1,1,416,547,21.3806479,"\int \frac{\cot ^5(d+e x)}{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx","Integrate[Cot[d + e*x]^5/Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2],x]","\frac{\sqrt{\frac{a \cos (2 (d+e x))-a-b \sin (2 (d+e x))-c \cos (2 (d+e x))-c}{\cos (2 (d+e x))-1}} \left(\frac{16 a c-15 b^2+32 c^2}{24 c^3}+\frac{5 b \cot (d+e x)}{12 c^2}-\frac{\csc ^2(d+e x)}{3 c}\right)}{e}+\frac{\tan (d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)} \left(b \left(5 b^2-4 c (3 a+2 c)\right) \tanh ^{-1}\left(\frac{b \tan (d+e x)+2 c}{2 \sqrt{c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right)-\frac{8 i c^{7/2} \tan ^{-1}\left(\frac{(b+2 i a) \tan (d+e x)+i b+2 c}{2 \sqrt{a-i b-c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right)}{\sqrt{a-i b-c}}+\frac{8 c^{7/2} \tanh ^{-1}\left(\frac{(2 a+i b) \tan (d+e x)+b+2 i c}{2 \sqrt{a+i b-c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right)}{\sqrt{a+i b-c}}\right)}{16 c^{7/2} e \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}","-\frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left(\frac{-\sqrt{a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}+\frac{\sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left(\frac{\sqrt{a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}+\frac{b \left(5 b^2-12 a c\right) \tanh ^{-1}\left(\frac{b+2 c \cot (d+e x)}{2 \sqrt{c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{16 c^{7/2} e}-\frac{\left(-16 a c+15 b^2-10 b c \cot (d+e x)\right) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{24 c^3 e}-\frac{b \tanh ^{-1}\left(\frac{b+2 c \cot (d+e x)}{2 \sqrt{c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{2 c^{3/2} e}-\frac{\cot ^2(d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{3 c e}+\frac{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c e}",1,"(((-15*b^2 + 16*a*c + 32*c^2)/(24*c^3) + (5*b*Cot[d + e*x])/(12*c^2) - Csc[d + e*x]^2/(3*c))*Sqrt[(-a - c + a*Cos[2*(d + e*x)] - c*Cos[2*(d + e*x)] - b*Sin[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)])])/e + ((((-8*I)*c^(7/2)*ArcTan[(I*b + 2*c + ((2*I)*a + b)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])])/Sqrt[a - I*b - c] + (8*c^(7/2)*ArcTanh[(b + (2*I)*c + (2*a + I*b)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])])/Sqrt[a + I*b - c] + b*(5*b^2 - 4*c*(3*a + 2*c))*ArcTanh[(2*c + b*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x])/(16*c^(7/2)*e*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])","C",0
2,1,352,384,22.2818946,"\int \frac{\cot ^3(d+e x)}{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx","Integrate[Cot[d + e*x]^3/Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2],x]","-\frac{\sqrt{\frac{a \cos (2 (d+e x))-a-b \sin (2 (d+e x))-c \cos (2 (d+e x))-c}{\cos (2 (d+e x))-1}}}{c e}-\frac{\tan (d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)} \left(-\frac{i c^{3/2} \tan ^{-1}\left(\frac{(b+2 i a) \tan (d+e x)+i b+2 c}{2 \sqrt{a-i b-c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right)}{\sqrt{a-i b-c}}+\frac{c^{3/2} \tanh ^{-1}\left(\frac{(2 a+i b) \tan (d+e x)+b+2 i c}{2 \sqrt{a+i b-c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right)}{\sqrt{a+i b-c}}-b \tanh ^{-1}\left(\frac{b \tan (d+e x)+2 c}{2 \sqrt{c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right)\right)}{2 c^{3/2} e \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}","\frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left(\frac{-\sqrt{a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}-\frac{\sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left(\frac{\sqrt{a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}+\frac{b \tanh ^{-1}\left(\frac{b+2 c \cot (d+e x)}{2 \sqrt{c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{2 c^{3/2} e}-\frac{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c e}",1,"-(Sqrt[(-a - c + a*Cos[2*(d + e*x)] - c*Cos[2*(d + e*x)] - b*Sin[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)])]/(c*e)) - ((((-I)*c^(3/2)*ArcTan[(I*b + 2*c + ((2*I)*a + b)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])])/Sqrt[a - I*b - c] + (c^(3/2)*ArcTanh[(b + (2*I)*c + (2*a + I*b)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])])/Sqrt[a + I*b - c] - b*ArcTanh[(2*c + b*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x])/(2*c^(3/2)*e*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])","C",0
3,1,253,294,11.4309607,"\int \frac{\cot (d+e x)}{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx","Integrate[Cot[d + e*x]/Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2],x]","\frac{\tan (d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)} \left(\sqrt{a-i b-c} \tanh ^{-1}\left(\frac{(2 a+i b) \tan (d+e x)+b+2 i c}{2 \sqrt{a+i b-c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right)-i \sqrt{a+i b-c} \tan ^{-1}\left(\frac{(b+2 i a) \tan (d+e x)+i b+2 c}{2 \sqrt{a-i b-c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right)\right)}{2 e \sqrt{a-i b-c} \sqrt{a+i b-c} \sqrt{a \tan ^2(d+e x)+b \tan (d+e x)+c}}","\frac{\sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left(\frac{\sqrt{a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}-\frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left(\frac{-\sqrt{a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}",1,"(((-I)*Sqrt[a + I*b - c]*ArcTan[(I*b + 2*c + ((2*I)*a + b)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])] + Sqrt[a - I*b - c]*ArcTanh[(b + (2*I)*c + (2*a + I*b)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + I*b - c]*e*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])","C",1
4,1,64621,349,35.8568732,"\int \frac{\tan (d+e x)}{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx","Integrate[Tan[d + e*x]/Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2],x]","\text{Result too large to show}","\frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left(\frac{-\sqrt{a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}-\frac{\sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left(\frac{\sqrt{a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}+\frac{\tanh ^{-1}\left(\frac{2 a+b \cot (d+e x)}{2 \sqrt{a} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{\sqrt{a} e}",1,"Result too large to show","C",0
