1,1,547,0,1.0856767,"\int \frac{\cot ^5(d+e x)}{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx","Int[Cot[d + e*x]^5/Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2],x]","-\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{\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 \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{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}","-\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{\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 \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{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,"-((Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Cot[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e)) + (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Cot[d + e*x])/(Sqrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) - (b*ArcTanh[(b + 2*c*Cot[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(2*c^(3/2)*e) + (b*(5*b^2 - 12*a*c)*ArcTanh[(b + 2*c*Cot[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(16*c^(7/2)*e) + Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]/(c*e) - (Cot[d + e*x]^2*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])/(3*c*e) - ((15*b^2 - 16*a*c - 10*b*c*Cot[d + e*x])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])/(24*c^3*e)","A",15,10,33,0.3030,1,"{3701, 6725, 640, 621, 206, 742, 779, 1036, 1030, 208}"
2,1,384,0,0.7146925,"\int \frac{\cot ^3(d+e x)}{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx","Int[Cot[d + e*x]^3/Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2],x]","\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}","\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 - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Cot[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) - (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Cot[d + e*x])/(Sqrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) + (b*ArcTanh[(b + 2*c*Cot[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(2*c^(3/2)*e) - Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]/(c*e)","A",11,8,33,0.2424,1,"{3701, 6725, 640, 621, 206, 1036, 1030, 208}"
3,1,294,0,0.288611,"\int \frac{\cot (d+e x)}{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx","Int[Cot[d + e*x]/Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2],x]","\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{\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,"-((Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Cot[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e)) + (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Cot[d + e*x])/(Sqrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e)","A",6,4,31,0.1290,1,"{3701, 1036, 1030, 208}"
4,1,349,0,0.6878575,"\int \frac{\tan (d+e x)}{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx","Int[Tan[d + e*x]/Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2],x]","\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}","\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,"ArcTanh[(2*a + b*Cot[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])]/(Sqrt[a]*e) + (Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Cot[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) - (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Cot[d + e*x])/(Sqrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e)","A",10,7,31,0.2258,1,"{3701, 6725, 724, 206, 1036, 1030, 208}"
5,1,501,0,0.8166332,"\int \frac{\tan ^3(d+e x)}{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx","Int[Tan[d + e*x]^3/Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2],x]","-\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{\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{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}","-\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{\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{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,"-(ArcTanh[(2*a + b*Cot[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])]/(Sqrt[a]*e)) + ((3*b^2 - 4*a*c)*ArcTanh[(2*a + b*Cot[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(8*a^(5/2)*e) - (Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Cot[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) + (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Cot[d + e*x])/(Sqrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) - (3*b*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x])/(4*a^2*e) + (Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x]^2)/(2*a*e)","A",14,9,33,0.2727,1,"{3701, 6725, 744, 806, 724, 206, 1036, 1030, 208}"
