1,1,690,975,6.2458202,"\int \tan ^5(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx","Integrate[Tan[d + e*x]^5*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","\frac{\frac{b \left(\frac{(b+2 c \tan (d+e x)) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c}-\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{8 c^{3/2}}\right)}{2 c}+\frac{\frac{\left(-8 a c+\frac{35 b^2}{4}-\frac{21}{2} b c \tan (d+e x)\right) \left(a+b \tan (d+e x)+c \tan ^2(d+e x)\right)^{3/2}}{12 c^2}+\frac{\left(15 a b c-\frac{35 b^3}{4}\right) \left(\frac{(b+2 c \tan (d+e x)) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c}-\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{8 c^{3/2}}\right)}{8 c^2}}{5 c}+\frac{\tan ^2(d+e x) \left(a+b \tan (d+e x)+c \tan ^2(d+e x)\right)^{3/2}}{5 c}-\frac{\left(a+b \tan (d+e x)+c \tan ^2(d+e x)\right)^{3/2}}{3 c}+\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}+\frac{1}{4} \left(\frac{(b-2 i c) \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{c}}-2 \sqrt{a-i b-c} \tanh ^{-1}\left(\frac{2 a+(b-2 i c) \tan (d+e x)-i b}{2 \sqrt{a-i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)\right)+\frac{1}{4} \left(\frac{(b+2 i c) \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{c}}-2 \sqrt{a+i b-c} \tanh ^{-1}\left(\frac{2 a+(b+2 i c) \tan (d+e x)+i b}{2 \sqrt{a+i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)\right)}{e}","\frac{\left(c \tan ^2(d+e x)+b \tan (d+e x)+a\right)^{3/2} \tan ^2(d+e x)}{5 c e}+\frac{\left(35 b^2-42 c \tan (d+e x) b-32 a c\right) \left(c \tan ^2(d+e x)+b \tan (d+e x)+a\right)^{3/2}}{240 c^3 e}-\frac{\left(c \tan ^2(d+e x)+b \tan (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} \tan (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 \tan ^2(d+e x)+b \tan (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 \tan (d+e x)}{2 \sqrt{c} \sqrt{c \tan ^2(d+e x)+b \tan (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 \tan (d+e x)}{2 \sqrt{c} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right)}{16 c^{5/2} e}+\frac{b \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{c \tan ^2(d+e x)+b \tan (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} \tan (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 \tan ^2(d+e x)+b \tan (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 \tan (d+e x)) \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}{128 c^4 e}+\frac{b (b+2 c \tan (d+e x)) \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}{8 c^2 e}+\frac{\sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}{e}",1,"((-2*Sqrt[a - I*b - c]*ArcTanh[(2*a - I*b + (b - (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])] + ((b - (2*I)*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[c])/4 + (-2*Sqrt[a + I*b - c]*ArcTanh[(2*a + I*b + (b + (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])] + ((b + (2*I)*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[c])/4 + Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2] - (a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2)/(3*c) + (Tan[d + e*x]^2*(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2))/(5*c) + (b*(-1/8*((b^2 - 4*a*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/c^(3/2) + ((b + 2*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(4*c)))/(2*c) + ((((35*b^2)/4 - 8*a*c - (21*b*c*Tan[d + e*x])/2)*(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2))/(12*c^2) + (((-35*b^3)/4 + 15*a*b*c)*(-1/8*((b^2 - 4*a*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/c^(3/2) + ((b + 2*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(4*c)))/(8*c^2))/(5*c))/e","C",0
2,1,582,889,5.4121546,"\int \tan ^4(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx","Integrate[Tan[d + e*x]^4*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","\frac{\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{c^{3/2}}-\frac{\left(\frac{5 b^2}{2}-2 a c\right) \left(\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)-2 \sqrt{c} (b+2 c \tan (d+e x)) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}\right)}{8 c^{7/2}}-\frac{5 b \left(a+b \tan (d+e x)+c \tan ^2(d+e x)\right)^{3/2}}{3 c^2}+\frac{2 \tan (d+e x) \left(a+b \tan (d+e x)+c \tan ^2(d+e x)\right)^{3/2}}{c}-\frac{2 (b+2 c \tan (d+e x)) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{c}-4 i \sqrt{a-i b-c} \tanh ^{-1}\left(\frac{2 a+(b-2 i c) \tan (d+e x)-i b}{2 \sqrt{a-i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)+4 i \sqrt{a+i b-c} \tanh ^{-1}\left(\frac{2 a+(b+2 i c) \tan (d+e x)+i b}{2 \sqrt{a+i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)+\frac{2 (2 c-i b) \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{c}}+\frac{2 (2 c+i b) \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{c}}}{8 e}","\frac{\tan (d+e x) \left(c \tan ^2(d+e x)+b \tan (d+e x)+a\right)^{3/2}}{4 c e}-\frac{5 b \left(c \tan ^2(d+e x)+b \tan (d+e x)+a\right)^{3/2}}{24 c^2 e}+\frac{\left(5 b^2-4 a c\right) (b+2 c \tan (d+e x)) \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}{64 c^3 e}-\frac{(b+2 c \tan (d+e x)) \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}{4 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 \sqrt{a^2-2 c a+b^2+c^2}-\left(b^2+(a-c) \left(a-c+\sqrt{a^2-2 c a+b^2+c^2}\right)\right) \tan (d+e x)}{\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 \tan ^2(d+e x)+b \tan (d+e x)+a}}\right)}{\sqrt{2} \sqrt[4]{a^2-2 c a+b^2+c^2} e}+\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right)}{8 c^{3/2} e}-\frac{\left(b^2-4 a c\right) \left(5 b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right)}{128 c^{7/2} e}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right)}{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{\sqrt{a^2-2 c a+b^2+c^2} b+\left(b^2+(a-c) \left(a-c-\sqrt{a^2-2 c a+b^2+c^2}\right)\right) \tan (d+e x)}{\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 \tan ^2(d+e x)+b \tan (d+e x)+a}}\right)}{\sqrt{2} \sqrt[4]{a^2-2 c a+b^2+c^2} e}",1,"((-4*I)*Sqrt[a - I*b - c]*ArcTanh[(2*a - I*b + (b - (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])] + (4*I)*Sqrt[a + I*b - c]*ArcTanh[(2*a + I*b + (b + (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])] + (2*((-I)*b + 2*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[c] + (2*(I*b + 2*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[c] + ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/c^(3/2) - (2*(b + 2*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/c - (5*b*(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2))/(3*c^2) + (2*Tan[d + e*x]*(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2))/c - (((5*b^2)/2 - 2*a*c)*((b^2 - 4*a*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])] - 2*Sqrt[c]*(b + 2*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]))/(8*c^(7/2)))/(8*e)","C",0
3,1,451,748,2.5266397,"\int \tan ^3(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx","Integrate[Tan[d + e*x]^3*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","\frac{\frac{3 b \left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{4 c^{5/2}}-\frac{3 b (b+2 c \tan (d+e x)) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{2 c^2}+\frac{4 \left(a+b \tan (d+e x)+c \tan ^2(d+e x)\right)^{3/2}}{c}-12 \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}+6 \sqrt{a-i b-c} \tanh ^{-1}\left(\frac{2 a+(b-2 i c) \tan (d+e x)-i b}{2 \sqrt{a-i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)+6 \sqrt{a+i b-c} \tanh ^{-1}\left(\frac{2 a+(b+2 i c) \tan (d+e x)+i b}{2 \sqrt{a+i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)-\frac{3 (b-2 i c) \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{c}}-\frac{3 (b+2 i c) \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{c}}}{12 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} \tan (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 \tan (d+e x)+c \tan ^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} \tan (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 \tan (d+e x)+c \tan ^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 \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{16 c^{5/2} e}-\frac{b (b+2 c \tan (d+e x)) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{8 c^2 e}+\frac{\left(a+b \tan (d+e x)+c \tan ^2(d+e x)\right)^{3/2}}{3 c e}-\frac{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}-\frac{b \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{2 \sqrt{c} e}",1,"(6*Sqrt[a - I*b - c]*ArcTanh[(2*a - I*b + (b - (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])] + 6*Sqrt[a + I*b - c]*ArcTanh[(2*a + I*b + (b + (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])] - (3*(b - (2*I)*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[c] - (3*(b + (2*I)*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[c] + (3*b*(b^2 - 4*a*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(4*c^(5/2)) - 12*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2] - (3*b*(b + 2*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(2*c^2) + (4*(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2))/c)/(12*e)","C",0
4,1,405,676,0.6421265,"\int \tan ^2(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx","Integrate[Tan[d + e*x]^2*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","\frac{-\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{8 c^{3/2}}+\frac{(b+2 c \tan (d+e x)) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c}+\frac{1}{4} i \left(2 \sqrt{a-i b-c} \tanh ^{-1}\left(\frac{2 a+(b-2 i c) \tan (d+e x)-i b}{2 \sqrt{a-i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)-\frac{(b-2 i c) \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{c}}\right)-\frac{1}{4} i \left(2 \sqrt{a+i b-c} \tanh ^{-1}\left(\frac{2 a+(b+2 i c) \tan (d+e x)+i b}{2 \sqrt{a+i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)-\frac{(b+2 i c) \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{c}}\right)}{e}","\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} \tan ^{-1}\left(\frac{b \sqrt{a^2-2 a c+b^2+c^2}-\left(-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\right) \tan (d+e x)}{\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 \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{2} e \sqrt[4]{a^2-2 a c+b^2+c^2}}+\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} \tanh ^{-1}\left(\frac{\left(-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\right) \tan (d+e x)+b \sqrt{a^2-2 a c+b^2+c^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 \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{2} e \sqrt[4]{a^2-2 a c+b^2+c^2}}-\frac{\left(b^2-4 c (a-2 c)\right) \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{8 c^{3/2} e}+\frac{(b+2 c \tan (d+e x)) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c e}",1,"(-1/8*((b^2 - 4*a*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/c^(3/2) + (I/4)*(2*Sqrt[a - I*b - c]*ArcTanh[(2*a - I*b + (b - (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])] - ((b - (2*I)*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[c]) - (I/4)*(2*Sqrt[a + I*b - c]*ArcTanh[(2*a + I*b + (b + (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])] - ((b + (2*I)*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[c]) + ((b + 2*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(4*c))/e","C",0
5,1,250,601,0.3068704,"\int \tan (d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx","Integrate[Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","\frac{2 \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}-\sqrt{a-i b-c} \tanh ^{-1}\left(\frac{2 a+(b-2 i c) \tan (d+e x)-i b}{2 \sqrt{a-i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)-\sqrt{a+i b-c} \tanh ^{-1}\left(\frac{2 a+(b+2 i c) \tan (d+e x)+i b}{2 \sqrt{a+i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)+\frac{b \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{c}}}{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} \tan (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 \tan (d+e x)+c \tan ^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} \tan (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 \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{2} e \sqrt[4]{a^2-2 a c+b^2+c^2}}+\frac{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}+\frac{b \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{2 \sqrt{c} e}",1,"(-(Sqrt[a - I*b - c]*ArcTanh[(2*a - I*b + (b - (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])]) - Sqrt[a + I*b - c]*ArcTanh[(2*a + I*b + (b + (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])] + (b*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[c] + 2*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(2*e)","C",0