5,1,134907,501,40.1389028,"\int \frac{\tan ^3(d+e x)}{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx","Integrate[Tan[d + e*x]^3/Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2],x]","\text{Result too large to show}","\frac{\left(3 b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b \cot (d+e x)}{2 \sqrt{a} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{8 a^{5/2} e}-\frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left(\frac{-\sqrt{a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}+\frac{\sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left(\frac{\sqrt{a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}-\frac{3 b \tan (d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{4 a^2 e}+\frac{\tan ^2(d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{2 a e}-\frac{\tanh ^{-1}\left(\frac{2 a+b \cot (d+e x)}{2 \sqrt{a} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{\sqrt{a} e}",1,"Result too large to show","C",0
6,1,4224,976,38.8902567,"\int \cot ^5(d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx","Integrate[Cot[d + e*x]^5*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2],x]","\text{Result too large to show}","-\frac{\left(c \cot ^2(d+e x)+b \cot (d+e x)+a\right)^{3/2} \cot ^2(d+e x)}{5 c e}-\frac{\left(35 b^2-42 c \cot (d+e x) b-32 a c\right) \left(c \cot ^2(d+e x)+b \cot (d+e x)+a\right)^{3/2}}{240 c^3 e}+\frac{\left(c \cot ^2(d+e x)+b \cot (d+e x)+a\right)^{3/2}}{3 c e}-\frac{\sqrt{a^2-\left(2 c+\sqrt{a^2-2 c a+b^2+c^2}\right) a+b^2+c \left(c+\sqrt{a^2-2 c a+b^2+c^2}\right)} \tan ^{-1}\left(\frac{b^2-\sqrt{a^2-2 c a+b^2+c^2} \cot (d+e x) b+(a-c) \left(a-c-\sqrt{a^2-2 c a+b^2+c^2}\right)}{\sqrt{2} \sqrt[4]{a^2-2 c a+b^2+c^2} \sqrt{a^2-\left(2 c+\sqrt{a^2-2 c a+b^2+c^2}\right) a+b^2+c \left(c+\sqrt{a^2-2 c a+b^2+c^2}\right)} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right)}{\sqrt{2} \sqrt[4]{a^2-2 c a+b^2+c^2} e}-\frac{b \left(7 b^2-12 a c\right) \left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c \cot (d+e x)}{2 \sqrt{c} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right)}{256 c^{9/2} e}+\frac{b \left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c \cot (d+e x)}{2 \sqrt{c} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right)}{16 c^{5/2} e}-\frac{b \tanh ^{-1}\left(\frac{b+2 c \cot (d+e x)}{2 \sqrt{c} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right)}{2 \sqrt{c} e}+\frac{\sqrt{a^2-\left(2 c-\sqrt{a^2-2 c a+b^2+c^2}\right) a+b^2+c \left(c-\sqrt{a^2-2 c a+b^2+c^2}\right)} \tanh ^{-1}\left(\frac{b^2+\sqrt{a^2-2 c a+b^2+c^2} \cot (d+e x) b+(a-c) \left(a-c+\sqrt{a^2-2 c a+b^2+c^2}\right)}{\sqrt{2} \sqrt[4]{a^2-2 c a+b^2+c^2} \sqrt{a^2-\left(2 c-\sqrt{a^2-2 c a+b^2+c^2}\right) a+b^2+c \left(c-\sqrt{a^2-2 c a+b^2+c^2}\right)} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right)}{\sqrt{2} \sqrt[4]{a^2-2 c a+b^2+c^2} e}+\frac{b \left(7 b^2-12 a c\right) (b+2 c \cot (d+e x)) \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}{128 c^4 e}-\frac{b (b+2 c \cot (d+e x)) \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}{8 c^2 e}-\frac{\sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}{e}",1,"((-1/1920*(-105*b^4 + 460*a*b^2*c - 256*a^2*c^2 + 296*b^2*c^2 - 768*a*c^3 + 2944*c^4)/c^4 + ((-35*b^3*Cos[d + e*x] + 116*a*b*c*Cos[d + e*x] + 104*b*c^2*Cos[d + e*x])*Csc[d + e*x])/(960*c^3) + ((7*b^2 - 16*a*c + 176*c^2)*Csc[d + e*x]^2)/(240*c^2) - (b*Cot[d + e*x]*Csc[d + e*x]^2)/(40*c) - Csc[d + e*x]^4/5)*Sqrt[(-a - c + a*Cos[2*(d + e*x)] - c*Cos[2*(d + e*x)] - b*Sin[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)])])/e + (Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*(b*(7*b^4 - 8*b^2*c*(5*a + 2*c) + 16*c^2*(3*a^2 + 4*a*c + 8*c^2))*Log[Tan[d + e*x]] - 128*Sqrt[a - I*b - c]*c^(9/2)*Log[(-2*c - (2*I)*a*Tan[d + e*x] - b*(I + Tan[d + e*x]) + (2*I)*Sqrt[a - I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])/(128*(a - I*b - c)^(3/2)*c^4*(-I + Tan[d + e*x]))] - b*(7*b^4 - 8*b^2*c*(5*a + 2*c) + 16*c^2*(3*a^2 + 4*a*c + 8*c^2))*Log[2*c + b*Tan[d + e*x] + 2*Sqrt[c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])]] + 128*Sqrt[a + I*b - c]*c^(9/2)*Log[(2*c + b*(-I + Tan[d + e*x]) - (2*I)*(a*Tan[d + e*x] + Sqrt[a + I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])]))/(128*(a + I*b - c)^(3/2)*c^4*(I + Tan[d + e*x]))])*((7*b^5*Sqrt[-(a/(-1 + Cos[2*(d + e*x)])) - c/(-1 + Cos[2*(d + e*x)]) + (a*Cos[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)]) - (c*Cos[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)]) - (b*Sin[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)])])/(128*c^4*(a + c - a*Cos[2*(d + e*x)] + c*Cos[2*(d + e*x)] + b*Sin[2*(d + e*x)])) - (5*a*b^3*Sqrt[-(a/(-1 + Cos[2*(d + e*x)])) - c/(-1 + Cos[2*(d + e*x)]) + (a*Cos[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)]) - (c*Cos[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)]) - (b*Sin[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)])])/(16*c^3*(a + c - a*Cos[2*(d + e*x)] + c*Cos[2*(d + e*x)] + b*Sin[2*(d + e*x)])) + (3*a^2*b*Sqrt[-(a/(-1 + Cos[2*(d + e*x)])) - c/(-1 + Cos[2*(d + e*x)]) + (a*Cos[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)]) - (c*Cos[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)]) - (b*Sin[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)])])/(8*c^2*(a + c - a*Cos[2*(d + e*x)] + c*Cos[2*(d + e*x)] + b*Sin[2*(d + e*x)])) - (b^3*Sqrt[-(a/(-1 + Cos[2*(d + e*x)])) - c/(-1 + Cos[2*(d + e*x)]) + (a*Cos[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)]) - (c*Cos[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)]) - (b*Sin[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)])])/(8*c^2*(a + c - a*Cos[2*(d + e*x)] + c*Cos[2*(d + e*x)] + b*Sin[2*(d + e*x)])) + (a*b*Sqrt[-(a/(-1 + Cos[2*(d + e*x)])) - c/(-1 + Cos[2*(d + e*x)]) + (a*Cos[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)]) - (c*Cos[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)]) - (b*Sin[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)])])/(2*c*(a + c - a*Cos[2*(d + e*x)] + c*Cos[2*(d + e*x)] + b*Sin[2*(d + e*x)])) + (b*Cos[2*(d + e*x)]*Sqrt[-(a/(-1 + Cos[2*(d + e*x)])) - c/(-1 + Cos[2*(d + e*x)]) + (a*Cos[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)]) - (c*Cos[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)]) - (b*Sin[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)])])/(a + c - a*Cos[2*(d + e*x)] + c*Cos[2*(d + e*x)] + b*Sin[2*(d + e*x)]) + (a*Sin[2*(d + e*x)]*Sqrt[-(a/(-1 + Cos[2*(d + e*x)])) - c/(-1 + Cos[2*(d + e*x)]) + (a*Cos[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)]) - (c*Cos[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)]) - (b*Sin[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)])])/(a + c - a*Cos[2*(d + e*x)] + c*Cos[2*(d + e*x)] + b*Sin[2*(d + e*x)]) - (c*Sin[2*(d + e*x)]*Sqrt[-(a/(-1 + Cos[2*(d + e*x)])) - c/(-1 + Cos[2*(d + e*x)]) + (a*Cos[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)]) - (c*Cos[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)]) - (b*Sin[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)])])/(a + c - a*Cos[2*(d + e*x)] + c*Cos[2*(d + e*x)] + b*Sin[2*(d + e*x)]))*Tan[d + e*x])/(256*c^(9/2)*e*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])]*(-1/512*(Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*(b*(7*b^4 - 8*b^2*c*(5*a + 2*c) + 16*c^2*(3*a^2 + 4*a*c + 8*c^2))*Log[Tan[d + e*x]] - 128*Sqrt[a - I*b - c]*c^(9/2)*Log[(-2*c - (2*I)*a*Tan[d + e*x] - b*(I + Tan[d + e*x]) + (2*I)*Sqrt[a - I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])/(128*(a - I*b - c)^(3/2)*c^4*(-I + Tan[d + e*x]))] - b*(7*b^4 - 8*b^2*c*(5*a + 2*c) + 16*c^2*(3*a^2 + 4*a*c + 8*c^2))*Log[2*c + b*Tan[d + e*x] + 2*Sqrt[c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])]] + 128*Sqrt[a + I*b - c]*c^(9/2)*Log[(2*c + b*(-I + Tan[d + e*x]) - (2*I)*(a*Tan[d + e*x] + Sqrt[a + I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])]))/(128*(a + I*b - c)^(3/2)*c^4*(I + Tan[d + e*x]))])*Tan[d + e*x]*(a*Sec[d + e*x]^2*Tan[d + e*x] + Sec[d + e*x]^2*(b + a*Tan[d + e*x])))/(c^(9/2)*(c + Tan[d + e*x]*(b + a*Tan[d + e*x]))^(3/2)) + (Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*(b*(7*b^4 - 8*b^2*c*(5*a + 2*c) + 16*c^2*(3*a^2 + 4*a*c + 8*c^2))*Log[Tan[d + e*x]] - 128*Sqrt[a - I*b - c]*c^(9/2)*Log[(-2*c - (2*I)*a*Tan[d + e*x] - b*(I + Tan[d + e*x]) + (2*I)*Sqrt[a - I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])/(128*(a - I*b - c)^(3/2)*c^4*(-I + Tan[d + e*x]))] - b*(7*b^4 - 8*b^2*c*(5*a + 2*c) + 16*c^2*(3*a^2 + 4*a*c + 8*c^2))*Log[2*c + b*Tan[d + e*x] + 2*Sqrt[c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])]] + 128*Sqrt[a + I*b - c]*c^(9/2)*Log[(2*c + b*(-I + Tan[d + e*x]) - (2*I)*(a*Tan[d + e*x] + Sqrt[a + I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])]))/(128*(a + I*b - c)^(3/2)*c^4*(I + Tan[d + e*x]))])*Sec[d + e*x]^2)/(256*c^(9/2)*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])]) + ((-(b*Csc[d + e*x]^2) - 2*c*Cot[d + e*x]*Csc[d + e*x]^2)*(b*(7*b^4 - 8*b^2*c*(5*a + 2*c) + 16*c^2*(3*a^2 + 4*a*c + 8*c^2))*Log[Tan[d + e*x]] - 128*Sqrt[a - I*b - c]*c^(9/2)*Log[(-2*c - (2*I)*a*Tan[d + e*x] - b*(I + Tan[d + e*x]) + (2*I)*Sqrt[a - I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])/(128*(a - I*b - c)^(3/2)*c^4*(-I + Tan[d + e*x]))] - b*(7*b^4 - 8*b^2*c*(5*a + 2*c) + 16*c^2*(3*a^2 + 4*a*c + 8*c^2))*Log[2*c + b*Tan[d + e*x] + 2*Sqrt[c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])]] + 128*Sqrt[a + I*b - c]*c^(9/2)*Log[(2*c + b*(-I + Tan[d + e*x]) - (2*I)*(a*Tan[d + e*x] + Sqrt[a + I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])]))/(128*(a + I*b - c)^(3/2)*c^4*(I + Tan[d + e*x]))])*Tan[d + e*x])/(512*c^(9/2)*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])]) + (Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x]*(b*(7*b^4 - 8*b^2*c*(5*a + 2*c) + 16*c^2*(3*a^2 + 4*a*c + 8*c^2))*Csc[d + e*x]*Sec[d + e*x] - (b*(7*b^4 - 8*b^2*c*(5*a + 2*c) + 16*c^2*(3*a^2 + 4*a*c + 8*c^2))*(b*Sec[d + e*x]^2 + (Sqrt[c]*(a*Sec[d + e*x]^2*Tan[d + e*x] + Sec[d + e*x]^2*(b + a*Tan[d + e*x])))/Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])]))/(2*c + b*Tan[d + e*x] + 2*Sqrt[c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])]) - (16384*(a - I*b - c)^2*c^(17/2)*(-I + Tan[d + e*x])*(((-2*I)*a*Sec[d + e*x]^2 - b*Sec[d + e*x]^2 + (I*Sqrt[a - I*b - c]*(a*Sec[d + e*x]^2*Tan[d + e*x] + Sec[d + e*x]^2*(b + a*Tan[d + e*x])))/Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])/(128*(a - I*b - c)^(3/2)*c^4*(-I + Tan[d + e*x])) - (Sec[d + e*x]^2*(-2*c - (2*I)*a*Tan[d + e*x] - b*(I + Tan[d + e*x]) + (2*I)*Sqrt[a - I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])]))/(128*(a - I*b - c)^(3/2)*c^4*(-I + Tan[d + e*x])^2)))/(-2*c - (2*I)*a*Tan[d + e*x] - b*(I + Tan[d + e*x]) + (2*I)*Sqrt[a - I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])]) + (16384*(a + I*b - c)^2*c^(17/2)*(I + Tan[d + e*x])*((b*Sec[d + e*x]^2 - (2*I)*(a*Sec[d + e*x]^2 + (Sqrt[a + I*b - c]*(a*Sec[d + e*x]^2*Tan[d + e*x] + Sec[d + e*x]^2*(b + a*Tan[d + e*x])))/(2*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])))/(128*(a + I*b - c)^(3/2)*c^4*(I + Tan[d + e*x])) - (Sec[d + e*x]^2*(2*c + b*(-I + Tan[d + e*x]) - (2*I)*(a*Tan[d + e*x] + Sqrt[a + I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])))/(128*(a + I*b - c)^(3/2)*c^4*(I + Tan[d + e*x])^2)))/(2*c + b*(-I + Tan[d + e*x]) - (2*I)*(a*Tan[d + e*x] + Sqrt[a + I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])]))))/(256*c^(9/2)*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])))","C",0
7,1,411,747,27.0927065,"\int \cot ^3(d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx","Integrate[Cot[d + e*x]^3*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2],x]","\frac{\sqrt{\frac{a \cos (2 (d+e x))-a-b \sin (2 (d+e x))-c \cos (2 (d+e x))-c}{\cos (2 (d+e x))-1}} \left(\frac{-8 a c+3 b^2+32 c^2}{24 c^2}-\frac{b \cot (d+e x)}{12 c}-\frac{1}{3} \csc ^2(d+e x)\right)}{e}+\frac{\tan (d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)} \left(-b \left(b^2-4 c (a+2 c)\right) \tanh ^{-1}\left(\frac{b \tan (d+e x)+2 c}{2 \sqrt{c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right)+8 i c^{5/2} \sqrt{a-i b-c} \tan ^{-1}\left(\frac{(b+2 i a) \tan (d+e x)+i b+2 c}{2 \sqrt{a-i b-c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right)-8 c^{5/2} \sqrt{a+i b-c} \tanh ^{-1}\left(\frac{(2 a+i b) \tan (d+e x)+b+2 i c}{2 \sqrt{a+i b-c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right)\right)}{16 c^{5/2} e \sqrt{a \tan ^2(d+e x)+b \tan (d+e x)+c}}","\frac{\sqrt{-a \left(\sqrt{a^2-2 a c+b^2+c^2}+2 c\right)+c \left(\sqrt{a^2-2 a c+b^2+c^2}+c\right)+a^2+b^2} \tan ^{-1}\left(\frac{-b \sqrt{a^2-2 a c+b^2+c^2} \cot (d+e x)+(a-c) \left(-\sqrt{a^2-2 a c+b^2+c^2}+a-c\right)+b^2}{\sqrt{2} \sqrt[4]{a^2-2 a c+b^2+c^2} \sqrt{-a \left(\sqrt{a^2-2 a c+b^2+c^2}+2 c\right)+c \left(\sqrt{a^2-2 a c+b^2+c^2}+c\right)+a^2+b^2} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{\sqrt{2} e \sqrt[4]{a^2-2 a c+b^2+c^2}}-\frac{\sqrt{-a \left(2 c-\sqrt{a^2-2 a c+b^2+c^2}\right)+c \left(c-\sqrt{a^2-2 a c+b^2+c^2}\right)+a^2+b^2} \tanh ^{-1}\left(\frac{b \sqrt{a^2-2 a c+b^2+c^2} \cot (d+e x)+(a-c) \left(\sqrt{a^2-2 a c+b^2+c^2}+a-c\right)+b^2}{\sqrt{2} \sqrt[4]{a^2-2 a c+b^2+c^2} \sqrt{-a \left(2 c-\sqrt{a^2-2 a c+b^2+c^2}\right)+c \left(c-\sqrt{a^2-2 a c+b^2+c^2}\right)+a^2+b^2} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{\sqrt{2} e \sqrt[4]{a^2-2 a c+b^2+c^2}}-\frac{b \left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c \cot (d+e x)}{2 \sqrt{c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{16 c^{5/2} e}+\frac{b (b+2 c \cot (d+e x)) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{8 c^2 e}-\frac{\left(a+b \cot (d+e x)+c \cot ^2(d+e x)\right)^{3/2}}{3 c e}+\frac{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{e}+\frac{b \tanh ^{-1}\left(\frac{b+2 c \cot (d+e x)}{2 \sqrt{c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{2 \sqrt{c} e}",1,"(((3*b^2 - 8*a*c + 32*c^2)/(24*c^2) - (b*Cot[d + e*x])/(12*c) - Csc[d + e*x]^2/3)*Sqrt[(-a - c + a*Cos[2*(d + e*x)] - c*Cos[2*(d + e*x)] - b*Sin[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)])])/e + (((8*I)*Sqrt[a - I*b - c]*c^(5/2)*ArcTan[(I*b + 2*c + ((2*I)*a + b)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])] - 8*Sqrt[a + I*b - c]*c^(5/2)*ArcTanh[(b + (2*I)*c + (2*a + I*b)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])] - b*(b^2 - 4*c*(a + 2*c))*ArcTanh[(2*c + b*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x])/(16*c^(5/2)*e*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])","C",0