6,1,976,0,24.3122571,"\int \cot ^5(d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx","Int[Cot[d + e*x]^5*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2],x]","-\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}","-\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,"-((Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Cot[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e)) - (b*ArcTanh[(b + 2*c*Cot[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(2*Sqrt[c]*e) + (b*(b^2 - 4*a*c)*ArcTanh[(b + 2*c*Cot[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(16*c^(5/2)*e) - (b*(7*b^2 - 12*a*c)*(b^2 - 4*a*c)*ArcTanh[(b + 2*c*Cot[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(256*c^(9/2)*e) + (Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTanh[(b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) + b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Cot[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) - Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]/e - (b*(b + 2*c*Cot[d + e*x])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])/(8*c^2*e) + (b*(7*b^2 - 12*a*c)*(b + 2*c*Cot[d + e*x])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])/(128*c^4*e) + (a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2)/(3*c*e) - (Cot[d + e*x]^2*(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2))/(5*c*e) - ((35*b^2 - 32*a*c - 42*b*c*Cot[d + e*x])*(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2))/(240*c^3*e)","A",21,14,33,0.4242,1,"{3701, 6725, 640, 612, 621, 206, 742, 779, 1021, 1078, 1036, 1030, 208, 205}"
7,1,747,0,23.6938388,"\int \cot ^3(d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx","Int[Cot[d + e*x]^3*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2],x]","\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}","\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,"(Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Cot[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) + (b*ArcTanh[(b + 2*c*Cot[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(2*Sqrt[c]*e) - (b*(b^2 - 4*a*c)*ArcTanh[(b + 2*c*Cot[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(16*c^(5/2)*e) - (Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTanh[(b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) + b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Cot[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) + Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]/e + (b*(b + 2*c*Cot[d + e*x])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])/(8*c^2*e) - (a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2)/(3*c*e)","A",16,12,33,0.3636,1,"{3701, 6725, 640, 612, 621, 206, 1021, 1078, 1036, 1030, 208, 205}"
8,1,602,0,23.1815655,"\int \cot (d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx","Int[Cot[d + e*x]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2],x]","-\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}","-\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^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Cot[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e)) - (b*ArcTanh[(b + 2*c*Cot[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(2*Sqrt[c]*e) + (Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTanh[(b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) + b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Cot[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) - Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]/e","A",10,9,31,0.2903,1,"{3701, 1021, 1078, 621, 206, 1036, 1030, 208, 205}"
9,1,570,0,23.622114,"\int \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan (d+e x) \, dx","Int[Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x],x]","\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}","\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,"(Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Cot[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) + (Sqrt[a]*ArcTanh[(2*a + b*Cot[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/e - (Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTanh[(b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) + b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Cot[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e)","A",18,13,31,0.4194,1,"{3701, 6725, 734, 843, 621, 206, 724, 1021, 1078, 1036, 1030, 208, 205}"
10,1,691,0,23.7119724,"\int \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan ^3(d+e x) \, dx","Int[Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x]^3,x]","-\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{\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{\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}","-\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{\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{\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,"-((Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Cot[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e)) - (Sqrt[a]*ArcTanh[(2*a + b*Cot[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/e - ((b^2 - 4*a*c)*ArcTanh[(2*a + b*Cot[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(8*a^(3/2)*e) + (Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTanh[(b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) + b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Cot[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) + ((2*a + b*Cot[d + e*x])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x]^2)/(4*a*e)","A",21,14,33,0.4242,1,"{3701, 6725, 720, 724, 206, 734, 843, 621, 1021, 1078, 1036, 1030, 208, 205}"