6,1,228,574,0.1975565,"\int \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx","Integrate[Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","\frac{-i \sqrt{a-i b-c} \tanh ^{-1}\left(\frac{2 a+(b-2 i c) \tan (d+e x)-i b}{2 \sqrt{a-i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)+i \sqrt{a+i b-c} \tanh ^{-1}\left(\frac{2 a+(b+2 i c) \tan (d+e x)+i b}{2 \sqrt{a+i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)+2 \sqrt{c} \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{2 e}","-\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} \tan ^{-1}\left(\frac{b \sqrt{a^2-2 a c+b^2+c^2}-\left((a-c) \left(\sqrt{a^2-2 a c+b^2+c^2}+a-c\right)+b^2\right) \tan (d+e x)}{\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 \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{2} e \sqrt[4]{a^2-2 a c+b^2+c^2}}-\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} \tanh ^{-1}\left(\frac{\left((a-c) \left(-\sqrt{a^2-2 a c+b^2+c^2}+a-c\right)+b^2\right) \tan (d+e x)+b \sqrt{a^2-2 a c+b^2+c^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 \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{2} e \sqrt[4]{a^2-2 a c+b^2+c^2}}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{e}",1,"((-I)*Sqrt[a - I*b - c]*ArcTanh[(2*a - I*b + (b - (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])] + I*Sqrt[a + I*b - c]*ArcTanh[(2*a + I*b + (b + (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])] + 2*Sqrt[c]*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(2*e)","C",0
7,1,223,571,0.3613988,"\int \cot (d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx","Integrate[Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","\frac{-2 \sqrt{a} \tanh ^{-1}\left(\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)+\sqrt{a-i b-c} \tanh ^{-1}\left(\frac{2 a+(b-2 i c) \tan (d+e x)-i b}{2 \sqrt{a-i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)+\sqrt{a+i b-c} \tanh ^{-1}\left(\frac{2 a+(b+2 i c) \tan (d+e x)+i b}{2 \sqrt{a+i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{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} \tan (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 \tan (d+e x)+c \tan ^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} \tan (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 \tan (d+e x)+c \tan ^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 \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{e}",1,"(-2*Sqrt[a]*ArcTanh[(2*a + b*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])] + Sqrt[a - I*b - c]*ArcTanh[(2*a - I*b + (b - (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])] + Sqrt[a + I*b - c]*ArcTanh[(2*a + I*b + (b + (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(2*e)","C",0
8,1,261,612,1.2192327,"\int \cot ^2(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx","Integrate[Cot[d + e*x]^2*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","-\frac{\frac{b \tanh ^{-1}\left(\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{a}}-i \sqrt{a-i b-c} \tanh ^{-1}\left(\frac{2 a+(b-2 i c) \tan (d+e x)-i b}{2 \sqrt{a-i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)+i \sqrt{a+i b-c} \tanh ^{-1}\left(\frac{2 a+(b+2 i c) \tan (d+e x)+i b}{2 \sqrt{a+i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)+2 \cot (d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{2 e}","\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} \tan ^{-1}\left(\frac{b \sqrt{a^2-2 a c+b^2+c^2}-\left((a-c) \left(\sqrt{a^2-2 a c+b^2+c^2}+a-c\right)+b^2\right) \tan (d+e x)}{\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 \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{2} e \sqrt[4]{a^2-2 a c+b^2+c^2}}+\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} \tanh ^{-1}\left(\frac{\left((a-c) \left(-\sqrt{a^2-2 a c+b^2+c^2}+a-c\right)+b^2\right) \tan (d+e x)+b \sqrt{a^2-2 a c+b^2+c^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 \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{2} e \sqrt[4]{a^2-2 a c+b^2+c^2}}-\frac{b \tanh ^{-1}\left(\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{2 \sqrt{a} e}-\frac{\cot (d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}",1,"-1/2*((b*ArcTanh[(2*a + b*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[a] - I*Sqrt[a - I*b - c]*ArcTanh[(2*a - I*b + (b - (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])] + I*Sqrt[a + I*b - c]*ArcTanh[(2*a + I*b + (b + (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])] + 2*Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/e","C",0
9,1,289,690,1.9199495,"\int \cot ^3(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx","Integrate[Cot[d + e*x]^3*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","\frac{\left(8 a^2-4 a c+b^2\right) \tanh ^{-1}\left(\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)-2 \sqrt{a} \left(2 a \sqrt{a-i b-c} \tanh ^{-1}\left(\frac{2 a+(b-2 i c) \tan (d+e x)-i b}{2 \sqrt{a-i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)+2 a \sqrt{a+i b-c} \tanh ^{-1}\left(\frac{2 a+(b+2 i c) \tan (d+e x)+i b}{2 \sqrt{a+i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)+\cot (d+e x) (2 a \cot (d+e x)+b) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}\right)}{8 a^{3/2} e}","\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^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} \tan (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 \tan (d+e x)+c \tan ^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} \tan (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 \tan (d+e x)+c \tan ^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 \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{e}-\frac{\cot ^2(d+e x) (2 a+b \tan (d+e x)) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 a e}",1,"((8*a^2 + b^2 - 4*a*c)*ArcTanh[(2*a + b*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])] - 2*Sqrt[a]*(2*a*Sqrt[a - I*b - c]*ArcTanh[(2*a - I*b + (b - (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])] + 2*a*Sqrt[a + I*b - c]*ArcTanh[(2*a + I*b + (b + (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])] + Cot[d + e*x]*(b + 2*a*Cot[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]))/(8*a^(3/2)*e)","C",0
10,1,456,548,6.135039,"\int \frac{\tan ^5(d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx","Integrate[Tan[d + e*x]^5/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","\frac{\frac{\frac{\left(9 a b c-\frac{15 b^3}{4}\right) \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{4 c^{5/2}}+\frac{\left(-4 a c+\frac{15 b^2}{4}-\frac{5}{2} b c \tan (d+e x)\right) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{2 c^2}}{3 c}+\frac{b \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{2 c^{3/2}}+\frac{\tan ^2(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{3 c}-\frac{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{c}-\frac{2 \sqrt{a+i b-c} \tanh ^{-1}\left(\frac{2 a-(-b-2 i c) \tan (d+e x)+i b}{2 \sqrt{a+i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{4 a+4 i b-4 c}-\frac{2 \sqrt{a-i b-c} \tanh ^{-1}\left(\frac{2 a-(-b+2 i c) \tan (d+e x)-i b}{2 \sqrt{a-i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{4 a-4 i b-4 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 \tan (d+e x)-c}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \tan (d+e x)+c \tan ^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 \tan (d+e x)-c}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \tan (d+e x)+c \tan ^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 \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{16 c^{7/2} e}+\frac{\left(-16 a c+15 b^2-10 b c \tan (d+e x)\right) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{24 c^3 e}+\frac{b \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{2 c^{3/2} e}+\frac{\tan ^2(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{3 c e}-\frac{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{c e}",1,"((-2*Sqrt[a + I*b - c]*ArcTanh[(2*a + I*b - (-b - (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(4*a + (4*I)*b - 4*c) - (2*Sqrt[a - I*b - c]*ArcTanh[(2*a - I*b - (-b + (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(4*a - (4*I)*b - 4*c) + (b*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(2*c^(3/2)) - Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]/c + (Tan[d + e*x]^2*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(3*c) + ((((-15*b^3)/4 + 9*a*b*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(4*c^(5/2)) + (((15*b^2)/4 - 4*a*c - (5*b*c*Tan[d + e*x])/2)*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(2*c^2))/(3*c))/e","C",1
11,1,283,495,2.7567092,"\int \frac{\tan ^4(d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx","Integrate[Tan[d + e*x]^4/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","\frac{\frac{\left(3 b^2-4 c (a+2 c)\right) \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{c^{5/2}}+\frac{2 (2 c \tan (d+e x)-3 b) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{c^2}-\frac{4 i \tanh ^{-1}\left(\frac{2 a+(b-2 i c) \tan (d+e x)-i b}{2 \sqrt{a-i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{a-i b-c}}+\frac{4 i \tanh ^{-1}\left(\frac{2 a+(b+2 i c) \tan (d+e x)+i b}{2 \sqrt{a+i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{a+i b-c}}}{8 e}","\frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tan ^{-1}\left(\frac{b-\left(-\sqrt{a^2-2 a c+b^2+c^2}+a-c\right) \tan (d+e x)}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \tan (d+e x)+c \tan ^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} \tan ^{-1}\left(\frac{b-\left(\sqrt{a^2-2 a c+b^2+c^2}+a-c\right) \tan (d+e x)}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \tan (d+e x)+c \tan ^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{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{8 c^{5/2} e}-\frac{3 b \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c^2 e}+\frac{\tan (d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{2 c e}-\frac{\tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{c} e}",1,"(((-4*I)*ArcTanh[(2*a - I*b + (b - (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[a - I*b - c] + ((4*I)*ArcTanh[(2*a + I*b + (b + (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[a + I*b - c] + ((3*b^2 - 4*c*(a + 2*c))*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/c^(5/2) + (2*(-3*b + 2*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/c^2)/(8*e)","C",1
12,1,252,383,0.941813,"\int \frac{\tan ^3(d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx","Integrate[Tan[d + e*x]^3/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","\frac{-\frac{b \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{c^{3/2}}+\frac{2 \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{c}+\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 i c) \tan (d+e x)-i b}{2 \sqrt{a-i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{a-i b-c}}+\frac{\tanh ^{-1}\left(\frac{2 a+(b+2 i c) \tan (d+e x)+i b}{2 \sqrt{a+i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{a+i b-c}}}{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 \tan (d+e x)-c}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \tan (d+e x)+c \tan ^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 \tan (d+e x)-c}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \tan (d+e x)+c \tan ^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 \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{2 c^{3/2} e}+\frac{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{c e}",1,"(ArcTanh[(2*a - I*b + (b - (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])]/Sqrt[a - I*b - c] + ArcTanh[(2*a + I*b + (b + (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])]/Sqrt[a + I*b - c] - (b*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/c^(3/2) + (2*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/c)/(2*e)","C",1
13,1,228,352,0.2047833,"\int \frac{\tan ^2(d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx","Integrate[Tan[d + e*x]^2/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","\frac{\frac{i \tanh ^{-1}\left(\frac{2 a+(b-2 i c) \tan (d+e x)-i b}{2 \sqrt{a-i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{2 \sqrt{a-i b-c}}-\frac{i \tanh ^{-1}\left(\frac{2 a+(b+2 i c) \tan (d+e x)+i b}{2 \sqrt{a+i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{2 \sqrt{a+i b-c}}+\frac{\tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{c}}}{e}","-\frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tan ^{-1}\left(\frac{b-\left(-\sqrt{a^2-2 a c+b^2+c^2}+a-c\right) \tan (d+e x)}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \tan (d+e x)+c \tan ^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} \tan ^{-1}\left(\frac{b-\left(\sqrt{a^2-2 a c+b^2+c^2}+a-c\right) \tan (d+e x)}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}+\frac{\tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{c} e}",1,"(((I/2)*ArcTanh[(2*a - I*b + (b - (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[a - I*b - c] - ((I/2)*ArcTanh[(2*a + I*b + (b + (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[a + I*b - c] + ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])]/Sqrt[c])/e","C",1
14,1,173,294,0.1148833,"\int \frac{\tan (d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx","Integrate[Tan[d + e*x]/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","\frac{-\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 i c) \tan (d+e x)-i b}{2 \sqrt{a-i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{2 \sqrt{a-i b-c}}-\frac{\tanh ^{-1}\left(\frac{2 a+(b+2 i c) \tan (d+e x)+i b}{2 \sqrt{a+i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{2 \sqrt{a+i b-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 \tan (d+e x)-c}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \tan (d+e x)+c \tan ^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 \tan (d+e x)-c}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}",1,"(-1/2*ArcTanh[(2*a - I*b + (b - (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])]/Sqrt[a - I*b - c] - ArcTanh[(2*a + I*b + (b + (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])]/(2*Sqrt[a + I*b - c]))/e","C",1