8,1,350,602,23.7946841,"\int \cot (d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx","Integrate[Cot[d + e*x]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2],x]","-\frac{\sqrt{\frac{a \cos (2 (d+e x))-a-b \sin (2 (d+e x))-c \cos (2 (d+e x))-c}{\cos (2 (d+e x))-1}}}{e}+\frac{\tan (d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)} \left(-i \sqrt{c} \sqrt{a-i b-c} \tan ^{-1}\left(\frac{(b+2 i a) \tan (d+e x)+i b+2 c}{2 \sqrt{a-i b-c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right)+\sqrt{c} \sqrt{a+i b-c} \tanh ^{-1}\left(\frac{(2 a+i b) \tan (d+e x)+b+2 i c}{2 \sqrt{a+i b-c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right)-b \tanh ^{-1}\left(\frac{b \tan (d+e x)+2 c}{2 \sqrt{c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right)\right)}{2 \sqrt{c} e \sqrt{a \tan ^2(d+e x)+b \tan (d+e x)+c}}","-\frac{\sqrt{-a \left(\sqrt{a^2-2 a c+b^2+c^2}+2 c\right)+c \left(\sqrt{a^2-2 a c+b^2+c^2}+c\right)+a^2+b^2} \tan ^{-1}\left(\frac{-b \sqrt{a^2-2 a c+b^2+c^2} \cot (d+e x)+(a-c) \left(-\sqrt{a^2-2 a c+b^2+c^2}+a-c\right)+b^2}{\sqrt{2} \sqrt[4]{a^2-2 a c+b^2+c^2} \sqrt{-a \left(\sqrt{a^2-2 a c+b^2+c^2}+2 c\right)+c \left(\sqrt{a^2-2 a c+b^2+c^2}+c\right)+a^2+b^2} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{\sqrt{2} e \sqrt[4]{a^2-2 a c+b^2+c^2}}+\frac{\sqrt{-a \left(2 c-\sqrt{a^2-2 a c+b^2+c^2}\right)+c \left(c-\sqrt{a^2-2 a c+b^2+c^2}\right)+a^2+b^2} \tanh ^{-1}\left(\frac{b \sqrt{a^2-2 a c+b^2+c^2} \cot (d+e x)+(a-c) \left(\sqrt{a^2-2 a c+b^2+c^2}+a-c\right)+b^2}{\sqrt{2} \sqrt[4]{a^2-2 a c+b^2+c^2} \sqrt{-a \left(2 c-\sqrt{a^2-2 a c+b^2+c^2}\right)+c \left(c-\sqrt{a^2-2 a c+b^2+c^2}\right)+a^2+b^2} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{\sqrt{2} e \sqrt[4]{a^2-2 a c+b^2+c^2}}-\frac{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{e}-\frac{b \tanh ^{-1}\left(\frac{b+2 c \cot (d+e x)}{2 \sqrt{c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{2 \sqrt{c} e}",1,"-(Sqrt[(-a - c + a*Cos[2*(d + e*x)] - c*Cos[2*(d + e*x)] - b*Sin[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)])]/e) + (((-I)*Sqrt[a - I*b - c]*Sqrt[c]*ArcTan[(I*b + 2*c + ((2*I)*a + b)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])] + Sqrt[a + I*b - c]*Sqrt[c]*ArcTanh[(b + (2*I)*c + (2*a + I*b)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])] - b*ArcTanh[(2*c + b*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x])/(2*Sqrt[c]*e*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])","C",0
9,1,278,570,16.8691772,"\int \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan (d+e x) \, dx","Integrate[Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x],x]","\frac{\tan (d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)} \left(i \sqrt{a-i b-c} \tan ^{-1}\left(\frac{(b+2 i a) \tan (d+e x)+i b+2 c}{2 \sqrt{a-i b-c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right)-\sqrt{a+i b-c} \tanh ^{-1}\left(\frac{(2 a+i b) \tan (d+e x)+b+2 i c}{2 \sqrt{a+i b-c} \sqrt{a \tan ^2(d+e x)+b \tan (d+e x)+c}}\right)+2 \sqrt{a} \tanh ^{-1}\left(\frac{2 a \tan (d+e x)+b}{2 \sqrt{a} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right)\right)}{2 e \sqrt{a \tan ^2(d+e x)+b \tan (d+e x)+c}}","\frac{\sqrt{-a \left(\sqrt{a^2-2 a c+b^2+c^2}+2 c\right)+c \left(\sqrt{a^2-2 a c+b^2+c^2}+c\right)+a^2+b^2} \tan ^{-1}\left(\frac{-b \sqrt{a^2-2 a c+b^2+c^2} \cot (d+e x)+(a-c) \left(-\sqrt{a^2-2 a c+b^2+c^2}+a-c\right)+b^2}{\sqrt{2} \sqrt[4]{a^2-2 a c+b^2+c^2} \sqrt{-a \left(\sqrt{a^2-2 a c+b^2+c^2}+2 c\right)+c \left(\sqrt{a^2-2 a c+b^2+c^2}+c\right)+a^2+b^2} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{\sqrt{2} e \sqrt[4]{a^2-2 a c+b^2+c^2}}-\frac{\sqrt{-a \left(2 c-\sqrt{a^2-2 a c+b^2+c^2}\right)+c \left(c-\sqrt{a^2-2 a c+b^2+c^2}\right)+a^2+b^2} \tanh ^{-1}\left(\frac{b \sqrt{a^2-2 a c+b^2+c^2} \cot (d+e x)+(a-c) \left(\sqrt{a^2-2 a c+b^2+c^2}+a-c\right)+b^2}{\sqrt{2} \sqrt[4]{a^2-2 a c+b^2+c^2} \sqrt{-a \left(2 c-\sqrt{a^2-2 a c+b^2+c^2}\right)+c \left(c-\sqrt{a^2-2 a c+b^2+c^2}\right)+a^2+b^2} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{\sqrt{2} e \sqrt[4]{a^2-2 a c+b^2+c^2}}+\frac{\sqrt{a} \tanh ^{-1}\left(\frac{2 a+b \cot (d+e x)}{2 \sqrt{a} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{e}",1,"((I*Sqrt[a - I*b - c]*ArcTan[(I*b + 2*c + ((2*I)*a + b)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])] - Sqrt[a + I*b - c]*ArcTanh[(b + (2*I)*c + (2*a + I*b)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])] + 2*Sqrt[a]*ArcTanh[(b + 2*a*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x])/(2*e*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])","C",1
10,1,465721,691,49.0854186,"\int \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan ^3(d+e x) \, dx","Integrate[Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x]^3,x]","\text{Result too large to show}","-\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b \cot (d+e x)}{2 \sqrt{a} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{8 a^{3/2} e}-\frac{\sqrt{-a \left(\sqrt{a^2-2 a c+b^2+c^2}+2 c\right)+c \left(\sqrt{a^2-2 a c+b^2+c^2}+c\right)+a^2+b^2} \tan ^{-1}\left(\frac{-b \sqrt{a^2-2 a c+b^2+c^2} \cot (d+e x)+(a-c) \left(-\sqrt{a^2-2 a c+b^2+c^2}+a-c\right)+b^2}{\sqrt{2} \sqrt[4]{a^2-2 a c+b^2+c^2} \sqrt{-a \left(\sqrt{a^2-2 a c+b^2+c^2}+2 c\right)+c \left(\sqrt{a^2-2 a c+b^2+c^2}+c\right)+a^2+b^2} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{\sqrt{2} e \sqrt[4]{a^2-2 a c+b^2+c^2}}+\frac{\sqrt{-a \left(2 c-\sqrt{a^2-2 a c+b^2+c^2}\right)+c \left(c-\sqrt{a^2-2 a c+b^2+c^2}\right)+a^2+b^2} \tanh ^{-1}\left(\frac{b \sqrt{a^2-2 a c+b^2+c^2} \cot (d+e x)+(a-c) \left(\sqrt{a^2-2 a c+b^2+c^2}+a-c\right)+b^2}{\sqrt{2} \sqrt[4]{a^2-2 a c+b^2+c^2} \sqrt{-a \left(2 c-\sqrt{a^2-2 a c+b^2+c^2}\right)+c \left(c-\sqrt{a^2-2 a c+b^2+c^2}\right)+a^2+b^2} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{\sqrt{2} e \sqrt[4]{a^2-2 a c+b^2+c^2}}+\frac{\tan ^2(d+e x) (2 a+b \cot (d+e x)) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{4 a e}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{2 a+b \cot (d+e x)}{2 \sqrt{a} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{e}",1,"Result too large to show","C",0