11,1,1189,0,6.2886347,"\int \frac{\cot ^7(d+e x)}{\left(a+b \cot (d+e x)+c \cot ^2(d+e x)\right)^{3/2}} \, dx","Int[Cot[d + e*x]^7/(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2),x]","-\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}}","-\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,"(-3*b*ArcTanh[(b + 2*c*Cot[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(2*c^(5/2)*e) + (5*b*(7*b^2 - 12*a*c)*ArcTanh[(b + 2*c*Cot[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(16*c^(9/2)*e) + (Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) - (Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) - (2*(2*a + b*Cot[d + e*x]))/((b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]) + (2*Cot[d + e*x]^2*(2*a + b*Cot[d + e*x]))/((b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]) - (2*Cot[d + e*x]^4*(2*a + b*Cot[d + e*x]))/((b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]) + (2*(a*(b^2 - 2*(a - c)*c) + b*c*(a + c)*Cot[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]) - ((7*b^2 - 16*a*c)*Cot[d + e*x]^2*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])/(3*c^2*(b^2 - 4*a*c)*e) + (2*b*Cot[d + e*x]^3*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])/(c*(b^2 - 4*a*c)*e) + ((3*b^2 - 8*a*c - 2*b*c*Cot[d + e*x])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])/(c^2*(b^2 - 4*a*c)*e) - ((105*b^4 - 460*a*b^2*c + 256*a^2*c^2 - 2*b*c*(35*b^2 - 116*a*c)*Cot[d + e*x])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])/(24*c^4*(b^2 - 4*a*c)*e)","A",20,12,33,0.3636,1,"{3701, 6725, 636, 738, 779, 621, 206, 832, 1018, 1036, 1030, 208}"
12,1,865,0,4.7739726,"\int \frac{\cot ^5(d+e x)}{\left(a+b \cot (d+e x)+c \cot ^2(d+e x)\right)^{3/2}} \, dx","Int[Cot[d + e*x]^5/(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2),x]","-\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}}","-\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,"(3*b*ArcTanh[(b + 2*c*Cot[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(2*c^(5/2)*e) - (Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) + (Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) + (2*(2*a + b*Cot[d + e*x]))/((b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]) - (2*Cot[d + e*x]^2*(2*a + b*Cot[d + e*x]))/((b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]) - (2*(a*(b^2 - 2*(a - c)*c) + b*c*(a + c)*Cot[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]) - ((3*b^2 - 8*a*c - 2*b*c*Cot[d + e*x])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])/(c^2*(b^2 - 4*a*c)*e)","A",14,11,33,0.3333,1,"{3701, 6725, 636, 738, 779, 621, 206, 1018, 1036, 1030, 208}"
13,1,686,0,4.5976637,"\int \frac{\cot ^3(d+e x)}{\left(a+b \cot (d+e x)+c \cot ^2(d+e x)\right)^{3/2}} \, dx","Int[Cot[d + e*x]^3/(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2),x]","\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)}}","\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,"(Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) - (Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) - (2*(2*a + b*Cot[d + e*x]))/((b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]) + (2*(a*(b^2 - 2*(a - c)*c) + b*c*(a + c)*Cot[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])","A",10,7,33,0.2121,1,"{3701, 6725, 636, 1018, 1036, 1030, 208}"
14,1,635,0,3.7570323,"\int \frac{\cot (d+e x)}{\left(a+b \cot (d+e x)+c \cot ^2(d+e x)\right)^{3/2}} \, dx","Int[Cot[d + e*x]/(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2),x]","-\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)}}","-\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,"-((Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e)) + (Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) - (2*(a*(b^2 - 2*(a - c)*c) + b*c*(a + c)*Cot[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])","A",7,5,31,0.1613,1,"{3701, 1018, 1036, 1030, 208}"
15,1,749,0,4.4762339,"\int \frac{\tan (d+e x)}{\left(a+b \cot (d+e x)+c \cot ^2(d+e x)\right)^{3/2}} \, dx","Int[Tan[d + e*x]/(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2),x]","\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{\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{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)}}","\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{\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{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,"ArcTanh[(2*a + b*Cot[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])]/(a^(3/2)*e) + (Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) - (Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) - (2*(b^2 - 2*a*c + b*c*Cot[d + e*x]))/(a*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]) + (2*(a*(b^2 - 2*(a - c)*c) + b*c*(a + c)*Cot[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])","A",13,10,31,0.3226,1,"{3701, 6725, 740, 12, 724, 206, 1018, 1036, 1030, 208}"
16,1,1008,0,4.8506089,"\int \frac{\tan ^3(d+e x)}{\left(a+b \cot (d+e x)+c \cot ^2(d+e x)\right)^{3/2}} \, dx","Int[Tan[d + e*x]^3/(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2),x]","\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}}","\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,"-(ArcTanh[(2*a + b*Cot[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])]/(a^(3/2)*e)) + (3*(5*b^2 - 4*a*c)*ArcTanh[(2*a + b*Cot[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(8*a^(7/2)*e) - (Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) + (Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) + (2*(b^2 - 2*a*c + b*c*Cot[d + e*x]))/(a*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]) - (2*(a*(b^2 - 2*(a - c)*c) + b*c*(a + c)*Cot[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]) - (b*(15*b^2 - 52*a*c)*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x])/(4*a^3*(b^2 - 4*a*c)*e) - (2*(b^2 - 2*a*c + b*c*Cot[d + e*x])*Tan[d + e*x]^2)/(a*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]) + ((5*b^2 - 12*a*c)*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x]^2)/(2*a^2*(b^2 - 4*a*c)*e)","A",18,12,33,0.3636,1,"{3701, 6725, 740, 834, 806, 724, 206, 12, 1018, 1036, 1030, 208}"