15,1,173,298,0.1410594,"\int \frac{1}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx","Integrate[1/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","-\frac{i \left(\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 i c) \tan (d+e x)-i b}{2 \sqrt{a-i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{a-i b-c}}-\frac{\tanh ^{-1}\left(\frac{2 a+(b+2 i c) \tan (d+e x)+i b}{2 \sqrt{a+i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{a+i b-c}}\right)}{2 e}","\frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tan ^{-1}\left(\frac{b-\left(-\sqrt{a^2-2 a c+b^2+c^2}+a-c\right) \tan (d+e x)}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \tan (d+e x)+c \tan ^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} \tan ^{-1}\left(\frac{b-\left(\sqrt{a^2-2 a c+b^2+c^2}+a-c\right) \tan (d+e x)}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}",1,"((-1/2*I)*(ArcTanh[(2*a - I*b + (b - (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])]/Sqrt[a - I*b - c] - ArcTanh[(2*a + I*b + (b + (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])]/Sqrt[a + I*b - c]))/e","C",1
16,1,223,350,0.4035177,"\int \frac{\cot (d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx","Integrate[Cot[d + e*x]/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","\frac{-\frac{2 \tanh ^{-1}\left(\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{a}}+\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 i c) \tan (d+e x)-i b}{2 \sqrt{a-i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{a-i b-c}}+\frac{\tanh ^{-1}\left(\frac{2 a+(b+2 i c) \tan (d+e x)+i b}{2 \sqrt{a+i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{a+i b-c}}}{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 \tan (d+e x)-c}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \tan (d+e x)+c \tan ^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 \tan (d+e x)-c}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \tan (d+e x)+c \tan ^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 \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{a} e}",1,"((-2*ArcTanh[(2*a + b*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[a] + ArcTanh[(2*a - I*b + (b - (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])]/Sqrt[a - I*b - c] + ArcTanh[(2*a + I*b + (b + (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])]/Sqrt[a + I*b - c])/(2*e)","C",1
17,1,264,395,1.5747473,"\int \frac{\cot ^2(d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx","Integrate[Cot[d + e*x]^2/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","\frac{\frac{b \tanh ^{-1}\left(\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{a^{3/2}}+\frac{i \tanh ^{-1}\left(\frac{2 a+(b-2 i c) \tan (d+e x)-i b}{2 \sqrt{a-i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{a-i b-c}}-\frac{i \tanh ^{-1}\left(\frac{2 a+(b+2 i c) \tan (d+e x)+i b}{2 \sqrt{a+i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{a+i b-c}}-\frac{2 \cot (d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{a}}{2 e}","\frac{b \tanh ^{-1}\left(\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{2 a^{3/2} e}-\frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tan ^{-1}\left(\frac{b-\left(-\sqrt{a^2-2 a c+b^2+c^2}+a-c\right) \tan (d+e x)}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \tan (d+e x)+c \tan ^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} \tan ^{-1}\left(\frac{b-\left(\sqrt{a^2-2 a c+b^2+c^2}+a-c\right) \tan (d+e x)}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}-\frac{\cot (d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{a e}",1,"((b*ArcTanh[(2*a + b*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/a^(3/2) + (I*ArcTanh[(2*a - I*b + (b - (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[a - I*b - c] - (I*ArcTanh[(2*a + I*b + (b + (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[a + I*b - c] - (2*Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/a)/(2*e)","C",1
18,1,315,500,5.9761738,"\int \frac{\cot ^3(d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx","Integrate[Cot[d + e*x]^3/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","\frac{\frac{3 b \cot (d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{a^2}+\frac{\left(8 a^2+4 a c-3 b^2\right) \tanh ^{-1}\left(\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{2 a^{5/2}}-\frac{2 \tanh ^{-1}\left(\frac{2 a+(b-2 i c) \tan (d+e x)-i b}{2 \sqrt{a-i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{a-i b-c}}-\frac{2 \tanh ^{-1}\left(\frac{2 a+(b+2 i c) \tan (d+e x)+i b}{2 \sqrt{a+i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{a+i b-c}}-\frac{2 \cot ^2(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{a}}{4 e}","-\frac{\left(3 b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^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 \tan (d+e x)-c}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \tan (d+e x)+c \tan ^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 \tan (d+e x)-c}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}+\frac{3 b \cot (d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 a^2 e}+\frac{\tanh ^{-1}\left(\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{a} e}-\frac{\cot ^2(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{2 a e}",1,"(((8*a^2 - 3*b^2 + 4*a*c)*ArcTanh[(2*a + b*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(2*a^(5/2)) - (2*ArcTanh[(2*a - I*b + (b - (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[a - I*b - c] - (2*ArcTanh[(2*a + I*b + (b + (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[a + I*b - c] + (3*b*Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/a^2 - (2*Cot[d + e*x]^2*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/a)/(4*e)","C",1
19,1,884,1190,25.1352243,"\int \frac{\tan ^7(d+e x)}{\left(a+b \tan (d+e x)+c \tan ^2(d+e x)\right)^{3/2}} \, dx","Integrate[Tan[d + e*x]^7/(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2),x]","\frac{\frac{8 \log \left(\frac{-2 a-i b-(b+2 i c) \tan (d+e x)-2 \sqrt{a+i b-c} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}{8 (a-i b-c) \sqrt{a+i b-c} c^4 (\tan (d+e x)-i)}\right)}{(a+i b-c)^{3/2}}+\frac{8 \log \left(\frac{-2 a+i b-b \tan (d+e x)+2 i c \tan (d+e x)-2 \sqrt{a-i b-c} \sqrt{a+\tan (d+e x) (b+c \tan (d+e x))}}{8 \sqrt{a-i b-c} (a+i b-c) c^4 (\tan (d+e x)+i)}\right)}{(a-i b-c)^{3/2}}+\frac{b \left(-35 b^2+24 c^2+60 a c\right) \log \left(b+2 c \tan (d+e x)+2 \sqrt{c} \sqrt{a+\tan (d+e x) (b+c \tan (d+e x))}\right)}{c^{9/2}}}{16 e}+\frac{\sqrt{\frac{\cos (2 (d+e x)) a+a+c-c \cos (2 (d+e x))+b \sin (2 (d+e x))}{\cos (2 (d+e x))+1}} \left(\frac{\sec ^2(d+e x)}{3 c^2}-\frac{105 a b^6-57 c b^6+105 a^3 b^4-25 c^3 b^4+407 a c^2 b^4-727 a^2 c b^4+32 c^5 b^2+44 a c^4 b^2-740 a^2 c^3 b^2+1364 a^3 c^2 b^2-460 a^4 c b^2-128 a c^6+224 a^2 c^5+96 a^3 c^4-448 a^4 c^3+256 a^5 c^2}{24 (a-c) (a-i b-c) (a+i b-c) c^4 \left(4 a c-b^2\right)}+\frac{2 \left(\sin (2 (d+e x)) b^7+2 a b^6+2 a^2 \sin (2 (d+e x)) b^5-7 a c \sin (2 (d+e x)) b^5+2 a^3 b^4-12 a^2 c b^4+a^4 \sin (2 (d+e x)) b^3+14 a^2 c^2 \sin (2 (d+e x)) b^3-10 a^3 c \sin (2 (d+e x)) b^3+18 a^3 c^2 b^2-8 a^4 c b^2-7 a^3 c^3 \sin (2 (d+e x)) b+10 a^4 c^2 \sin (2 (d+e x)) b-3 a^5 c \sin (2 (d+e x)) b-4 a^4 c^3+4 a^5 c^2\right)}{(a-c) (a-i b-c) (a+i b-c) c^3 \left(4 a c-b^2\right) (\cos (2 (d+e x)) a+a+c-c \cos (2 (d+e x))+b \sin (2 (d+e x)))}-\frac{11 b \tan (d+e x)}{12 c^3}\right)}{e}","\frac{2 (2 a+b \tan (d+e x)) \tan ^4(d+e x)}{\left(b^2-4 a c\right) e \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}-\frac{2 b \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan ^3(d+e x)}{c \left(b^2-4 a c\right) e}+\frac{\left(7 b^2-16 a c\right) \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan ^2(d+e x)}{3 c^2 \left(b^2-4 a c\right) e}-\frac{2 (2 a+b \tan (d+e x)) \tan ^2(d+e x)}{\left(b^2-4 a c\right) e \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}-\frac{5 b \left(7 b^2-12 a c\right) \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right)}{16 c^{9/2} e}+\frac{3 b \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{c \tan ^2(d+e x)+b \tan (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) \tan (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 \tan ^2(d+e x)+b \tan (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) \tan (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 \tan ^2(d+e x)+b \tan (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 \tan (d+e x) b-8 a c\right) \sqrt{c \tan ^2(d+e x)+b \tan (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) \tan (d+e x) b+256 a^2 c^2\right) \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}{24 c^4 \left(b^2-4 a c\right) e}+\frac{2 (2 a+b \tan (d+e x))}{\left(b^2-4 a c\right) e \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}-\frac{2 \left(a \left(b^2-2 (a-c) c\right)+b c (a+c) \tan (d+e x)\right)}{\left(b^2+(a-c)^2\right) \left(b^2-4 a c\right) e \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}",1,"((8*Log[(-2*a - I*b - (b + (2*I)*c)*Tan[d + e*x] - 2*Sqrt[a + I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(8*(a - I*b - c)*Sqrt[a + I*b - c]*c^4*(-I + Tan[d + e*x]))])/(a + I*b - c)^(3/2) + (8*Log[(-2*a + I*b - b*Tan[d + e*x] + (2*I)*c*Tan[d + e*x] - 2*Sqrt[a - I*b - c]*Sqrt[a + Tan[d + e*x]*(b + c*Tan[d + e*x])])/(8*Sqrt[a - I*b - c]*(a + I*b - c)*c^4*(I + Tan[d + e*x]))])/(a - I*b - c)^(3/2) + (b*(-35*b^2 + 60*a*c + 24*c^2)*Log[b + 2*c*Tan[d + e*x] + 2*Sqrt[c]*Sqrt[a + Tan[d + e*x]*(b + c*Tan[d + e*x])]])/c^(9/2))/(16*e) + (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)])]*(-1/24*(105*a^3*b^4 + 105*a*b^6 - 460*a^4*b^2*c - 727*a^2*b^4*c - 57*b^6*c + 256*a^5*c^2 + 1364*a^3*b^2*c^2 + 407*a*b^4*c^2 - 448*a^4*c^3 - 740*a^2*b^2*c^3 - 25*b^4*c^3 + 96*a^3*c^4 + 44*a*b^2*c^4 + 224*a^2*c^5 + 32*b^2*c^5 - 128*a*c^6)/((a - c)*(a - I*b - c)*(a + I*b - c)*c^4*(-b^2 + 4*a*c)) + Sec[d + e*x]^2/(3*c^2) + (2*(2*a^3*b^4 + 2*a*b^6 - 8*a^4*b^2*c - 12*a^2*b^4*c + 4*a^5*c^2 + 18*a^3*b^2*c^2 - 4*a^4*c^3 + a^4*b^3*Sin[2*(d + e*x)] + 2*a^2*b^5*Sin[2*(d + e*x)] + b^7*Sin[2*(d + e*x)] - 3*a^5*b*c*Sin[2*(d + e*x)] - 10*a^3*b^3*c*Sin[2*(d + e*x)] - 7*a*b^5*c*Sin[2*(d + e*x)] + 10*a^4*b*c^2*Sin[2*(d + e*x)] + 14*a^2*b^3*c^2*Sin[2*(d + e*x)] - 7*a^3*b*c^3*Sin[2*(d + e*x)]))/((a - c)*(a - I*b - c)*(a + I*b - c)*c^3*(-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)])) - (11*b*Tan[d + e*x])/(12*c^3)))/e","C",0
20,1,611,864,21.3013039,"\int \frac{\tan ^5(d+e x)}{\left(a+b \tan (d+e x)+c \tan ^2(d+e x)\right)^{3/2}} \, dx","Integrate[Tan[d + e*x]^5/(a + b*Tan[d + e*x] + c*Tan[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{-\frac{3 b \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+\tan (d+e x) (b+c \tan (d+e x))}}\right)}{c^{5/2}}-\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 i c) \tan (d+e x)-i b}{2 \sqrt{a-i b-c} \sqrt{a+\tan (d+e x) (b+c \tan (d+e x))}}\right)}{(a-i b-c)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{2 a+(b+2 i c) \tan (d+e x)+i b}{2 \sqrt{a+i b-c} \sqrt{a+\tan (d+e x) (b+c \tan (d+e x))}}\right)}{(a+i b-c)^{3/2}}}{2 e}","\frac{2 (2 a+b \tan (d+e x)) \tan ^2(d+e x)}{\left(b^2-4 a c\right) e \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}-\frac{3 b \tanh ^{-1}\left(\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{c \tan ^2(d+e x)+b \tan (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) \tan (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 \tan ^2(d+e x)+b \tan (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) \tan (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 \tan ^2(d+e x)+b \tan (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 \tan (d+e x) b-8 a c\right) \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}{c^2 \left(b^2-4 a c\right) e}-\frac{2 (2 a+b \tan (d+e x))}{\left(b^2-4 a c\right) e \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}+\frac{2 \left(a \left(b^2-2 (a-c) c\right)+b c (a+c) \tan (d+e x)\right)}{\left(b^2+(a-c)^2\right) \left(b^2-4 a c\right) e \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}",1,"(-(ArcTanh[(2*a - I*b + (b - (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + Tan[d + e*x]*(b + c*Tan[d + e*x])])]/(a - I*b - c)^(3/2)) - ArcTanh[(2*a + I*b + (b + (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[a + Tan[d + e*x]*(b + c*Tan[d + e*x])])]/(a + I*b - c)^(3/2) - (3*b*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + Tan[d + e*x]*(b + c*Tan[d + e*x])])])/c^(5/2))/(2*e) + (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","C",0