11,1,5618,1189,40.5203615,"\int \frac{\cot ^7(d+e x)}{\left(a+b \cot (d+e x)+c \cot ^2(d+e x)\right)^{3/2}} \, dx","Integrate[Cot[d + e*x]^7/(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2),x]","\text{Result too large to show}","-\frac{2 (2 a+b \cot (d+e x)) \cot ^4(d+e x)}{\left(b^2-4 a c\right) e \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}+\frac{2 b \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot ^3(d+e x)}{c \left(b^2-4 a c\right) e}-\frac{\left(7 b^2-16 a c\right) \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot ^2(d+e x)}{3 c^2 \left(b^2-4 a c\right) e}+\frac{2 (2 a+b \cot (d+e x)) \cot ^2(d+e x)}{\left(b^2-4 a c\right) e \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}+\frac{5 b \left(7 b^2-12 a c\right) \tanh ^{-1}\left(\frac{b+2 c \cot (d+e x)}{2 \sqrt{c} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right)}{16 c^{9/2} e}-\frac{3 b \tanh ^{-1}\left(\frac{b+2 c \cot (d+e x)}{2 \sqrt{c} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right)}{2 c^{5/2} e}+\frac{\sqrt{2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2+(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \tanh ^{-1}\left(\frac{b^2-\left(2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}\right) \cot (d+e x) b-(a-c) \left(a-c+\sqrt{a^2-2 c a+b^2+c^2}\right)}{\sqrt{2} \sqrt{2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2+(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right)}{\sqrt{2} \left(a^2-2 c a+b^2+c^2\right)^{3/2} e}-\frac{\sqrt{2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2-(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \tanh ^{-1}\left(\frac{b^2-\left(2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}\right) \cot (d+e x) b-(a-c) \left(a-c-\sqrt{a^2-2 c a+b^2+c^2}\right)}{\sqrt{2} \sqrt{2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2-(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right)}{\sqrt{2} \left(a^2-2 c a+b^2+c^2\right)^{3/2} e}+\frac{\left(3 b^2-2 c \cot (d+e x) b-8 a c\right) \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}{c^2 \left(b^2-4 a c\right) e}-\frac{\left(105 b^4-460 a c b^2-2 c \left(35 b^2-116 a c\right) \cot (d+e x) b+256 a^2 c^2\right) \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}{24 c^4 \left(b^2-4 a c\right) e}-\frac{2 (2 a+b \cot (d+e x))}{\left(b^2-4 a c\right) e \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}+\frac{2 \left(a \left(b^2-2 (a-c) c\right)+b c (a+c) \cot (d+e x)\right)}{\left(b^2+(a-c)^2\right) \left(b^2-4 a c\right) e \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}",1,"Result too large to show","C",0
12,1,737,865,28.0674382,"\int \frac{\cot ^5(d+e x)}{\left(a+b \cot (d+e x)+c \cot ^2(d+e x)\right)^{3/2}} \, dx","Integrate[Cot[d + e*x]^5/(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2),x]","\frac{\sqrt{\frac{a \cos (2 (d+e x))-a-b \sin (2 (d+e x))-c \cos (2 (d+e x))-c}{\cos (2 (d+e x))-1}} \left(-\frac{8 a^4 c-3 a^3 b^2-16 a^3 c^2+15 a^2 b^2 c+12 a^2 c^3-3 a b^4-7 a b^2 c^2-4 a c^4+b^4 c+b^2 c^3}{c^2 (a-c) (a-i b-c) (a+i b-c) \left(4 a c-b^2\right)}-\frac{2 \left(-a^4 b \sin (2 (d+e x))+4 a^4 c-2 a^3 b^2+6 a^3 b c \sin (2 (d+e x))-4 a^3 c^2-2 a^2 b^3 \sin (2 (d+e x))+8 a^2 b^2 c-5 a^2 b c^2 \sin (2 (d+e x))-2 a b^4+5 a b^3 c \sin (2 (d+e x))+b^5 (-\sin (2 (d+e x)))\right)}{c (a-c) (a-i b-c) (a+i b-c) \left(4 a c-b^2\right) (a \cos (2 (d+e x))-a-b \sin (2 (d+e x))-c \cos (2 (d+e x))-c)}\right)}{e}-\frac{\tan (d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)} \left(-3 b \left(a^2-2 a c+b^2+c^2\right) \tanh ^{-1}\left(\frac{b \tan (d+e x)+2 c}{2 \sqrt{c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right)+\frac{i c^{5/2} (a+i b-c) \tan ^{-1}\left(\frac{(b+2 i a) \tan (d+e x)+i b+2 c}{2 \sqrt{a-i b-c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right)}{\sqrt{a-i b-c}}+\frac{c^{5/2} (-a+i b+c) \tanh ^{-1}\left(\frac{(2 a+i b) \tan (d+e x)+b+2 i c}{2 \sqrt{a+i b-c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right)}{\sqrt{a+i b-c}}\right)}{2 c^{5/2} e \left(a^2-2 a c+b^2+c^2\right) \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}","-\frac{2 (2 a+b \cot (d+e x)) \cot ^2(d+e x)}{\left(b^2-4 a c\right) e \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}+\frac{3 b \tanh ^{-1}\left(\frac{b+2 c \cot (d+e x)}{2 \sqrt{c} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right)}{2 c^{5/2} e}-\frac{\sqrt{2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2+(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \tanh ^{-1}\left(\frac{b^2-\left(2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}\right) \cot (d+e x) b-(a-c) \left(a-c+\sqrt{a^2-2 c a+b^2+c^2}\right)}{\sqrt{2} \sqrt{2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2+(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right)}{\sqrt{2} \left(a^2-2 c a+b^2+c^2\right)^{3/2} e}+\frac{\sqrt{2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2-(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \tanh ^{-1}\left(\frac{b^2-\left(2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}\right) \cot (d+e x) b-(a-c) \left(a-c-\sqrt{a^2-2 c a+b^2+c^2}\right)}{\sqrt{2} \sqrt{2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2-(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right)}{\sqrt{2} \left(a^2-2 c a+b^2+c^2\right)^{3/2} e}-\frac{\left(3 b^2-2 c \cot (d+e x) b-8 a c\right) \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}{c^2 \left(b^2-4 a c\right) e}+\frac{2 (2 a+b \cot (d+e x))}{\left(b^2-4 a c\right) e \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}-\frac{2 \left(a \left(b^2-2 (a-c) c\right)+b c (a+c) \cot (d+e x)\right)}{\left(b^2+(a-c)^2\right) \left(b^2-4 a c\right) e \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}",1,"(Sqrt[(-a - c + a*Cos[2*(d + e*x)] - c*Cos[2*(d + e*x)] - b*Sin[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)])]*(-((-3*a^3*b^2 - 3*a*b^4 + 8*a^4*c + 15*a^2*b^2*c + b^4*c - 16*a^3*c^2 - 7*a*b^2*c^2 + 12*a^2*c^3 + b^2*c^3 - 4*a*c^4)/((a - c)*(a - I*b - c)*(a + I*b - c)*c^2*(-b^2 + 4*a*c))) - (2*(-2*a^3*b^2 - 2*a*b^4 + 4*a^4*c + 8*a^2*b^2*c - 4*a^3*c^2 - a^4*b*Sin[2*(d + e*x)] - 2*a^2*b^3*Sin[2*(d + e*x)] - b^5*Sin[2*(d + e*x)] + 6*a^3*b*c*Sin[2*(d + e*x)] + 5*a*b^3*c*Sin[2*(d + e*x)] - 5*a^2*b*c^2*Sin[2*(d + e*x)]))/((a - c)*(a - I*b - c)*(a + I*b - c)*c*(-b^2 + 4*a*c)*(-a - c + a*Cos[2*(d + e*x)] - c*Cos[2*(d + e*x)] - b*Sin[2*(d + e*x)]))))/e - (((I*(a + I*b - c)*c^(5/2)*ArcTan[(I*b + 2*c + ((2*I)*a + b)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])])/Sqrt[a - I*b - c] + (c^(5/2)*(-a + I*b + c)*ArcTanh[(b + (2*I)*c + (2*a + I*b)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])])/Sqrt[a + I*b - c] - 3*b*(a^2 + b^2 - 2*a*c + c^2)*ArcTanh[(2*c + b*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x])/(2*c^(5/2)*(a^2 + b^2 - 2*a*c + c^2)*e*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])","C",0