17,1,182,0,0.3886185,"\int \frac{\cot ^5(d+e x)}{\sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx","Int[Cot[d + e*x]^5/Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]","\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}}","\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,"ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*Sqrt[a - b + c]*e) + ((b + 2*c)*ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(4*c^(3/2)*e) - Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]/(2*c*e)","A",8,7,35,0.2000,1,"{3701, 1251, 1653, 843, 621, 206, 724}"
18,1,141,0,0.2136225,"\int \frac{\cot ^3(d+e x)}{\sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx","Int[Cot[d + e*x]^3/Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],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 \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}","-\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,"-ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*Sqrt[a - b + c]*e) - ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*Sqrt[c]*e)","A",7,6,35,0.1714,1,"{3701, 1251, 843, 621, 206, 724}"
19,1,79,0,0.1164623,"\int \frac{\cot (d+e x)}{\sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx","Int[Cot[d + e*x]/Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],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 \sqrt{a-b+c}}","\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,"ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*Sqrt[a - b + c]*e)","A",4,4,33,0.1212,1,"{3701, 1247, 724, 206}"
20,1,142,0,0.2437111,"\int \frac{\tan (d+e x)}{\sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx","Int[Tan[d + e*x]/Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]","\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}}","\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,"ArcTanh[(2*a + b*Cot[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*Sqrt[a]*e) - ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*Sqrt[a - b + c]*e)","A",8,5,33,0.1515,1,"{3701, 1251, 960, 724, 206}"
21,1,249,0,0.3185047,"\int \frac{\tan ^3(d+e x)}{\sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx","Int[Tan[d + e*x]^3/Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]","-\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}}","-\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,"-ArcTanh[(2*a + b*Cot[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*Sqrt[a]*e) - (b*ArcTanh[(2*a + b*Cot[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(4*a^(3/2)*e) + ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*Sqrt[a - b + c]*e) + (Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]*Tan[d + e*x]^2)/(2*a*e)","A",11,6,35,0.1714,1,"{3701, 1251, 960, 730, 724, 206}"
22,1,270,0,0.5561281,"\int \cot ^5(d+e x) \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx","Int[Cot[d + e*x]^5*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]","-\frac{\left(-4 b c (a-2 c)-8 c^2 (a+2 c)+2 b^2 c+b^3\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}","-\frac{\left(-4 b c (a-2 c)-8 c^2 (a+2 c)+2 b^2 c+b^3\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,"(Sqrt[a - b + c]*ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(2*e) - ((b^3 + 2*b^2*c - 4*b*(a - 2*c)*c - 8*c^2*(a + 2*c))*ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(32*c^(5/2)*e) + (((b - 2*c)*(b + 4*c) + 2*c*(b + 2*c)*Cot[d + e*x]^2)*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])/(16*c^2*e) - (a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4)^(3/2)/(6*c*e)","A",9,8,35,0.2286,1,"{3701, 1251, 1653, 814, 843, 621, 206, 724}"
23,1,209,0,0.3411413,"\int \cot ^3(d+e x) \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx","Int[Cot[d + e*x]^3*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]","\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}","\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,"-(Sqrt[a - b + c]*ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(2*e) + ((b^2 + 4*b*c - 4*c*(a + 2*c))*ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(16*c^(3/2)*e) - ((b - 4*c + 2*c*Cot[d + e*x]^2)*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])/(8*c*e)","A",8,7,35,0.2000,1,"{3701, 1251, 814, 843, 621, 206, 724}"
24,1,179,0,0.2210577,"\int \cot (d+e x) \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx","Int[Cot[d + e*x]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]","-\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}","-\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,"(Sqrt[a - b + c]*ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(2*e) - ((b - 2*c)*ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(4*Sqrt[c]*e) - Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]/(2*e)","A",8,7,33,0.2121,1,"{3701, 1247, 734, 843, 621, 206, 724}"
25,1,203,0,0.2698065,"\int \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan (d+e x) \, dx","Int[Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]*Tan[d + e*x],x]","\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{\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,"(Sqrt[a]*ArcTanh[(2*a + b*Cot[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(2*e) - (Sqrt[a - b + c]*ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(2*e) - (Sqrt[c]*ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(2*e)","A",10,7,33,0.2121,1,"{3701, 1251, 895, 724, 206, 843, 621}"