21,1,286,686,17.3329799,"\int \frac{\tan ^3(d+e x)}{\left(a+b \tan (d+e x)+c \tan ^2(d+e x)\right)^{3/2}} \, dx","Integrate[Tan[d + e*x]^3/(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2),x]","\frac{-\frac{4 \sqrt{2} \left(b \left(a^2-3 a c+b^2\right) \tan (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{\sec ^2(d+e x) ((a-c) \cos (2 (d+e x))+a+b \sin (2 (d+e x))+c)}}+\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 i c) \tan (d+e x)-i b}{2 \sqrt{a-i b-c} \sqrt{a+\tan (d+e x) (b+c \tan (d+e x))}}\right)}{(a-i b-c)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{2 a+(b+2 i c) \tan (d+e x)+i b}{2 \sqrt{a+i b-c} \sqrt{a+\tan (d+e x) (b+c \tan (d+e x))}}\right)}{(a+i b-c)^{3/2}}}{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) \tan (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 \tan (d+e x)+c \tan ^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) \tan (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 \tan (d+e x)+c \tan ^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 \tan (d+e x))}{e \left(b^2-4 a c\right) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}-\frac{2 \left(a \left(b^2-2 c (a-c)\right)+b c (a+c) \tan (d+e x)\right)}{e \left((a-c)^2+b^2\right) \left(b^2-4 a c\right) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}",1,"(ArcTanh[(2*a - I*b + (b - (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + Tan[d + e*x]*(b + c*Tan[d + e*x])])]/(a - I*b - c)^(3/2) + ArcTanh[(2*a + I*b + (b + (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[a + Tan[d + e*x]*(b + c*Tan[d + e*x])])]/(a + I*b - c)^(3/2) - (4*Sqrt[2]*(a*(2*a^2 + b^2 - 2*a*c) + b*(a^2 + b^2 - 3*a*c)*Tan[d + e*x]))/((a - I*b - c)*(a + I*b - c)*(-b^2 + 4*a*c)*Sqrt[Sec[d + e*x]^2*(a + c + (a - c)*Cos[2*(d + e*x)] + b*Sin[2*(d + e*x)])]))/(2*e)","C",0
22,1,328,638,4.9918745,"\int \frac{\tan ^2(d+e x)}{\left(a+b \tan (d+e x)+c \tan ^2(d+e x)\right)^{3/2}} \, dx","Integrate[Tan[d + e*x]^2/(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2),x]","\frac{\frac{\left(-4 i a^2 c+a \left(i b^2+4 b c+4 i c^2\right)-b^2 (b+i c)\right) \tanh ^{-1}\left(\frac{2 a+(b-2 i c) \tan (d+e x)-i b}{2 \sqrt{a-i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{a-i b-c}}+\frac{i \left(4 a^2 c-a \left(b^2+4 i b c+4 c^2\right)+b^2 (c+i b)\right) \tanh ^{-1}\left(\frac{2 a+(b+2 i c) \tan (d+e x)+i b}{2 \sqrt{a+i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{a+i b-c}}-\frac{4 \left(c \left(2 a^2-2 a c+b^2\right) \tan (d+e x)+a b (a+c)\right)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}}{2 e \left((a-c)^2+b^2\right) \left(b^2-4 a c\right)}","-\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} \tan ^{-1}\left(\frac{\left(b^2-(a-c) \left(-\sqrt{a^2-2 a c+b^2+c^2}+a-c\right)\right) \tan (d+e x)+b \left(\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c\right)}{\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 \tan (d+e x)+c \tan ^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} \tan ^{-1}\left(\frac{\left(b^2-(a-c) \left(\sqrt{a^2-2 a c+b^2+c^2}+a-c\right)\right) \tan (d+e x)+b \left(-\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c\right)}{\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 \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{2} e \left(a^2-2 a c+b^2+c^2\right)^{3/2}}-\frac{2 \left(c \left(2 a^2-2 a c+b^2\right) \tan (d+e x)+a b (a+c)\right)}{e \left((a-c)^2+b^2\right) \left(b^2-4 a c\right) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}",1,"(((-(b^2*(b + I*c)) - (4*I)*a^2*c + a*(I*b^2 + 4*b*c + (4*I)*c^2))*ArcTanh[(2*a - I*b + (b - (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[a - I*b - c] + (I*(4*a^2*c + b^2*(I*b + c) - a*(b^2 + (4*I)*b*c + 4*c^2))*ArcTanh[(2*a + I*b + (b + (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[a + I*b - c] - (4*(a*b*(a + c) + c*(2*a^2 + b^2 - 2*a*c)*Tan[d + e*x]))/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(2*(b^2 + (a - c)^2)*(b^2 - 4*a*c)*e)","C",0
23,1,318,635,4.751053,"\int \frac{\tan (d+e x)}{\left(a+b \tan (d+e x)+c \tan ^2(d+e x)\right)^{3/2}} \, dx","Integrate[Tan[d + e*x]/(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2),x]","\frac{\frac{\left(4 a^2 c-a \left(b^2-4 i b c+4 c^2\right)+b^2 (c-i b)\right) \tanh ^{-1}\left(\frac{2 a+(b-2 i c) \tan (d+e x)-i b}{2 \sqrt{a-i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{a-i b-c}}+\frac{\left(4 a^2 c-a \left(b^2+4 i b c+4 c^2\right)+b^2 (c+i b)\right) \tanh ^{-1}\left(\frac{2 a+(b+2 i c) \tan (d+e x)+i b}{2 \sqrt{a+i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{\sqrt{a+i b-c}}+\frac{4 \left(a \left(2 c (c-a)+b^2\right)+b c (a+c) \tan (d+e x)\right)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}}{2 e \left((a-c)^2+b^2\right) \left(b^2-4 a c\right)}","\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) \tan (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 \tan (d+e x)+c \tan ^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) \tan (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 \tan (d+e x)+c \tan ^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) \tan (d+e x)\right)}{e \left((a-c)^2+b^2\right) \left(b^2-4 a c\right) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}",1,"(((4*a^2*c + b^2*((-I)*b + c) - a*(b^2 - (4*I)*b*c + 4*c^2))*ArcTanh[(2*a - I*b + (b - (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[a - I*b - c] + ((4*a^2*c + b^2*(I*b + c) - a*(b^2 + (4*I)*b*c + 4*c^2))*ArcTanh[(2*a + I*b + (b + (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[a + I*b - c] + (4*(a*(b^2 + 2*c*(-a + c)) + b*c*(a + c)*Tan[d + e*x]))/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(2*(b^2 + (a - c)^2)*(b^2 - 4*a*c)*e)","C",0
24,1,450,750,5.1922976,"\int \frac{\cot (d+e x)}{\left(a+b \tan (d+e x)+c \tan ^2(d+e x)\right)^{3/2}} \, dx","Integrate[Cot[d + e*x]/(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2),x]","\frac{2 \left(\frac{\left(2 a c-\frac{b^2}{2}\right) \tanh ^{-1}\left(\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{a^{3/2}}+\frac{-a b^2-b c (a+c) \tan (d+e x)+2 a c (a-c)}{\left(a^2-2 a c+b^2+c^2\right) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac{-\frac{\left(4 a^2 c-a \left(b^2-4 i b c+4 c^2\right)+b^2 (c-i b)\right) \tanh ^{-1}\left(\frac{2 a+(b-2 i c) \tan (d+e x)-i b}{2 \sqrt{a-i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{4 \sqrt{a-i b-c}}-\frac{\left(4 a^2 c-a \left(b^2+4 i b c+4 c^2\right)+b^2 (c+i b)\right) \tanh ^{-1}\left(\frac{2 a+(b+2 i c) \tan (d+e x)+i b}{2 \sqrt{a+i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{4 \sqrt{a+i b-c}}}{(a-c)^2+b^2}+\frac{-2 a c+b^2+b c \tan (d+e x)}{a \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{e \left(b^2-4 a c\right)}","-\frac{\tanh ^{-1}\left(\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^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) \tan (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 \tan (d+e x)+c \tan ^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) \tan (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 \tan (d+e x)+c \tan ^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 \tan (d+e x)\right)}{a e \left(b^2-4 a c\right) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}-\frac{2 \left(a \left(b^2-2 c (a-c)\right)+b c (a+c) \tan (d+e x)\right)}{e \left((a-c)^2+b^2\right) \left(b^2-4 a c\right) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}",1,"(2*(((-1/2*b^2 + 2*a*c)*ArcTanh[(2*a + b*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/a^(3/2) + (-1/4*((4*a^2*c + b^2*((-I)*b + c) - a*(b^2 - (4*I)*b*c + 4*c^2))*ArcTanh[(2*a - I*b + (b - (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[a - I*b - c] - ((4*a^2*c + b^2*(I*b + c) - a*(b^2 + (4*I)*b*c + 4*c^2))*ArcTanh[(2*a + I*b + (b + (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(4*Sqrt[a + I*b - c]))/(b^2 + (a - c)^2) + (b^2 - 2*a*c + b*c*Tan[d + e*x])/(a*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) + (-(a*b^2) + 2*a*(a - c)*c - b*c*(a + c)*Tan[d + e*x])/((a^2 + b^2 - 2*a*c + c^2)*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])))/((b^2 - 4*a*c)*e)","C",0
25,1,583,829,6.1707499,"\int \frac{\cot ^2(d+e x)}{\left(a+b \tan (d+e x)+c \tan ^2(d+e x)\right)^{3/2}} \, dx","Integrate[Cot[d + e*x]^2/(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2),x]","\frac{-\frac{2 \left(\frac{\left(\frac{1}{2} b \left(8 a c-3 b^2\right)+2 a b c\right) \tanh ^{-1}\left(\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{2 a^{3/2}}+\frac{\left(3 b^2-8 a c\right) \cot (d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{2 a}\right)}{a \left(b^2-4 a c\right)}+\frac{2 \left(-\frac{4 \sqrt{a-i b-c} \left(-\frac{1}{4} b \left(b^2-4 a c\right)+\frac{1}{4} i (a-c) \left(b^2-4 a c\right)\right) \tanh ^{-1}\left(\frac{-2 a-(b-2 i c) \tan (d+e x)+i b}{2 \sqrt{a-i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{4 a-4 i b-4 c}-\frac{4 \sqrt{a+i b-c} \left(-\frac{1}{4} b \left(b^2-4 a c\right)-\frac{1}{4} i (a-c) \left(b^2-4 a c\right)\right) \tanh ^{-1}\left(\frac{-2 a-(b+2 i c) \tan (d+e x)-i b}{2 \sqrt{a+i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{4 a+4 i b-4 c}\right)}{\left((a-c)^2+b^2\right) \left(b^2-4 a c\right)}-\frac{2 \cot (d+e x) \left(2 a c-b^2-b c \tan (d+e x)\right)}{a \left(b^2-4 a c\right) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}-\frac{2 \left(c \left(2 a c-b^2-2 c^2\right) \tan (d+e x)+b c (3 a-c)-b^3\right)}{\left((a-c)^2+b^2\right) \left(b^2-4 a c\right) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}}{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}} \tan ^{-1}\left(\frac{b \left(2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}\right)+\left(b^2-(a-c) \left(a-c-\sqrt{a^2-2 c a+b^2+c^2}\right)\right) \tan (d+e x)}{\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 \tan ^2(d+e x)+b \tan (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}} \tan ^{-1}\left(\frac{b \left(2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}\right)+\left(b^2-(a-c) \left(a-c+\sqrt{a^2-2 c a+b^2+c^2}\right)\right) \tan (d+e x)}{\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 \tan ^2(d+e x)+b \tan (d+e x)+a}}\right)}{\sqrt{2} \left(a^2-2 c a+b^2+c^2\right)^{3/2} e}+\frac{3 b \tanh ^{-1}\left(\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right)}{2 a^{5/2} e}-\frac{\left(3 b^2-8 a c\right) \cot (d+e x) \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}{a^2 \left(b^2-4 a c\right) e}+\frac{2 \cot (d+e x) \left(b^2+c \tan (d+e x) b-2 a c\right)}{a \left(b^2-4 a c\right) e \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}+\frac{2 \left(b \left(b^2-(3 a-c) c\right)+c \left(b^2-2 (a-c) c\right) \tan (d+e x)\right)}{\left(b^2+(a-c)^2\right) \left(b^2-4 a c\right) e \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}",1,"((2*((-4*Sqrt[a - I*b - c]*(-1/4*(b*(b^2 - 4*a*c)) + (I/4)*(a - c)*(b^2 - 4*a*c))*ArcTanh[(-2*a + I*b - (b - (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(4*a - (4*I)*b - 4*c) - (4*Sqrt[a + I*b - c]*(-1/4*(b*(b^2 - 4*a*c)) - (I/4)*(a - c)*(b^2 - 4*a*c))*ArcTanh[(-2*a - I*b - (b + (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(4*a + (4*I)*b - 4*c)))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)) - (2*Cot[d + e*x]*(-b^2 + 2*a*c - b*c*Tan[d + e*x]))/(a*(b^2 - 4*a*c)*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) - (2*(-b^3 + b*(3*a - c)*c + c*(-b^2 + 2*a*c - 2*c^2)*Tan[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) - (2*(((2*a*b*c + (b*(-3*b^2 + 8*a*c))/2)*ArcTanh[(2*a + b*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(2*a^(3/2)) + ((3*b^2 - 8*a*c)*Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(2*a)))/(a*(b^2 - 4*a*c)))/e","C",0