13,1,382,686,27.2210218,"\int \frac{\cot ^3(d+e x)}{\left(a+b \cot (d+e x)+c \cot ^2(d+e x)\right)^{3/2}} \, dx","Integrate[Cot[d + e*x]^3/(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2),x]","\frac{\frac{4 \sqrt{2} \left(b \left(a^2-3 a c+b^2\right) \cot (d+e x)+a \left(2 a^2-2 a c+b^2\right)\right)}{(a-i b-c) (a+i b-c) \left(4 a c-b^2\right) \sqrt{\csc ^2(d+e x) ((c-a) \cos (2 (d+e x))+a+b \sin (2 (d+e x))+c)}}-\frac{\tan (d+e x) \left(\frac{(i a+b-i c) \tan ^{-1}\left(\frac{(b-2 i a) \tan (d+e x)-i b+2 c}{2 \sqrt{a+i b-c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right)}{\sqrt{a+i b-c}}+\frac{(-i a+b+i c) \tan ^{-1}\left(\frac{(b+2 i a) \tan (d+e x)+i b+2 c}{2 \sqrt{a-i b-c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right)}{\sqrt{a-i b-c}}\right) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{\left(a^2-2 a c+b^2+c^2\right) \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}}{2 e}","\frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \tanh ^{-1}\left(\frac{-b \left(-\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c\right) \cot (d+e x)-(a-c) \left(\sqrt{a^2-2 a c+b^2+c^2}+a-c\right)+b^2}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{\sqrt{2} e \left(a^2-2 a c+b^2+c^2\right)^{3/2}}-\frac{\sqrt{\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{-(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \tanh ^{-1}\left(\frac{-b \left(\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c\right) \cot (d+e x)-(a-c) \left(-\sqrt{a^2-2 a c+b^2+c^2}+a-c\right)+b^2}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{-(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{\sqrt{2} e \left(a^2-2 a c+b^2+c^2\right)^{3/2}}-\frac{2 (2 a+b \cot (d+e x))}{e \left(b^2-4 a c\right) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}+\frac{2 \left(a \left(b^2-2 c (a-c)\right)+b c (a+c) \cot (d+e x)\right)}{e \left((a-c)^2+b^2\right) \left(b^2-4 a c\right) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}",1,"((4*Sqrt[2]*(a*(2*a^2 + b^2 - 2*a*c) + b*(a^2 + b^2 - 3*a*c)*Cot[d + e*x]))/((a - I*b - c)*(a + I*b - c)*(-b^2 + 4*a*c)*Sqrt[Csc[d + e*x]^2*(a + c + (-a + c)*Cos[2*(d + e*x)] + b*Sin[2*(d + e*x)])]) - ((((I*a + b - I*c)*ArcTan[((-I)*b + 2*c + ((-2*I)*a + b)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])])/Sqrt[a + I*b - c] + (((-I)*a + b + I*c)*ArcTan[(I*b + 2*c + ((2*I)*a + b)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])])/Sqrt[a - I*b - c])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x])/((a^2 + b^2 - 2*a*c + c^2)*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])]))/(2*e)","C",1
14,1,375,635,26.6381049,"\int \frac{\cot (d+e x)}{\left(a+b \cot (d+e x)+c \cot ^2(d+e x)\right)^{3/2}} \, dx","Integrate[Cot[d + e*x]/(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2),x]","\frac{\frac{\tan (d+e x) \left(\frac{(i a+b-i c) \tan ^{-1}\left(\frac{(b-2 i a) \tan (d+e x)-i b+2 c}{2 \sqrt{a+i b-c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right)}{\sqrt{a+i b-c}}+\frac{(-i a+b+i c) \tan ^{-1}\left(\frac{(b+2 i a) \tan (d+e x)+i b+2 c}{2 \sqrt{a-i b-c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right)}{\sqrt{a-i b-c}}\right) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{\left(a^2-2 a c+b^2+c^2\right) \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}-\frac{4 \sqrt{2} \left(-a b^2-b c (a+c) \cot (d+e x)+2 a c (a-c)\right)}{(a-i b-c) (a+i b-c) \left(4 a c-b^2\right) \sqrt{\csc ^2(d+e x) ((c-a) \cos (2 (d+e x))+a+b \sin (2 (d+e x))+c)}}}{2 e}","-\frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \tanh ^{-1}\left(\frac{-b \left(-\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c\right) \cot (d+e x)-(a-c) \left(\sqrt{a^2-2 a c+b^2+c^2}+a-c\right)+b^2}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{\sqrt{2} e \left(a^2-2 a c+b^2+c^2\right)^{3/2}}+\frac{\sqrt{\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{-(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \tanh ^{-1}\left(\frac{-b \left(\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c\right) \cot (d+e x)-(a-c) \left(-\sqrt{a^2-2 a c+b^2+c^2}+a-c\right)+b^2}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{-(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{\sqrt{2} e \left(a^2-2 a c+b^2+c^2\right)^{3/2}}-\frac{2 \left(a \left(b^2-2 c (a-c)\right)+b c (a+c) \cot (d+e x)\right)}{e \left((a-c)^2+b^2\right) \left(b^2-4 a c\right) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}",1,"((-4*Sqrt[2]*(-(a*b^2) + 2*a*(a - c)*c - b*c*(a + c)*Cot[d + e*x]))/((a - I*b - c)*(a + I*b - c)*(-b^2 + 4*a*c)*Sqrt[Csc[d + e*x]^2*(a + c + (-a + c)*Cos[2*(d + e*x)] + b*Sin[2*(d + e*x)])]) + ((((I*a + b - I*c)*ArcTan[((-I)*b + 2*c + ((-2*I)*a + b)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])])/Sqrt[a + I*b - c] + (((-I)*a + b + I*c)*ArcTan[(I*b + 2*c + ((2*I)*a + b)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])])/Sqrt[a - I*b - c])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x])/((a^2 + b^2 - 2*a*c + c^2)*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])]))/(2*e)","C",1
15,1,558961,749,50.1928778,"\int \frac{\tan (d+e x)}{\left(a+b \cot (d+e x)+c \cot ^2(d+e x)\right)^{3/2}} \, dx","Integrate[Tan[d + e*x]/(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2),x]","\text{Result too large to show}","\frac{\tanh ^{-1}\left(\frac{2 a+b \cot (d+e x)}{2 \sqrt{a} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{a^{3/2} e}+\frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \tanh ^{-1}\left(\frac{-b \left(-\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c\right) \cot (d+e x)-(a-c) \left(\sqrt{a^2-2 a c+b^2+c^2}+a-c\right)+b^2}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{\sqrt{2} e \left(a^2-2 a c+b^2+c^2\right)^{3/2}}-\frac{\sqrt{\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{-(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \tanh ^{-1}\left(\frac{-b \left(\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c\right) \cot (d+e x)-(a-c) \left(-\sqrt{a^2-2 a c+b^2+c^2}+a-c\right)+b^2}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{-(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right)}{\sqrt{2} e \left(a^2-2 a c+b^2+c^2\right)^{3/2}}-\frac{2 \left(-2 a c+b^2+b c \cot (d+e x)\right)}{a e \left(b^2-4 a c\right) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}+\frac{2 \left(a \left(b^2-2 c (a-c)\right)+b c (a+c) \cot (d+e x)\right)}{e \left((a-c)^2+b^2\right) \left(b^2-4 a c\right) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}",1,"Result too large to show","C",0