26,1,435,0,0.5367726,"\int \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan ^3(d+e x) \, dx","Int[Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]*Tan[d + e*x]^3,x]","\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}","\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,"-(Sqrt[a]*ArcTanh[(2*a + b*Cot[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(2*e) + (b*ArcTanh[(2*a + b*Cot[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(4*Sqrt[a]*e) + (Sqrt[a - b + c]*ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(2*e) + (b*ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(4*Sqrt[c]*e) - ((b - 2*c)*ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(4*Sqrt[c]*e) - (Sqrt[c]*ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(2*e) + (Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]*Tan[d + e*x]^2)/(2*e)","A",22,9,35,0.2571,1,"{3701, 1251, 960, 732, 843, 621, 206, 724, 734}"
27,1,236,0,0.5587811,"\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","Int[Cot[d + e*x]^7/(a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4)^(3/2),x]","-\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}}","-\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,"-ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*(a - b + c)^(3/2)*e) - ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*c^(3/2)*e) - (a*(b^2 - a*(b + 2*c)) + (b^3 + 2*a^2*c - a*b*(b + 3*c))*Cot[d + e*x]^2)/(c*(a - b + c)*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])","A",8,7,35,0.2000,1,"{3701, 1251, 1646, 843, 621, 206, 724}"
28,1,160,0,0.394066,"\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","Int[Cot[d + e*x]^5/(a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4)^(3/2),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}}-\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)}}","\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,"ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*(a - b + c)^(3/2)*e) - (a*(2*a - b) + ((a - b)*b + 2*a*c)*Cot[d + e*x]^2)/((a - b + c)*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])","A",6,6,35,0.1714,1,"{3701, 1251, 1646, 12, 724, 206}"
29,1,153,0,0.2844331,"\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","Int[Cot[d + e*x]^3/(a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4)^(3/2),x]","\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}}","\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,"-ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*(a - b + c)^(3/2)*e) + (a*(b - 2*c) + (2*a - b)*c*Cot[d + e*x]^2)/((a - b + c)*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])","A",6,6,35,0.1714,1,"{3701, 1251, 822, 12, 724, 206}"
30,1,156,0,0.2202567,"\int \frac{\cot (d+e x)}{\left(a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right)^{3/2}} \, dx","Int[Cot[d + e*x]/(a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4)^(3/2),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}}-\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}}-\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,"ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*(a - b + c)^(3/2)*e) - (b^2 - 2*a*c - b*c + (b - 2*c)*c*Cot[d + e*x]^2)/((a - b + c)*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])","A",6,6,33,0.1818,1,"{3701, 1247, 740, 12, 724, 206}"
31,1,280,0,0.383294,"\int \frac{\tan (d+e x)}{\left(a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right)^{3/2}} \, dx","Int[Tan[d + e*x]/(a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4)^(3/2),x]","\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}}","\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,"ArcTanh[(2*a + b*Cot[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*a^(3/2)*e) - ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*(a - b + c)^(3/2)*e) - (b^2 - 2*a*c + b*c*Cot[d + e*x]^2)/(a*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]) + (b^2 - 2*a*c - b*c + (b - 2*c)*c*Cot[d + e*x]^2)/((a - b + c)*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])","A",12,7,33,0.2121,1,"{3701, 1251, 960, 740, 12, 724, 206}"
32,1,478,0,0.5585666,"\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","Int[Tan[d + e*x]^3/(a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4)^(3/2),x]","\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{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{-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}}","\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{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{-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,"-ArcTanh[(2*a + b*Cot[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*a^(3/2)*e) - (3*b*ArcTanh[(2*a + b*Cot[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(4*a^(5/2)*e) + ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*(a - b + c)^(3/2)*e) + (b^2 - 2*a*c + b*c*Cot[d + e*x]^2)/(a*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]) - (b^2 - 2*a*c - b*c + (b - 2*c)*c*Cot[d + e*x]^2)/((a - b + c)*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]) - ((b^2 - 2*a*c + b*c*Cot[d + e*x]^2)*Tan[d + e*x]^2)/(a*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]) + ((3*b^2 - 8*a*c)*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]*Tan[d + e*x]^2)/(2*a^2*(b^2 - 4*a*c)*e)","A",16,8,35,0.2286,1,"{3701, 1251, 960, 740, 806, 724, 206, 12}"