26,1,786,1007,6.2080567,"\int \frac{\cot ^3(d+e x)}{\left(a+b \tan (d+e x)+c \tan ^2(d+e x)\right)^{3/2}} \, dx","Integrate[Cot[d + e*x]^3/(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2),x]","\frac{-\frac{2 \left(2 a c-\frac{b^2}{2}\right) \tanh ^{-1}\left(\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{a^{3/2} \left(b^2-4 a c\right)}-\frac{2 \left(-\frac{\frac{\left(a c \left(5 b^2-12 a c\right)-\frac{1}{4} b^2 \left(15 b^2-52 a c\right)\right) \tanh ^{-1}\left(\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{2 a^{3/2}}+\frac{b \left(15 b^2-52 a c\right) \cot (d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 a}}{2 a}-\frac{\left(12 a c-5 b^2\right) \cot ^2(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 a}\right)}{a \left(b^2-4 a c\right)}-\frac{2 \left(c (-a b-b c) \tan (d+e x)-a \left(-2 a c+b^2+2 c^2\right)\right)}{\left(b^2-4 a c\right) \left((c-a)^2+b^2\right) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac{2 \left(2 a c-b^2-b c \tan (d+e x)\right)}{a \left(b^2-4 a c\right) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}-\frac{2 \left(-\frac{4 \sqrt{a+i b-c} \left(-\frac{1}{4} (a-c) \left(b^2-4 a c\right)+\frac{1}{4} i b \left(b^2-4 a c\right)\right) \tanh ^{-1}\left(\frac{2 a-(-b-2 i c) \tan (d+e x)+i b}{2 \sqrt{a+i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{4 a+4 i b-4 c}-\frac{4 \sqrt{a-i b-c} \left(-\frac{1}{4} (a-c) \left(b^2-4 a c\right)-\frac{1}{4} i b \left(b^2-4 a c\right)\right) \tanh ^{-1}\left(\frac{2 a-(-b+2 i c) \tan (d+e x)-i b}{2 \sqrt{a-i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right)}{4 a-4 i b-4 c}\right)}{\left(b^2-4 a c\right) \left((c-a)^2+b^2\right)}-\frac{2 \cot ^2(d+e x) \left(2 a c-b^2-b c \tan (d+e x)\right)}{a \left(b^2-4 a c\right) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}}{e}","-\frac{\left(5 b^2-12 a c\right) \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a} \cot ^2(d+e x)}{2 a^2 \left(b^2-4 a c\right) e}+\frac{2 \left(b^2+c \tan (d+e x) b-2 a c\right) \cot ^2(d+e x)}{a \left(b^2-4 a c\right) e \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}+\frac{b \left(15 b^2-52 a c\right) \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a} \cot (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 \tan (d+e x)}{2 \sqrt{a} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right)}{8 a^{7/2} e}+\frac{\tanh ^{-1}\left(\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{c \tan ^2(d+e x)+b \tan (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) \tan (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 \tan ^2(d+e x)+b \tan (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) \tan (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 \tan ^2(d+e x)+b \tan (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 \tan (d+e x) b-2 a c\right)}{a \left(b^2-4 a c\right) e \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}+\frac{2 \left(a \left(b^2-2 (a-c) c\right)+b c (a+c) \tan (d+e x)\right)}{\left(b^2+(a-c)^2\right) \left(b^2-4 a c\right) e \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}",1,"((-2*(-1/2*b^2 + 2*a*c)*ArcTanh[(2*a + b*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(a^(3/2)*(b^2 - 4*a*c)) - (2*((-4*Sqrt[a + I*b - c]*((I/4)*b*(b^2 - 4*a*c) - ((a - c)*(b^2 - 4*a*c))/4)*ArcTanh[(2*a + I*b - (-b - (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(4*a + (4*I)*b - 4*c) - (4*Sqrt[a - I*b - c]*((-1/4*I)*b*(b^2 - 4*a*c) - ((a - c)*(b^2 - 4*a*c))/4)*ArcTanh[(2*a - I*b - (-b + (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(4*a - (4*I)*b - 4*c)))/((b^2 - 4*a*c)*(b^2 + (-a + c)^2)) + (2*(-b^2 + 2*a*c - b*c*Tan[d + e*x]))/(a*(b^2 - 4*a*c)*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) - (2*Cot[d + e*x]^2*(-b^2 + 2*a*c - b*c*Tan[d + e*x]))/(a*(b^2 - 4*a*c)*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) - (2*(-(a*(b^2 - 2*a*c + 2*c^2)) + c*(-(a*b) - b*c)*Tan[d + e*x]))/((b^2 - 4*a*c)*(b^2 + (-a + c)^2)*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) - (2*(-1/4*((-5*b^2 + 12*a*c)*Cot[d + e*x]^2*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/a - (((-1/4*(b^2*(15*b^2 - 52*a*c)) + a*c*(5*b^2 - 12*a*c))*ArcTanh[(2*a + b*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(2*a^(3/2)) + (b*(15*b^2 - 52*a*c)*Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(4*a))/(2*a)))/(a*(b^2 - 4*a*c)))/e","C",0
27,1,467,270,6.1333153,"\int \tan ^5(d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx","Integrate[Tan[d + e*x]^5*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],x]","\frac{\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c \tan ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{8 c^{3/2}}-\frac{b \left(\frac{\left(b+2 c \tan ^2(d+e x)\right) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{4 c}-\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c \tan ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{8 c^{3/2}}\right)}{2 c}+\frac{\left(a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right)^{3/2}}{3 c}-\frac{\left(b+2 c \tan ^2(d+e x)\right) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{4 c}+\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}+\frac{1}{2} \left(\frac{4 (-2 a+2 b-2 c) \sqrt{a-b+c} \tanh ^{-1}\left(\frac{2 a-\left((2 c-b) \tan ^2(d+e x)\right)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{4 a-4 b+4 c}-\frac{(2 c-b) \tanh ^{-1}\left(\frac{b+2 c \tan ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{\sqrt{c}}\right)}{2 e}","\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 \tan ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{32 c^{5/2} e}-\frac{\left(2 c (b+2 c) \tan ^2(d+e x)+(b-2 c) (b+4 c)\right) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{16 c^2 e}+\frac{\left(a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right)^{3/2}}{6 c e}-\frac{\sqrt{a-b+c} \tanh ^{-1}\left(\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 e}",1,"(((b^2 - 4*a*c)*ArcTanh[(b + 2*c*Tan[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(8*c^(3/2)) + (-(((-b + 2*c)*ArcTanh[(b + 2*c*Tan[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/Sqrt[c]) + (4*(-2*a + 2*b - 2*c)*Sqrt[a - b + c]*ArcTanh[(2*a - b - (-b + 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(4*a - 4*b + 4*c))/2 + Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4] - ((b + 2*c*Tan[d + e*x]^2)*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(4*c) + (a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)^(3/2)/(3*c) - (b*(-1/8*((b^2 - 4*a*c)*ArcTanh[(b + 2*c*Tan[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/c^(3/2) + ((b + 2*c*Tan[d + e*x]^2)*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(4*c)))/(2*c))/(2*e)","A",1
28,1,208,209,1.5657668,"\int \tan ^3(d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx","Integrate[Tan[d + e*x]^3*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[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 \tan ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)+8 c^{3/2} \sqrt{a-b+c} \tanh ^{-1}\left(\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)+2 \sqrt{c} \left(b+2 c \tan ^2(d+e x)-4 c\right) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{16 c^{3/2} e}","-\frac{\left(-4 c (a+2 c)+b^2+4 b c\right) \tanh ^{-1}\left(\frac{b+2 c \tan ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{16 c^{3/2} e}+\frac{\left(b+2 c \tan ^2(d+e x)-4 c\right) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{8 c e}+\frac{\sqrt{a-b+c} \tanh ^{-1}\left(\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 e}",1,"(8*c^(3/2)*Sqrt[a - b + c]*ArcTanh[(2*a - b + (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])] - (b^2 + 4*b*c - 4*c*(a + 2*c))*ArcTanh[(b + 2*c*Tan[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])] + 2*Sqrt[c]*(b - 4*c + 2*c*Tan[d + e*x]^2)*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(16*c^(3/2)*e)","A",1
29,1,180,179,0.2995528,"\int \tan (d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx","Integrate[Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],x]","\frac{2 \sqrt{c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}-2 \sqrt{c} \sqrt{a-b+c} \tanh ^{-1}\left(\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)+(b-2 c) \tanh ^{-1}\left(\frac{b+2 c \tan ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{4 \sqrt{c} e}","\frac{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{2 e}-\frac{\sqrt{a-b+c} \tanh ^{-1}\left(\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 e}+\frac{(b-2 c) \tanh ^{-1}\left(\frac{b+2 c \tan ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{4 \sqrt{c} e}",1,"(-2*Sqrt[c]*Sqrt[a - b + c]*ArcTanh[(2*a - b + (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])] + (b - 2*c)*ArcTanh[(b + 2*c*Tan[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])] + 2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(4*Sqrt[c]*e)","A",1
30,1,192,203,1.1781,"\int \cot (d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx","Integrate[Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],x]","-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{2 a+b \tan ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)+\sqrt{a-b+c} \tanh ^{-1}\left(\frac{-2 a-\left((b-2 c) \tan ^2(d+e x)\right)+b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)-\sqrt{c} \tanh ^{-1}\left(\frac{b+2 c \tan ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 e}","-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{2 a+b \tan ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 e}+\frac{\sqrt{a-b+c} \tanh ^{-1}\left(\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 e}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{b+2 c \tan ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 e}",1,"-1/2*(Sqrt[a]*ArcTanh[(2*a + b*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])] + Sqrt[a - b + c]*ArcTanh[(-2*a + b - (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])] - Sqrt[c]*ArcTanh[(b + 2*c*Tan[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/e","A",1
31,1,187,435,1.1986824,"\int \cot ^3(d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx","Integrate[Cot[d + e*x]^3*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],x]","\frac{(2 a-b) \tanh ^{-1}\left(\frac{2 a+b \tan ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)-2 \sqrt{a} \left(\sqrt{a-b+c} \tanh ^{-1}\left(\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)+\cot ^2(d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}\right)}{4 \sqrt{a} e}","-\frac{b \tanh ^{-1}\left(\frac{2 a+b \tan ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{4 \sqrt{a} e}+\frac{\sqrt{a} \tanh ^{-1}\left(\frac{2 a+b \tan ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 e}-\frac{\sqrt{a-b+c} \tanh ^{-1}\left(\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 e}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{b+2 c \tan ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 e}-\frac{b \tanh ^{-1}\left(\frac{b+2 c \tan ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{4 \sqrt{c} e}+\frac{(b-2 c) \tanh ^{-1}\left(\frac{b+2 c \tan ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{4 \sqrt{c} e}-\frac{\cot ^2(d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{2 e}",1,"((2*a - b)*ArcTanh[(2*a + b*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])] - 2*Sqrt[a]*(Sqrt[a - b + c]*ArcTanh[(2*a - b + (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])] + Cot[d + e*x]^2*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]))/(4*Sqrt[a]*e)","A",1
32,1,639,1254,29.6078985,"\int \tan ^2(d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx","Integrate[Tan[d + e*x]^2*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],x]","\frac{\sqrt{\frac{4 a \cos (2 (d+e x))+a \cos (4 (d+e x))+3 a-b \cos (4 (d+e x))+b-4 c \cos (2 (d+e x))+c \cos (4 (d+e x))+3 c}{4 \cos (2 (d+e x))+\cos (4 (d+e x))+3}} \left(\frac{(b-3 c) \sin (2 (d+e x))}{6 c}+\frac{1}{3} \tan (d+e x)\right)}{e}+\frac{-\frac{4 (b-3 c) \tan (d+e x) \left(a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right)}{\tan ^2(d+e x)+1}+\frac{i \sqrt{2} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c \tan ^2(d+e x)}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{2 c \tan ^2(d+e x)}{b-\sqrt{b^2-4 a c}}+1} \left(\left(-b \left(\sqrt{b^2-4 a c}-3 c\right)+c \left(3 \sqrt{b^2-4 a c}-4 a-6 c\right)+b^2\right) F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)+(b-3 c) \left(\sqrt{b^2-4 a c}-b\right) E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)+6 c (a-b+c) \Pi \left(\frac{b+\sqrt{b^2-4 a c}}{2 c};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)\right)}{\sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}}}}{12 c e \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}","-\frac{\sqrt{a-b+c} \tan ^{-1}\left(\frac{\sqrt{a-b+c} \tan (d+e x)}{\sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}\right)}{2 e}+\frac{\tan (d+e x) \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}{3 e}-\frac{\sqrt{c} \tan (d+e x) \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}{e \left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right)}+\frac{b \tan (d+e x) \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}{3 \sqrt{c} e \left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right)}+\frac{\sqrt[4]{a} \sqrt[4]{c} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right) \sqrt{\frac{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right)^2}}}{e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}-\frac{\sqrt[4]{a} b E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right) \sqrt{\frac{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right)^2}}}{3 c^{3/4} e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}+\frac{\sqrt[4]{c} (a-b+c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right) \sqrt{\frac{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right)^2}}}{2 \sqrt[4]{a} \left(\sqrt{a}-\sqrt{c}\right) e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}-\frac{\left(b-c+\sqrt{a} \sqrt{c}\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right) \sqrt{\frac{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right)^2}}}{2 \sqrt[4]{a} \sqrt[4]{c} e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}+\frac{\sqrt[4]{a} \left(b+2 \sqrt{a} \sqrt{c}\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right) \sqrt{\frac{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right)^2}}}{6 c^{3/4} e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}-\frac{\left(\sqrt{a}+\sqrt{c}\right) (a-b+c) \Pi \left(-\frac{\left(\sqrt{a}-\sqrt{c}\right)^2}{4 \sqrt{a} \sqrt{c}};2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right) \sqrt{\frac{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right)^2}}}{4 \sqrt[4]{a} \left(\sqrt{a}-\sqrt{c}\right) \sqrt[4]{c} e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}",1,"(Sqrt[(3*a + b + 3*c + 4*a*Cos[2*(d + e*x)] - 4*c*Cos[2*(d + e*x)] + a*Cos[4*(d + e*x)] - b*Cos[4*(d + e*x)] + c*Cos[4*(d + e*x)])/(3 + 4*Cos[2*(d + e*x)] + Cos[4*(d + e*x)])]*(((b - 3*c)*Sin[2*(d + e*x)])/(6*c) + Tan[d + e*x]/3))/e + ((I*Sqrt[2]*((b - 3*c)*(-b + Sqrt[b^2 - 4*a*c])*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + (b^2 - b*(-3*c + Sqrt[b^2 - 4*a*c]) + c*(-4*a - 6*c + 3*Sqrt[b^2 - 4*a*c]))*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + 6*c*(a - b + c)*EllipticPi[(b + Sqrt[b^2 - 4*a*c])/(2*c), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[c/(b + Sqrt[b^2 - 4*a*c])] - (4*(b - 3*c)*Tan[d + e*x]*(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4))/(1 + Tan[d + e*x]^2))/(12*c*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])","C",0