16,1,930953,1008,55.0467565,"\int \frac{\tan ^3(d+e x)}{\left(a+b \cot (d+e x)+c \cot ^2(d+e x)\right)^{3/2}} \, dx","Integrate[Tan[d + e*x]^3/(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2),x]","\text{Result too large to show}","\frac{\left(5 b^2-12 a c\right) \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a} \tan ^2(d+e x)}{2 a^2 \left(b^2-4 a c\right) e}-\frac{2 \left(b^2+c \cot (d+e x) b-2 a c\right) \tan ^2(d+e x)}{a \left(b^2-4 a c\right) e \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}-\frac{b \left(15 b^2-52 a c\right) \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a} \tan (d+e x)}{4 a^3 \left(b^2-4 a c\right) e}+\frac{3 \left(5 b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b \cot (d+e x)}{2 \sqrt{a} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right)}{8 a^{7/2} e}-\frac{\tanh ^{-1}\left(\frac{2 a+b \cot (d+e x)}{2 \sqrt{a} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right)}{a^{3/2} e}-\frac{\sqrt{2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2+(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \tanh ^{-1}\left(\frac{b^2-\left(2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}\right) \cot (d+e x) b-(a-c) \left(a-c+\sqrt{a^2-2 c a+b^2+c^2}\right)}{\sqrt{2} \sqrt{2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2+(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right)}{\sqrt{2} \left(a^2-2 c a+b^2+c^2\right)^{3/2} e}+\frac{\sqrt{2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2-(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \tanh ^{-1}\left(\frac{b^2-\left(2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}\right) \cot (d+e x) b-(a-c) \left(a-c-\sqrt{a^2-2 c a+b^2+c^2}\right)}{\sqrt{2} \sqrt{2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2-(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right)}{\sqrt{2} \left(a^2-2 c a+b^2+c^2\right)^{3/2} e}+\frac{2 \left(b^2+c \cot (d+e x) b-2 a c\right)}{a \left(b^2-4 a c\right) e \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}-\frac{2 \left(a \left(b^2-2 (a-c) c\right)+b c (a+c) \cot (d+e x)\right)}{\left(b^2+(a-c)^2\right) \left(b^2-4 a c\right) e \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}",1,"Result too large to show","C",0
17,1,179905,182,34.6325544,"\int \frac{\cot ^5(d+e x)}{\sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx","Integrate[Cot[d + e*x]^5/Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]","\text{Result too large to show}","\frac{(b+2 c) \tanh ^{-1}\left(\frac{b+2 c \cot ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{4 c^{3/2} e}-\frac{\sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 c e}+\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{2 e \sqrt{a-b+c}}",1,"Result too large to show","C",0
18,1,64578,141,46.5498901,"\int \frac{\cot ^3(d+e x)}{\sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx","Integrate[Cot[d + e*x]^3/Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]","\text{Result too large to show}","-\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{2 e \sqrt{a-b+c}}-\frac{\tanh ^{-1}\left(\frac{b+2 c \cot ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{2 \sqrt{c} e}",1,"Result too large to show","C",0
19,1,24736,79,34.9346027,"\int \frac{\cot (d+e x)}{\sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx","Integrate[Cot[d + e*x]/Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]","\text{Result too large to show}","\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{2 e \sqrt{a-b+c}}",1,"Result too large to show","C",0
20,1,44361,142,29.3286649,"\int \frac{\tan (d+e x)}{\sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx","Integrate[Tan[d + e*x]/Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]","\text{Result too large to show}","\frac{\tanh ^{-1}\left(\frac{2 a+b \cot ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{2 \sqrt{a} e}-\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{2 e \sqrt{a-b+c}}",1,"Result too large to show","C",0
21,1,37459,249,32.539653,"\int \frac{\tan ^3(d+e x)}{\sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx","Integrate[Tan[d + e*x]^3/Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]","\text{Result too large to show}","-\frac{b \tanh ^{-1}\left(\frac{2 a+b \cot ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{4 a^{3/2} e}+\frac{\tan ^2(d+e x) \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 a e}-\frac{\tanh ^{-1}\left(\frac{2 a+b \cot ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{2 \sqrt{a} e}+\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{2 e \sqrt{a-b+c}}",1,"Result too large to show","C",0
22,1,539292,270,38.5464776,"\int \cot ^5(d+e x) \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx","Integrate[Cot[d + e*x]^5*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]","\text{Result too large to show}","-\frac{\left(-4 b c (a-2 c)-8 c^2 (a+2 c)+b^3+2 b^2 c\right) \tanh ^{-1}\left(\frac{b+2 c \cot ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{32 c^{5/2} e}+\frac{\left(2 c (b+2 c) \cot ^2(d+e x)+(b-2 c) (b+4 c)\right) \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{16 c^2 e}-\frac{\left(a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right)^{3/2}}{6 c e}+\frac{\sqrt{a-b+c} \tanh ^{-1}\left(\frac{2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{2 e}",1,"Result too large to show","C",0
23,1,412434,209,36.6173758,"\int \cot ^3(d+e x) \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx","Integrate[Cot[d + e*x]^3*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]","\text{Result too large to show}","\frac{\left(-4 c (a+2 c)+b^2+4 b c\right) \tanh ^{-1}\left(\frac{b+2 c \cot ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{16 c^{3/2} e}-\frac{\left(b+2 c \cot ^2(d+e x)-4 c\right) \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{8 c e}-\frac{\sqrt{a-b+c} \tanh ^{-1}\left(\frac{2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{2 e}",1,"Result too large to show","C",0
24,1,286262,179,35.0819702,"\int \cot (d+e x) \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx","Integrate[Cot[d + e*x]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]","\text{Result too large to show}","-\frac{\sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 e}+\frac{\sqrt{a-b+c} \tanh ^{-1}\left(\frac{2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{2 e}-\frac{(b-2 c) \tanh ^{-1}\left(\frac{b+2 c \cot ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{4 \sqrt{c} e}",1,"Result too large to show","C",0
