrt[Csc[a + b*Log[c*x^n]]],x]","\frac{2 x^{m+1} \sin \left(a+b \log \left(c x^n\right)-b n \log (x)\right)}{\sqrt{\csc \left(a+b \log \left(c x^n\right)\right)} \left(2 (m+1) \sin \left(a+b \log \left(c x^n\right)-b n \log (x)\right)+b n \cos \left(a+b \log \left(c x^n\right)-b n \log (x)\right)\right)}-\frac{2 e^{i a} b n x^{m+1} \left(c x^n\right)^{i b} \sqrt{2-2 e^{2 i a} \left(c x^n\right)^{2 i b}} \sqrt{\frac{i e^{i a} \left(c x^n\right)^{i b}}{-1+e^{2 i a} \left(c x^n\right)^{2 i b}}} \left((3 b n-2 i m-2 i) \, _2F_1\left(\frac{1}{2},-\frac{2 i m+b n+2 i}{4 b n};-\frac{2 i m-3 b n+2 i}{4 b n};e^{2 i a} \left(c x^n\right)^{2 i b}\right)+(b n+2 i m+2 i) x^{2 i b n} \, _2F_1\left(\frac{1}{2},-\frac{i \left(m+\frac{3 i b n}{2}+1\right)}{2 b n};-\frac{2 i m-7 b n+2 i}{4 b n};e^{2 i a} \left(c x^n\right)^{2 i b}\right)\right)}{(-i b n+2 m+2) (3 i b n+2 m+2) \left(e^{2 i a} (b n-2 i m-2 i) \left(c x^n\right)^{2 i b}+(b n+2 i m+2 i) x^{2 i b n}\right)}","\frac{2 x^{m+1} \, _2F_1\left(-\frac{1}{2},-\frac{2 i m+b n+2 i}{4 b n};-\frac{2 i m-3 b n+2 i}{4 b n};e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{(-i b n+2 m+2) \sqrt{1-e^{2 i a} \left(c x^n\right)^{2 i b}} \sqrt{\csc \left(a+b \log \left(c x^n\right)\right)}}",1,"(-2*b*E^(I*a)*n*x^(1 + m)*(c*x^n)^(I*b)*Sqrt[2 - 2*E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Sqrt[(I*E^(I*a)*(c*x^n)^(I*b))/(-1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))]*((2*I + (2*I)*m + b*n)*x^((2*I)*b*n)*Hypergeometric2F1[1/2, ((-1/2*I)*(1 + m + ((3*I)/2)*b*n))/(b*n), -1/4*(2*I + (2*I)*m - 7*b*n)/(b*n), E^((2*I)*a)*(c*x^n)^((2*I)*b)] + (-2*I - (2*I)*m + 3*b*n)*Hypergeometric2F1[1/2, -1/4*(2*I + (2*I)*m + b*n)/(b*n), -1/4*(2*I + (2*I)*m - 3*b*n)/(b*n), E^((2*I)*a)*(c*x^n)^((2*I)*b)]))/((2 + 2*m - I*b*n)*(2 + 2*m + (3*I)*b*n)*((2*I + (2*I)*m + b*n)*x^((2*I)*b*n) + E^((2*I)*a)*(-2*I - (2*I)*m + b*n)*(c*x^n)^((2*I)*b))) + (2*x^(1 + m)*Sin[a - b*n*Log[x] + b*Log[c*x^n]])/(Sqrt[Csc[a + b*Log[c*x^n]]]*(b*n*Cos[a - b*n*Log[x] + b*Log[c*x^n]] + 2*(1 + m)*Sin[a - b*n*Log[x] + b*Log[c*x^n]]))","B",0
327,1,218,130,2.3756806,"\int \frac{x^m}{\csc ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)} \, dx","Integrate[x^m/Csc[a + b*Log[c*x^n]]^(3/2),x]","\frac{2 x^{m+1} \left(3 e^{-2 i a} b^2 n^2 \left(c x^n\right)^{-2 i b} \left(-1+e^{2 i a} \left(c x^n\right)^{2 i b}\right) \csc ^2\left(a+b \log \left(c x^n\right)\right) \, _2F_1\left(1,\frac{2 i m+3 b n+2 i}{4 b n};\frac{2 i m+5 b n+2 i}{4 b n};e^{-2 i \left(a+b \log \left(c x^n\right)\right)}\right)+(-i b n+2 m+2) \left(-3 b n \cot \left(a+b \log \left(c x^n\right)\right)+2 m+2\right)\right)}{(-i b n+2 m+2) (-3 i b n+2 m+2) (3 i b n+2 m+2) \csc ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}","\frac{2 x^{m+1} \, _2F_1\left(-\frac{3}{2},-\frac{2 i m+3 b n+2 i}{4 b n};-\frac{2 i m-b n+2 i}{4 b n};e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{(-3 i b n+2 m+2) \left(1-e^{2 i a} \left(c x^n\right)^{2 i b}\right)^{3/2} \csc ^{\frac{3}{2}}\left(a+b \log \left(c x^n\right)\right)}",1,"(2*x^(1 + m)*((2 + 2*m - I*b*n)*(2 + 2*m - 3*b*n*Cot[a + b*Log[c*x^n]]) + (3*b^2*n^2*(-1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))*Csc[a + b*Log[c*x^n]]^2*Hypergeometric2F1[1, (2*I + (2*I)*m + 3*b*n)/(4*b*n), (2*I + (2*I)*m + 5*b*n)/(4*b*n), E^((-2*I)*(a + b*Log[c*x^n]))])/(E^((2*I)*a)*(c*x^n)^((2*I)*b))))/((2 + 2*m - I*b*n)*(2 + 2*m - (3*I)*b*n)*(2 + 2*m + (3*I)*b*n)*Csc[a + b*Log[c*x^n]]^(3/2))","A",0