33,1,428,829,1.9813777,"\int \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx","Integrate[Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],x]","\frac{i \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c \tan ^2(d+e x)}{\sqrt{b^2-4 a c}+b}} \sqrt{1-\frac{2 c \tan ^2(d+e x)}{\sqrt{b^2-4 a c}-b}} \left(-\left(\sqrt{b^2-4 a c}+b-2 c\right) F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)+\left(\sqrt{b^2-4 a c}-b\right) E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)-2 (a-b+c) \Pi \left(\frac{b+\sqrt{b^2-4 a c}}{2 c};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)\right)}{2 \sqrt{2} e \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}","\frac{\sqrt{a-b+c} \tan ^{-1}\left(\frac{\sqrt{a-b+c} \tan (d+e x)}{\sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}\right)}{2 e}+\frac{\sqrt{c} \tan (d+e x) \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}{e \left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right)}-\frac{\sqrt[4]{a} \sqrt[4]{c} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right) \sqrt{\frac{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right)^2}}}{e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}-\frac{\sqrt[4]{c} (a-b+c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right) \sqrt{\frac{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right)^2}}}{2 \sqrt[4]{a} \left(\sqrt{a}-\sqrt{c}\right) e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}+\frac{\left(b-c+\sqrt{a} \sqrt{c}\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right) \sqrt{\frac{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right)^2}}}{2 \sqrt[4]{a} \sqrt[4]{c} e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}+\frac{\left(\sqrt{a}+\sqrt{c}\right) (a-b+c) \Pi \left(-\frac{\left(\sqrt{a}-\sqrt{c}\right)^2}{4 \sqrt{a} \sqrt{c}};2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right) \sqrt{\frac{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right)^2}}}{4 \sqrt[4]{a} \left(\sqrt{a}-\sqrt{c}\right) \sqrt[4]{c} e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}",1,"((I/2)*((-b + Sqrt[b^2 - 4*a*c])*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - (b - 2*c + Sqrt[b^2 - 4*a*c])*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - 2*(a - b + c)*EllipticPi[(b + Sqrt[b^2 - 4*a*c])/(2*c), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 - (2*c*Tan[d + e*x]^2)/(-b + Sqrt[b^2 - 4*a*c])])/(Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])","C",1
34,1,1258,861,27.0421411,"\int \cot ^2(d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx","Integrate[Cot[d + e*x]^2*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],x]","\frac{\sqrt{\frac{4 \cos (2 (d+e x)) a+\cos (4 (d+e x)) a+3 a+b+3 c-4 c \cos (2 (d+e x))-b \cos (4 (d+e x))+c \cos (4 (d+e x))}{4 \cos (2 (d+e x))+\cos (4 (d+e x))+3}} \left(\frac{1}{2} \sin (2 (d+e x))-\cot (d+e x)\right)}{e}+\frac{i \sqrt{2} \left(\sqrt{b^2-4 a c}-b\right) \left(E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)-F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)\right) \sqrt{\frac{2 c \tan ^2(d+e x)+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c \tan ^2(d+e x)}{b-\sqrt{b^2-4 a c}}+1} \left(\tan ^2(d+e x)+1\right)-2 i \sqrt{2} c F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right) \sqrt{\frac{2 c \tan ^2(d+e x)+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c \tan ^2(d+e x)}{b-\sqrt{b^2-4 a c}}+1} \left(\tan ^2(d+e x)+1\right)+2 i \sqrt{2} a \Pi \left(\frac{b+\sqrt{b^2-4 a c}}{2 c};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right) \sqrt{\frac{2 c \tan ^2(d+e x)+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c \tan ^2(d+e x)}{b-\sqrt{b^2-4 a c}}+1} \left(\tan ^2(d+e x)+1\right)-2 i \sqrt{2} b \Pi \left(\frac{b+\sqrt{b^2-4 a c}}{2 c};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right) \sqrt{\frac{2 c \tan ^2(d+e x)+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c \tan ^2(d+e x)}{b-\sqrt{b^2-4 a c}}+1} \left(\tan ^2(d+e x)+1\right)+2 i \sqrt{2} c \Pi \left(\frac{b+\sqrt{b^2-4 a c}}{2 c};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right) \sqrt{\frac{2 c \tan ^2(d+e x)+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c \tan ^2(d+e x)}{b-\sqrt{b^2-4 a c}}+1} \left(\tan ^2(d+e x)+1\right)-4 \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x) \left(c \tan ^4(d+e x)+b \tan ^2(d+e x)+a\right)}{4 \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} e \left(\tan ^2(d+e x)+1\right) \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}","-\frac{\sqrt{a-b+c} \tan ^{-1}\left(\frac{\sqrt{a-b+c} \tan (d+e x)}{\sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}\right)}{2 e}-\frac{\cot (d+e x) \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}{e}+\frac{\sqrt{c} \tan (d+e x) \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}{e \left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right)}-\frac{\sqrt[4]{a} \sqrt[4]{c} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right) \sqrt{\frac{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right)^2}}}{e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}+\frac{\sqrt[4]{c} (a-b+c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right) \sqrt{\frac{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right)^2}}}{2 \sqrt[4]{a} \left(\sqrt{a}-\sqrt{c}\right) e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}+\frac{\left(\sqrt{a}+\sqrt{c}\right) \sqrt[4]{c} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right) \sqrt{\frac{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right)^2}}}{2 \sqrt[4]{a} e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}-\frac{\left(\sqrt{a}+\sqrt{c}\right) (a-b+c) \Pi \left(-\frac{\left(\sqrt{a}-\sqrt{c}\right)^2}{4 \sqrt{a} \sqrt{c}};2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right) \sqrt{\frac{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right)^2}}}{4 \sqrt[4]{a} \left(\sqrt{a}-\sqrt{c}\right) \sqrt[4]{c} e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}",1,"(Sqrt[(3*a + b + 3*c + 4*a*Cos[2*(d + e*x)] - 4*c*Cos[2*(d + e*x)] + a*Cos[4*(d + e*x)] - b*Cos[4*(d + e*x)] + c*Cos[4*(d + e*x)])/(3 + 4*Cos[2*(d + e*x)] + Cos[4*(d + e*x)])]*(-Cot[d + e*x] + Sin[2*(d + e*x)]/2))/e + (I*Sqrt[2]*(-b + Sqrt[b^2 - 4*a*c])*(EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])*(1 + Tan[d + e*x]^2)*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b - Sqrt[b^2 - 4*a*c])] - (2*I)*Sqrt[2]*c*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]*(1 + Tan[d + e*x]^2)*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b - Sqrt[b^2 - 4*a*c])] + (2*I)*Sqrt[2]*a*EllipticPi[(b + Sqrt[b^2 - 4*a*c])/(2*c), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]*(1 + Tan[d + e*x]^2)*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b - Sqrt[b^2 - 4*a*c])] - (2*I)*Sqrt[2]*b*EllipticPi[(b + Sqrt[b^2 - 4*a*c])/(2*c), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]*(1 + Tan[d + e*x]^2)*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b - Sqrt[b^2 - 4*a*c])] + (2*I)*Sqrt[2]*c*EllipticPi[(b + Sqrt[b^2 - 4*a*c])/(2*c), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]*(1 + Tan[d + e*x]^2)*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b - Sqrt[b^2 - 4*a*c])] - 4*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]*(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4))/(4*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*e*(1 + Tan[d + e*x]^2)*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])","C",0
35,1,1590,943,31.4154028,"\int \cot ^4(d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx","Integrate[Cot[d + e*x]^4*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],x]","\frac{\sqrt{\frac{4 \cos (2 (d+e x)) a+\cos (4 (d+e x)) a+3 a+b+3 c-4 c \cos (2 (d+e x))-b \cos (4 (d+e x))+c \cos (4 (d+e x))}{4 \cos (2 (d+e x))+\cos (4 (d+e x))+3}} \left(-\frac{1}{3} \cot (d+e x) \csc ^2(d+e x)+\frac{(4 a \cos (d+e x)-b \cos (d+e x)) \csc (d+e x)}{3 a}-\frac{(3 a-b) \sin (2 (d+e x))}{6 a}\right)}{e}+\frac{-6 i \sqrt{2} \Pi \left(\frac{b+\sqrt{b^2-4 a c}}{2 c};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right) \left(\tan ^2(d+e x)+1\right) \sqrt{\frac{2 c \tan ^2(d+e x)+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c \tan ^2(d+e x)}{b-\sqrt{b^2-4 a c}}+1} a^2+3 i \sqrt{2} \left(b-\sqrt{b^2-4 a c}\right) \left(E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)-F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)\right) \left(\tan ^2(d+e x)+1\right) \sqrt{\frac{2 c \tan ^2(d+e x)+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c \tan ^2(d+e x)}{b-\sqrt{b^2-4 a c}}+1} a+2 i \sqrt{2} c F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right) \left(\tan ^2(d+e x)+1\right) \sqrt{\frac{2 c \tan ^2(d+e x)+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c \tan ^2(d+e x)}{b-\sqrt{b^2-4 a c}}+1} a+6 i \sqrt{2} b \Pi \left(\frac{b+\sqrt{b^2-4 a c}}{2 c};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right) \left(\tan ^2(d+e x)+1\right) \sqrt{\frac{2 c \tan ^2(d+e x)+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c \tan ^2(d+e x)}{b-\sqrt{b^2-4 a c}}+1} a-6 i \sqrt{2} c \Pi \left(\frac{b+\sqrt{b^2-4 a c}}{2 c};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right) \left(\tan ^2(d+e x)+1\right) \sqrt{\frac{2 c \tan ^2(d+e x)+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c \tan ^2(d+e x)}{b-\sqrt{b^2-4 a c}}+1} a-4 (b-3 a) \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x) \left(c \tan ^4(d+e x)+b \tan ^2(d+e x)+a\right)+i \sqrt{2} b \left(\sqrt{b^2-4 a c}-b\right) \left(E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)-F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)\right) \left(\tan ^2(d+e x)+1\right) \sqrt{\frac{2 c \tan ^2(d+e x)+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c \tan ^2(d+e x)}{b-\sqrt{b^2-4 a c}}+1}}{12 a \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} e \left(\tan ^2(d+e x)+1\right) \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}","-\frac{\sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a} \cot ^3(d+e x)}{3 e}+\frac{(3 a-b) \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a} \cot (d+e x)}{3 a e}+\frac{\sqrt{a-b+c} \tan ^{-1}\left(\frac{\sqrt{a-b+c} \tan (d+e x)}{\sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}\right)}{2 e}-\frac{(3 a-b) \sqrt{c} \tan (d+e x) \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}{3 a e \left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right)}+\frac{(3 a-b) \sqrt[4]{c} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right) \sqrt{\frac{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right)^2}}}{3 a^{3/4} e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}-\frac{\sqrt[4]{c} (a-b+c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right) \sqrt{\frac{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right)^2}}}{2 \sqrt[4]{a} \left(\sqrt{a}-\sqrt{c}\right) e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}-\frac{\left(3 a+\sqrt{c} \sqrt{a}-b\right) \sqrt[4]{c} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right) \sqrt{\frac{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right)^2}}}{6 a^{3/4} e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}+\frac{\left(\sqrt{a}+\sqrt{c}\right) (a-b+c) \Pi \left(-\frac{\left(\sqrt{a}-\sqrt{c}\right)^2}{4 \sqrt{a} \sqrt{c}};2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right) \sqrt{\frac{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right)^2}}}{4 \sqrt[4]{a} \left(\sqrt{a}-\sqrt{c}\right) \sqrt[4]{c} e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}",1,"(Sqrt[(3*a + b + 3*c + 4*a*Cos[2*(d + e*x)] - 4*c*Cos[2*(d + e*x)] + a*Cos[4*(d + e*x)] - b*Cos[4*(d + e*x)] + c*Cos[4*(d + e*x)])/(3 + 4*Cos[2*(d + e*x)] + Cos[4*(d + e*x)])]*(((4*a*Cos[d + e*x] - b*Cos[d + e*x])*Csc[d + e*x])/(3*a) - (Cot[d + e*x]*Csc[d + e*x]^2)/3 - ((3*a - b)*Sin[2*(d + e*x)])/(6*a)))/e + ((3*I)*Sqrt[2]*a*(b - Sqrt[b^2 - 4*a*c])*(EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])*(1 + Tan[d + e*x]^2)*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b - Sqrt[b^2 - 4*a*c])] + I*Sqrt[2]*b*(-b + Sqrt[b^2 - 4*a*c])*(EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])*(1 + Tan[d + e*x]^2)*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b - Sqrt[b^2 - 4*a*c])] + (2*I)*Sqrt[2]*a*c*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]*(1 + Tan[d + e*x]^2)*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b - Sqrt[b^2 - 4*a*c])] - (6*I)*Sqrt[2]*a^2*EllipticPi[(b + Sqrt[b^2 - 4*a*c])/(2*c), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]*(1 + Tan[d + e*x]^2)*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b - Sqrt[b^2 - 4*a*c])] + (6*I)*Sqrt[2]*a*b*EllipticPi[(b + Sqrt[b^2 - 4*a*c])/(2*c), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]*(1 + Tan[d + e*x]^2)*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b - Sqrt[b^2 - 4*a*c])] - (6*I)*Sqrt[2]*a*c*EllipticPi[(b + Sqrt[b^2 - 4*a*c])/(2*c), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]*(1 + Tan[d + e*x]^2)*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b - Sqrt[b^2 - 4*a*c])] - 4*(-3*a + b)*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]*(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4))/(12*a*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*e*(1 + Tan[d + e*x]^2)*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])","C",0