25,1,253,203,12.8175414,"\int \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan (d+e x) \, dx","Integrate[Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]*Tan[d + e*x],x]","-\frac{\tan ^2(d+e x) \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \left(-\sqrt{a} \tanh ^{-1}\left(\frac{2 a \tan ^2(d+e x)+b}{2 \sqrt{a} \sqrt{a \tan ^4(d+e x)+b \tan ^2(d+e x)+c}}\right)+\sqrt{a-b+c} \tanh ^{-1}\left(\frac{(2 a-b) \tan ^2(d+e x)+b-2 c}{2 \sqrt{a-b+c} \sqrt{a \tan ^4(d+e x)+b \tan ^2(d+e x)+c}}\right)+\sqrt{c} \tanh ^{-1}\left(\frac{b \tan ^2(d+e x)+2 c}{2 \sqrt{c} \sqrt{a \tan ^4(d+e x)+b \tan ^2(d+e x)+c}}\right)\right)}{2 e \sqrt{a \tan ^4(d+e x)+b \tan ^2(d+e x)+c}}","\frac{\sqrt{a} \tanh ^{-1}\left(\frac{2 a+b \cot ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{2 e}-\frac{\sqrt{a-b+c} \tanh ^{-1}\left(\frac{2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{2 e}-\frac{\sqrt{c} \tanh ^{-1}\left(\frac{b+2 c \cot ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{2 e}",1,"-1/2*((-(Sqrt[a]*ArcTanh[(b + 2*a*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4])]) + Sqrt[a - b + c]*ArcTanh[(b - 2*c + (2*a - b)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4])] + Sqrt[c]*ArcTanh[(2*c + b*Tan[d + e*x]^2)/(2*Sqrt[c]*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4])])*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]*Tan[d + e*x]^2)/(e*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4])","A",1
26,1,215131,435,34.9839056,"\int \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan ^3(d+e x) \, dx","Integrate[Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]*Tan[d + e*x]^3,x]","\text{Result too large to show}","\frac{\tan ^2(d+e x) \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 e}+\frac{b \tanh ^{-1}\left(\frac{2 a+b \cot ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{4 \sqrt{a} e}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{2 a+b \cot ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{2 e}+\frac{\sqrt{a-b+c} \tanh ^{-1}\left(\frac{2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{2 e}-\frac{\sqrt{c} \tanh ^{-1}\left(\frac{b+2 c \cot ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{2 e}+\frac{b \tanh ^{-1}\left(\frac{b+2 c \cot ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{4 \sqrt{c} e}-\frac{(b-2 c) \tanh ^{-1}\left(\frac{b+2 c \cot ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{4 \sqrt{c} e}",1,"Result too large to show","C",0
27,1,243520,236,36.2690214,"\int \frac{\cot ^7(d+e x)}{\left(a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right)^{3/2}} \, dx","Integrate[Cot[d + e*x]^7/(a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4)^(3/2),x]","\text{Result too large to show}","-\frac{\left(2 a^2 c-a b (b+3 c)+b^3\right) \cot ^2(d+e x)+a \left(b^2-a (b+2 c)\right)}{c e (a-b+c) \left(b^2-4 a c\right) \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}-\frac{\tanh ^{-1}\left(\frac{b+2 c \cot ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{2 c^{3/2} e}-\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{2 e (a-b+c)^{3/2}}",1,"Result too large to show","C",0
28,1,25130,160,31.8550294,"\int \frac{\cot ^5(d+e x)}{\left(a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right)^{3/2}} \, dx","Integrate[Cot[d + e*x]^5/(a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4)^(3/2),x]","\text{Result too large to show}","\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{2 e (a-b+c)^{3/2}}-\frac{(b (a-b)+2 a c) \cot ^2(d+e x)+a (2 a-b)}{e (a-b+c) \left(b^2-4 a c\right) \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}",1,"Result too large to show","C",0
29,1,25123,153,31.6531635,"\int \frac{\cot ^3(d+e x)}{\left(a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right)^{3/2}} \, dx","Integrate[Cot[d + e*x]^3/(a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4)^(3/2),x]","\text{Result too large to show}","\frac{c (2 a-b) \cot ^2(d+e x)+a (b-2 c)}{e (a-b+c) \left(b^2-4 a c\right) \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}-\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{2 e (a-b+c)^{3/2}}",1,"Result too large to show","C",0
30,1,25149,156,7.6269947,"\int \frac{\cot (d+e x)}{\left(a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right)^{3/2}} \, dx","Integrate[Cot[d + e*x]/(a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4)^(3/2),x]","\text{Result too large to show}","\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{2 e (a-b+c)^{3/2}}-\frac{-2 a c+b^2+c (b-2 c) \cot ^2(d+e x)-b c}{e (a-b+c) \left(b^2-4 a c\right) \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}",1,"Result too large to show","C",0
31,1,181078,280,35.9603269,"\int \frac{\tan (d+e x)}{\left(a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right)^{3/2}} \, dx","Integrate[Tan[d + e*x]/(a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4)^(3/2),x]","\text{Result too large to show}","\frac{\tanh ^{-1}\left(\frac{2 a+b \cot ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{2 a^{3/2} e}-\frac{-2 a c+b^2+b c \cot ^2(d+e x)}{a e \left(b^2-4 a c\right) \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}+\frac{-2 a c+b^2+c (b-2 c) \cot ^2(d+e x)-b c}{e (a-b+c) \left(b^2-4 a c\right) \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}-\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{2 e (a-b+c)^{3/2}}",1,"Result too large to show","C",0
32,1,293889,478,37.9523188,"\int \frac{\tan ^3(d+e x)}{\left(a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right)^{3/2}} \, dx","Integrate[Tan[d + e*x]^3/(a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4)^(3/2),x]","\text{Result too large to show}","-\frac{3 b \tanh ^{-1}\left(\frac{2 a+b \cot ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{4 a^{5/2} e}-\frac{\tanh ^{-1}\left(\frac{2 a+b \cot ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{2 a^{3/2} e}+\frac{\left(3 b^2-8 a c\right) \tan ^2(d+e x) \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 a^2 e \left(b^2-4 a c\right)}+\frac{-2 a c+b^2+b c \cot ^2(d+e x)}{a e \left(b^2-4 a c\right) \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}-\frac{-2 a c+b^2+c (b-2 c) \cot ^2(d+e x)-b c}{e (a-b+c) \left(b^2-4 a c\right) \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}-\frac{\tan ^2(d+e x) \left(-2 a c+b^2+b c \cot ^2(d+e x)\right)}{a e \left(b^2-4 a c\right) \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}+\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right)}{2 e (a-b+c)^{3/2}}",1,"Result too large to show","C",0