328,1,169,139,1.6871538,"\int (e x)^m \csc ^p\left(d \left(a+b \log \left(c x^n\right)\right)\right) \, dx","Integrate[(e*x)^m*Csc[d*(a + b*Log[c*x^n])]^p,x]","\frac{x (e x)^m \left(2-2 e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)^p \left(\frac{i e^{i a d} \left(c x^n\right)^{i b d}}{-1+e^{2 i a d} \left(c x^n\right)^{2 i b d}}\right)^p \, _2F_1\left(p,-\frac{i (m+i b d n p+1)}{2 b d n};\frac{1}{2} \left(-\frac{i (m+1)}{b d n}+p+2\right);e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)}{i b d n p+m+1}","\frac{(e x)^{m+1} \left(1-e^{2 i a d} \left(c x^n\right)^{2 i b d}\right)^p \, _2F_1\left(p,-\frac{i m-b d n p+i}{2 b d n};\frac{1}{2} \left(-\frac{i (m+1)}{b d n}+p+2\right);e^{2 i a d} \left(c x^n\right)^{2 i b d}\right) \csc ^p\left(d \left(a+b \log \left(c x^n\right)\right)\right)}{e (i b d n p+m+1)}",1,"(x*(e*x)^m*(2 - 2*E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))^p*((I*E^(I*a*d)*(c*x^n)^(I*b*d))/(-1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)))^p*Hypergeometric2F1[p, ((-1/2*I)*(1 + m + I*b*d*n*p))/(b*d*n), (2 - (I*(1 + m))/(b*d*n) + p)/2, E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)])/(1 + m + I*b*d*n*p)","A",0
329,1,142,106,1.1095659,"\int x \csc ^p\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[x*Csc[a + b*Log[c*x^n]]^p,x]","-\frac{i x^2 \left(2-2 e^{2 i a} \left(c x^n\right)^{2 i b}\right)^p \left(\frac{i e^{i a} \left(c x^n\right)^{i b}}{-1+e^{2 i a} \left(c x^n\right)^{2 i b}}\right)^p \, _2F_1\left(\frac{p}{2}-\frac{i}{b n},p;\frac{p}{2}-\frac{i}{b n}+1;e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{b n p-2 i}","\frac{x^2 \left(1-e^{2 i a} \left(c x^n\right)^{2 i b}\right)^p \, _2F_1\left(p,\frac{1}{2} \left(p-\frac{2 i}{b n}\right);\frac{1}{2} \left(p-\frac{2 i}{b n}+2\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right) \csc ^p\left(a+b \log \left(c x^n\right)\right)}{2+i b n p}",1,"((-I)*x^2*(2 - 2*E^((2*I)*a)*(c*x^n)^((2*I)*b))^p*((I*E^(I*a)*(c*x^n)^(I*b))/(-1 + E^((2*I)*a)*(c*x^n)^((2*I)*b)))^p*Hypergeometric2F1[(-I)/(b*n) + p/2, p, 1 - I/(b*n) + p/2, E^((2*I)*a)*(c*x^n)^((2*I)*b)])/(-2*I + b*n*p)","A",0
330,1,142,107,0.8657022,"\int \csc ^p\left(a+b \log \left(c x^n\right)\right) \, dx","Integrate[Csc[a + b*Log[c*x^n]]^p,x]","-\frac{i x \left(2-2 e^{2 i a} \left(c x^n\right)^{2 i b}\right)^p \left(\frac{i e^{i a} \left(c x^n\right)^{i b}}{-1+e^{2 i a} \left(c x^n\right)^{2 i b}}\right)^p \, _2F_1\left(p,\frac{b n p-i}{2 b n};\frac{1}{2} \left(p-\frac{i}{b n}+2\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right)}{b n p-i}","\frac{x \left(1-e^{2 i a} \left(c x^n\right)^{2 i b}\right)^p \, _2F_1\left(p,-\frac{i-b n p}{2 b n};\frac{1}{2} \left(p-\frac{i}{b n}+2\right);e^{2 i a} \left(c x^n\right)^{2 i b}\right) \csc ^p\left(a+b \log \left(c x^n\right)\right)}{1+i b n p}",1,"((-I)*x*(2 - 2*E^((2*I)*a)*(c*x^n)^((2*I)*b))^p*((I*E^(I*a)*(c*x^n)^(I*b))/(-1 + E^((2*I)*a)*(c*x^n)^((2*I)*b)))^p*Hypergeometric2F1[p, (-I + b*n*p)/(2*b*n), (2 - I/(b*n) + p)/2, E^((2*I)*a)*(c*x^n)^((2*I)*b)])/(-I + b*n*p)","A",0