36,1,173,182,2.4878455,"\int \frac{\tan ^5(d+e x)}{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx","Integrate[Tan[d + e*x]^5/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],x]","-\frac{\frac{(b+2 c) \tanh ^{-1}\left(\frac{b+2 c \tan ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{c^{3/2}}-\frac{2 \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{c}+\frac{2 \tanh ^{-1}\left(\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{\sqrt{a-b+c}}}{4 e}","-\frac{(b+2 c) \tanh ^{-1}\left(\frac{b+2 c \tan ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{4 c^{3/2} e}+\frac{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{2 c e}-\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 e \sqrt{a-b+c}}",1,"-1/4*((2*ArcTanh[(2*a - b + (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/Sqrt[a - b + c] + ((b + 2*c)*ArcTanh[(b + 2*c*Tan[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/c^(3/2) - (2*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/c)/e","A",1
37,1,136,141,0.2514836,"\int \frac{\tan ^3(d+e x)}{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx","Integrate[Tan[d + e*x]^3/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],x]","\frac{\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{\sqrt{a-b+c}}+\frac{\tanh ^{-1}\left(\frac{b+2 c \tan ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{\sqrt{c}}}{2 e}","\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 e \sqrt{a-b+c}}+\frac{\tanh ^{-1}\left(\frac{b+2 c \tan ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 \sqrt{c} e}",1,"(ArcTanh[(2*a - b + (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])]/Sqrt[a - b + c] + ArcTanh[(b + 2*c*Tan[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])]/Sqrt[c])/(2*e)","A",1
38,1,79,79,0.1141803,"\int \frac{\tan (d+e x)}{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx","Integrate[Tan[d + e*x]/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],x]","-\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 e \sqrt{a-b+c}}","-\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 e \sqrt{a-b+c}}",1,"-1/2*ArcTanh[(2*a - b + (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])]/(Sqrt[a - b + c]*e)","A",1
39,1,136,142,0.7524063,"\int \frac{\cot (d+e x)}{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx","Integrate[Cot[d + e*x]/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],x]","-\frac{\frac{\tanh ^{-1}\left(\frac{2 a+b \tan ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{\sqrt{a}}+\frac{\tanh ^{-1}\left(\frac{-2 a-\left((b-2 c) \tan ^2(d+e x)\right)+b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{\sqrt{a-b+c}}}{2 e}","\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 e \sqrt{a-b+c}}-\frac{\tanh ^{-1}\left(\frac{2 a+b \tan ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 \sqrt{a} e}",1,"-1/2*(ArcTanh[(2*a + b*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])]/Sqrt[a] + ArcTanh[(-2*a + b - (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])]/Sqrt[a - b + c])/e","A",1
40,1,188,249,4.2467485,"\int \frac{\cot ^3(d+e x)}{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx","Integrate[Cot[d + e*x]^3/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],x]","\frac{(2 a+b) \tanh ^{-1}\left(\frac{2 a+b \tan ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)+2 \sqrt{a} \left(\cot ^2(d+e x) \left(-\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}\right)-\frac{a \tanh ^{-1}\left(\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{\sqrt{a-b+c}}\right)}{4 a^{3/2} e}","\frac{b \tanh ^{-1}\left(\frac{2 a+b \tan ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{4 a^{3/2} e}+\frac{\tanh ^{-1}\left(\frac{2 a+b \tan ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 \sqrt{a} e}-\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 e \sqrt{a-b+c}}-\frac{\cot ^2(d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{2 a e}",1,"((2*a + b)*ArcTanh[(2*a + b*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])] + 2*Sqrt[a]*(-((a*ArcTanh[(2*a - b + (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/Sqrt[a - b + c]) - Cot[d + e*x]^2*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]))/(4*a^(3/2)*e)","A",1
41,1,533,662,23.1581682,"\int \frac{\tan ^4(d+e x)}{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx","Integrate[Tan[d + e*x]^4/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],x]","\frac{\frac{\sin (2 (d+e x)) \sqrt{\sec ^4(d+e x) ((a-b+c) \cos (4 (d+e x))+4 (a-c) \cos (2 (d+e x))+3 a+b+3 c)}}{\sqrt{2}}+\frac{-4 \sin (d+e x) \cos (d+e x) \left(a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right)+\frac{i \sqrt{2} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c \tan ^2(d+e x)}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{2 c \tan ^2(d+e x)}{b-\sqrt{b^2-4 a c}}+1} \left(\left(-\sqrt{b^2-4 a c}+b+2 c\right) F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)+\left(\sqrt{b^2-4 a c}-b\right) E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)-2 c \Pi \left(\frac{b+\sqrt{b^2-4 a c}}{2 c};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)\right)}{\sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}}}}{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}}{4 c e}","\frac{\sqrt[4]{a} \left(\sqrt{a}-2 \sqrt{c}\right) \left(\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right) \sqrt{\frac{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left(\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{2 c^{3/4} e \left(\sqrt{a}-\sqrt{c}\right) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right) \sqrt{\frac{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left(\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{c^{3/4} e \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac{\tan (d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{\sqrt{c} e \left(\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{a-b+c} \tan (d+e x)}{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 e \sqrt{a-b+c}}+\frac{\left(\sqrt{a}+\sqrt{c}\right) \left(\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right) \sqrt{\frac{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left(\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right)^2}} \Pi \left(-\frac{\left(\sqrt{a}-\sqrt{c}\right)^2}{4 \sqrt{a} \sqrt{c}};2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{4 \sqrt[4]{a} \sqrt[4]{c} e \left(\sqrt{a}-\sqrt{c}\right) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}",1,"((Sqrt[(3*a + b + 3*c + 4*(a - c)*Cos[2*(d + e*x)] + (a - b + c)*Cos[4*(d + e*x)])*Sec[d + e*x]^4]*Sin[2*(d + e*x)])/Sqrt[2] + ((I*Sqrt[2]*((-b + Sqrt[b^2 - 4*a*c])*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + (b + 2*c - Sqrt[b^2 - 4*a*c])*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - 2*c*EllipticPi[(b + Sqrt[b^2 - 4*a*c])/(2*c), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[c/(b + Sqrt[b^2 - 4*a*c])] - 4*Cos[d + e*x]*Sin[d + e*x]*(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4))/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(4*c*e)","C",1
42,1,311,436,11.1881986,"\int \frac{\tan ^2(d+e x)}{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx","Integrate[Tan[d + e*x]^2/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],x]","-\frac{i \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c \tan ^2(d+e x)}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{2 c \tan ^2(d+e x)}{b-\sqrt{b^2-4 a c}}+1} \left(F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)-\Pi \left(\frac{b+\sqrt{b^2-4 a c}}{2 c};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)\right)}{\sqrt{2} e \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b+c} \tan (d+e x)}{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 e \sqrt{a-b+c}}+\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right) \sqrt{\frac{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left(\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{2 \sqrt[4]{c} e \left(\sqrt{a}-\sqrt{c}\right) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac{\left(\sqrt{a}+\sqrt{c}\right) \left(\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right) \sqrt{\frac{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left(\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right)^2}} \Pi \left(-\frac{\left(\sqrt{a}-\sqrt{c}\right)^2}{4 \sqrt{a} \sqrt{c}};2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{4 \sqrt[4]{a} \sqrt[4]{c} e \left(\sqrt{a}-\sqrt{c}\right) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}",1,"((-I)*(EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - EllipticPi[(b + Sqrt[b^2 - 4*a*c])/(2*c), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b - Sqrt[b^2 - 4*a*c])])/(Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])","C",1
43,1,235,436,0.7389134,"\int \frac{1}{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx","Integrate[1/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],x]","-\frac{i \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c \tan ^2(d+e x)}{\sqrt{b^2-4 a c}+b}} \sqrt{1-\frac{2 c \tan ^2(d+e x)}{\sqrt{b^2-4 a c}-b}} \Pi \left(\frac{b+\sqrt{b^2-4 a c}}{2 c};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)}{\sqrt{2} e \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a-b+c} \tan (d+e x)}{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 e \sqrt{a-b+c}}-\frac{\sqrt[4]{c} \left(\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right) \sqrt{\frac{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left(\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{2 \sqrt[4]{a} e \left(\sqrt{a}-\sqrt{c}\right) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac{\left(\sqrt{a}+\sqrt{c}\right) \left(\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right) \sqrt{\frac{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left(\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right)^2}} \Pi \left(-\frac{\left(\sqrt{a}-\sqrt{c}\right)^2}{4 \sqrt{a} \sqrt{c}};2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{4 \sqrt[4]{a} \sqrt[4]{c} e \left(\sqrt{a}-\sqrt{c}\right) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}",1,"((-I)*EllipticPi[(b + Sqrt[b^2 - 4*a*c])/(2*c), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 - (2*c*Tan[d + e*x]^2)/(-b + Sqrt[b^2 - 4*a*c])])/(Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])","C",1
44,1,683,707,28.1565177,"\int \frac{\cot ^2(d+e x)}{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx","Integrate[Cot[d + e*x]^2/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],x]","\frac{\left(\frac{\sin (2 (d+e x))}{2 a}-\frac{\cot (d+e x)}{a}\right) \sqrt{\frac{4 a \cos (2 (d+e x))+a \cos (4 (d+e x))+3 a-b \cos (4 (d+e x))+b-4 c \cos (2 (d+e x))+c \cos (4 (d+e x))+3 c}{4 \cos (2 (d+e x))+\cos (4 (d+e x))+3}}}{e}+\frac{\frac{2 i \sqrt{2} a \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c \tan ^2(d+e x)}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{2 c \tan ^2(d+e x)}{b-\sqrt{b^2-4 a c}}+1} \Pi \left(\frac{b+\sqrt{b^2-4 a c}}{2 c};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)}{\sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}}}+\frac{i \sqrt{2} \left(\sqrt{b^2-4 a c}-b\right) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c \tan ^2(d+e x)}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{2 c \tan ^2(d+e x)}{b-\sqrt{b^2-4 a c}}+1} \left(E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)-F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)\right)}{\sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}}}-\frac{4 \tan (d+e x) \left(a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right)}{\tan ^2(d+e x)+1}}{4 a e \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}","\frac{\sqrt[4]{c} \left(2 \sqrt{a}-\sqrt{c}\right) \left(\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right) \sqrt{\frac{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left(\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{2 a^{3/4} e \left(\sqrt{a}-\sqrt{c}\right) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac{\sqrt[4]{c} \left(\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right) \sqrt{\frac{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left(\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{a^{3/4} e \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac{\sqrt{c} \tan (d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{a e \left(\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b+c} \tan (d+e x)}{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 e \sqrt{a-b+c}}-\frac{\cot (d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{a e}-\frac{\left(\sqrt{a}+\sqrt{c}\right) \left(\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right) \sqrt{\frac{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left(\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right)^2}} \Pi \left(-\frac{\left(\sqrt{a}-\sqrt{c}\right)^2}{4 \sqrt{a} \sqrt{c}};2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{4 \sqrt[4]{a} \sqrt[4]{c} e \left(\sqrt{a}-\sqrt{c}\right) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}",1,"(Sqrt[(3*a + b + 3*c + 4*a*Cos[2*(d + e*x)] - 4*c*Cos[2*(d + e*x)] + a*Cos[4*(d + e*x)] - b*Cos[4*(d + e*x)] + c*Cos[4*(d + e*x)])/(3 + 4*Cos[2*(d + e*x)] + Cos[4*(d + e*x)])]*(-(Cot[d + e*x]/a) + Sin[2*(d + e*x)]/(2*a)))/e + ((I*Sqrt[2]*(-b + Sqrt[b^2 - 4*a*c])*(EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[c/(b + Sqrt[b^2 - 4*a*c])] + ((2*I)*Sqrt[2]*a*EllipticPi[(b + Sqrt[b^2 - 4*a*c])/(2*c), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[c/(b + Sqrt[b^2 - 4*a*c])] - (4*Tan[d + e*x]*(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4))/(1 + Tan[d + e*x]^2))/(4*a*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])","C",0
45,1,182725,235,34.8600199,"\int \frac{\tan ^7(d+e x)}{\left(a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right)^{3/2}} \, dx","Integrate[Tan[d + e*x]^7/(a + b*Tan[d + e*x]^2 + c*Tan[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) \tan ^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 \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac{\tanh ^{-1}\left(\frac{b+2 c \tan ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 c^{3/2} e}+\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 e (a-b+c)^{3/2}}",1,"Result too large to show","C",0
46,1,57597,159,36.2403168,"\int \frac{\tan ^5(d+e x)}{\left(a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right)^{3/2}} \, dx","Integrate[Tan[d + e*x]^5/(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)^(3/2),x]","\text{Result too large to show}","\frac{(b (a-b)+2 a c) \tan ^2(d+e x)+a (2 a-b)}{e (a-b+c) \left(b^2-4 a c\right) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 e (a-b+c)^{3/2}}",1,"Result too large to show","C",0
47,1,155,154,2.8954361,"\int \frac{\tan ^3(d+e x)}{\left(a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right)^{3/2}} \, dx","Integrate[Tan[d + e*x]^3/(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)^(3/2),x]","\frac{\frac{2 c (2 a-b) \tan ^2(d+e x)+2 a (b-2 c)}{(a-b+c) \left(4 a c-b^2\right) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{(a-b+c)^{3/2}}}{2 e}","\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 e (a-b+c)^{3/2}}-\frac{c (2 a-b) \tan ^2(d+e x)+a (b-2 c)}{e (a-b+c) \left(b^2-4 a c\right) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}",1,"(ArcTanh[(2*a - b + (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])]/(a - b + c)^(3/2) + (2*a*(b - 2*c) + 2*(2*a - b)*c*Tan[d + e*x]^2)/((a - b + c)*(-b^2 + 4*a*c)*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]))/(2*e)","A",1
48,1,156,155,2.9777571,"\int \frac{\tan (d+e x)}{\left(a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right)^{3/2}} \, dx","Integrate[Tan[d + e*x]/(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)^(3/2),x]","-\frac{\frac{2 \left(2 a c-b^2-c (b-2 c) \tan ^2(d+e x)+b c\right)}{(a-b+c) \left(b^2-4 a c\right) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{(a-b+c)^{3/2}}}{2 e}","\frac{-2 a c+b^2+c (b-2 c) \tan ^2(d+e x)-b c}{e (a-b+c) \left(b^2-4 a c\right) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 e (a-b+c)^{3/2}}",1,"-1/2*(ArcTanh[(2*a - b + (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])]/(a - b + c)^(3/2) + (2*(-b^2 + 2*a*c + b*c - (b - 2*c)*c*Tan[d + e*x]^2))/((a - b + c)*(b^2 - 4*a*c)*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]))/e","A",1
49,1,278,280,3.4493541,"\int \frac{\cot (d+e x)}{\left(a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right)^{3/2}} \, dx","Integrate[Cot[d + e*x]/(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)^(3/2),x]","\frac{\frac{\left(2 a c-\frac{b^2}{2}\right) \tanh ^{-1}\left(\frac{2 a+b \tan ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{a^{3/2}}+\frac{-2 a c+b^2+b c \tan ^2(d+e x)}{a \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac{-2 a c+b^2+c (b-2 c) \tan ^2(d+e x)-b c}{(a-b+c) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{-2 a-\left((b-2 c) \tan ^2(d+e x)\right)+b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 (a-b+c)^{3/2}}}{e \left(b^2-4 a c\right)}","-\frac{\tanh ^{-1}\left(\frac{2 a+b \tan ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 a^{3/2} e}+\frac{-2 a c+b^2+b c \tan ^2(d+e x)}{a e \left(b^2-4 a c\right) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac{-2 a c+b^2+c (b-2 c) \tan ^2(d+e x)-b c}{e (a-b+c) \left(b^2-4 a c\right) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 e (a-b+c)^{3/2}}",1,"(((-1/2*b^2 + 2*a*c)*ArcTanh[(2*a + b*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/a^(3/2) - ((b^2 - 4*a*c)*ArcTanh[(-2*a + b - (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(2*(a - b + c)^(3/2)) + (b^2 - 2*a*c + b*c*Tan[d + e*x]^2)/(a*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - (b^2 - 2*a*c - b*c + (b - 2*c)*c*Tan[d + e*x]^2)/((a - b + c)*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]))/((b^2 - 4*a*c)*e)","A",1
50,1,555,477,6.0692413,"\int \frac{\cot ^3(d+e x)}{\left(a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right)^{3/2}} \, dx","Integrate[Cot[d + e*x]^3/(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)^(3/2),x]","\frac{-\frac{2 \left(2 a c-\frac{b^2}{2}\right) \tanh ^{-1}\left(\frac{2 a+b \tan ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{a^{3/2} \left(b^2-4 a c\right)}-\frac{2 \left(\frac{\left(\frac{1}{2} b \left(8 a c-3 b^2\right)+2 a b c\right) \tanh ^{-1}\left(\frac{2 a+b \tan ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 a^{3/2}}+\frac{\left(3 b^2-8 a c\right) \cot ^2(d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{2 a}\right)}{a \left(b^2-4 a c\right)}+\frac{2 \left(2 a c-b^2-b c \tan ^2(d+e x)\right)}{a \left(b^2-4 a c\right) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac{2 \left(2 a c-b^2+c (2 c-b) \tan ^2(d+e x)+b c\right)}{(a-b+c) \left(b^2-4 a c\right) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac{8 \left(2 a c-\frac{b^2}{2}\right) \tanh ^{-1}\left(\frac{2 a-\left((2 c-b) \tan ^2(d+e x)\right)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{\sqrt{a-b+c} (4 a-4 b+4 c) \left(b^2-4 a c\right)}-\frac{2 \cot ^2(d+e x) \left(2 a c-b^2-b c \tan ^2(d+e x)\right)}{a \left(b^2-4 a c\right) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}}{2 e}","\frac{3 b \tanh ^{-1}\left(\frac{2 a+b \tan ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{4 a^{5/2} e}+\frac{\tanh ^{-1}\left(\frac{2 a+b \tan ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 a^{3/2} e}-\frac{\left(3 b^2-8 a c\right) \cot ^2(d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{2 a^2 e \left(b^2-4 a c\right)}-\frac{-2 a c+b^2+b c \tan ^2(d+e x)}{a e \left(b^2-4 a c\right) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac{-2 a c+b^2+c (b-2 c) \tan ^2(d+e x)-b c}{e (a-b+c) \left(b^2-4 a c\right) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac{\cot ^2(d+e x) \left(-2 a c+b^2+b c \tan ^2(d+e x)\right)}{a e \left(b^2-4 a c\right) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac{\tanh ^{-1}\left(\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right)}{2 e (a-b+c)^{3/2}}",1,"((-2*(-1/2*b^2 + 2*a*c)*ArcTanh[(2*a + b*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(a^(3/2)*(b^2 - 4*a*c)) + (8*(-1/2*b^2 + 2*a*c)*ArcTanh[(2*a - b - (-b + 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(Sqrt[a - b + c]*(4*a - 4*b + 4*c)*(b^2 - 4*a*c)) + (2*(-b^2 + 2*a*c - b*c*Tan[d + e*x]^2))/(a*(b^2 - 4*a*c)*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - (2*Cot[d + e*x]^2*(-b^2 + 2*a*c - b*c*Tan[d + e*x]^2))/(a*(b^2 - 4*a*c)*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - (2*(-b^2 + 2*a*c + b*c + c*(-b + 2*c)*Tan[d + e*x]^2))/((a - b + c)*(b^2 - 4*a*c)*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - (2*(((2*a*b*c + (b*(-3*b^2 + 8*a*c))/2)*ArcTanh[(2*a + b*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(2*a^(3/2)) + ((3*b^2 - 8*a*c)*Cot[d + e*x]^2*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(2*a)))/(a*(b^2 - 4*a*c)))/(2*e)","A",1
51,1,831,981,34.4529368,"\int \frac{\tan ^2(d+e x)}{\left(a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right)^{3/2}} \, dx","Integrate[Tan[d + e*x]^2/(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)^(3/2),x]","\frac{\sqrt{\frac{4 \cos (2 (d+e x)) a+\cos (4 (d+e x)) a+3 a+b+3 c-4 c \cos (2 (d+e x))-b \cos (4 (d+e x))+c \cos (4 (d+e x))}{4 \cos (2 (d+e x))+\cos (4 (d+e x))+3}} \left(\frac{(b-2 c) \sin (2 (d+e x))}{2 (-a+b-c) \left(b^2-4 a c\right)}+\frac{2 \sin (2 (d+e x)) b^2+\sin (4 (d+e x)) b^2-2 c \sin (4 (d+e x)) b-4 c^2 \sin (2 (d+e x))-4 a c \sin (2 (d+e x))+2 c^2 \sin (4 (d+e x))-2 a c \sin (4 (d+e x))}{(a-b+c) \left(4 a c-b^2\right) (-4 \cos (2 (d+e x)) a-\cos (4 (d+e x)) a-3 a-b-3 c+4 c \cos (2 (d+e x))+b \cos (4 (d+e x))-c \cos (4 (d+e x)))}\right)}{e}+\frac{\frac{i \sqrt{2} \left((b-2 c) \left(\sqrt{b^2-4 a c}-b\right) E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)+\left(b^2-\sqrt{b^2-4 a c} b+2 c \left(\sqrt{b^2-4 a c}-2 a\right)\right) F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)-2 \left(b^2-4 a c\right) \Pi \left(\frac{b+\sqrt{b^2-4 a c}}{2 c};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)\right) \sqrt{\frac{2 c \tan ^2(d+e x)+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c \tan ^2(d+e x)}{b-\sqrt{b^2-4 a c}}+1}}{\sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}}}-\frac{4 (b-2 c) \tan (d+e x) \left(c \tan ^4(d+e x)+b \tan ^2(d+e x)+a\right)}{\tan ^2(d+e x)+1}}{4 (a-b+c) \left(4 a c-b^2\right) e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b+c} \tan (d+e x)}{\sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}\right)}{2 (a-b+c)^{3/2} e}-\frac{(b-2 c) \sqrt{c} \tan (d+e x) \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}{(a-b+c) \left(b^2-4 a c\right) e \left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right)}+\frac{\sqrt[4]{a} (b-2 c) \sqrt[4]{c} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right) \sqrt{\frac{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right)^2}}}{(a-b+c) \left(b^2-4 a c\right) e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}+\frac{\sqrt[4]{c} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right) \sqrt{\frac{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right)^2}}}{2 \sqrt[4]{a} \left(\sqrt{a}-\sqrt{c}\right) (a-b+c) e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}-\frac{\left(\sqrt{a}-\sqrt{c}\right) \sqrt[4]{c} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right) \sqrt{\frac{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right)^2}}}{2 \sqrt[4]{a} \left(b-2 \sqrt{a} \sqrt{c}\right) (a-b+c) e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}-\frac{\left(\sqrt{a}+\sqrt{c}\right) \Pi \left(-\frac{\left(\sqrt{a}-\sqrt{c}\right)^2}{4 \sqrt{a} \sqrt{c}};2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \tan (d+e x)}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right) \sqrt{\frac{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left(\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right)^2}}}{4 \sqrt[4]{a} \left(\sqrt{a}-\sqrt{c}\right) \sqrt[4]{c} (a-b+c) e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}+\frac{\tan (d+e x) \left(b^2-c b+(b-2 c) c \tan ^2(d+e x)-2 a c\right)}{(a-b+c) \left(b^2-4 a c\right) e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}",1,"(Sqrt[(3*a + b + 3*c + 4*a*Cos[2*(d + e*x)] - 4*c*Cos[2*(d + e*x)] + a*Cos[4*(d + e*x)] - b*Cos[4*(d + e*x)] + c*Cos[4*(d + e*x)])/(3 + 4*Cos[2*(d + e*x)] + Cos[4*(d + e*x)])]*(((b - 2*c)*Sin[2*(d + e*x)])/(2*(-a + b - c)*(b^2 - 4*a*c)) + (2*b^2*Sin[2*(d + e*x)] - 4*a*c*Sin[2*(d + e*x)] - 4*c^2*Sin[2*(d + e*x)] + b^2*Sin[4*(d + e*x)] - 2*a*c*Sin[4*(d + e*x)] - 2*b*c*Sin[4*(d + e*x)] + 2*c^2*Sin[4*(d + e*x)])/((a - b + c)*(-b^2 + 4*a*c)*(-3*a - b - 3*c - 4*a*Cos[2*(d + e*x)] + 4*c*Cos[2*(d + e*x)] - a*Cos[4*(d + e*x)] + b*Cos[4*(d + e*x)] - c*Cos[4*(d + e*x)]))))/e + ((I*Sqrt[2]*((b - 2*c)*(-b + Sqrt[b^2 - 4*a*c])*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + (b^2 - b*Sqrt[b^2 - 4*a*c] + 2*c*(-2*a + Sqrt[b^2 - 4*a*c]))*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - 2*(b^2 - 4*a*c)*EllipticPi[(b + Sqrt[b^2 - 4*a*c])/(2*c), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[c/(b + Sqrt[b^2 - 4*a*c])] - (4*(b - 2*c)*Tan[d + e*x]*(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4))/(1 + Tan[d + e*x]^2))/(4*(a - b + c)*(-b^2 + 4*a*c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])","C",0