1,1,975,0,24.4627098,"\int \tan ^5(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx","Int[Tan[d + e*x]^5*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","\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}","\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,"(Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) + (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*Sqrt[c]*e) - (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])])/(16*c^(5/2)*e) + (b*(7*b^2 - 12*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])])/(256*c^(9/2)*e) - (Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTanh[(b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) + b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) + Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]/e + (b*(b + 2*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(8*c^2*e) - (b*(7*b^2 - 12*a*c)*(b + 2*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(128*c^4*e) - (a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2)/(3*c*e) + (Tan[d + e*x]^2*(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2))/(5*c*e) + ((35*b^2 - 32*a*c - 42*b*c*Tan[d + e*x])*(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2))/(240*c^3*e)","A",21,14,33,0.4242,1,"{3700, 6725, 640, 612, 621, 206, 742, 779, 1021, 1078, 1036, 1030, 208, 205}"
2,1,889,0,23.9946973,"\int \tan ^4(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx","Int[Tan[d + e*x]^4*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","\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}","\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,"-((Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b*Sqrt[a^2 + b^2 - 2*a*c + c^2] - (b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e)) + (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])])/e + ((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])])/(8*c^(3/2)*e) - ((b^2 - 4*a*c)*(5*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])])/(128*c^(7/2)*e) - (Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTanh[(b*Sqrt[a^2 + b^2 - 2*a*c + c^2] + (b^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) - ((b + 2*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(4*c*e) + ((5*b^2 - 4*a*c)*(b + 2*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(64*c^3*e) - (5*b*(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2))/(24*c^2*e) + (Tan[d + e*x]*(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2))/(4*c*e)","A",19,12,33,0.3636,1,"{3700, 6725, 612, 621, 206, 742, 640, 990, 1036, 1030, 208, 205}"
3,1,748,0,23.6300863,"\int \tan ^3(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx","Int[Tan[d + e*x]^3*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","-\frac{\sqrt{-a \left(\sqrt{a^2-2 a c+b^2+c^2}+2 c\right)+c \left(\sqrt{a^2-2 a c+b^2+c^2}+c\right)+a^2+b^2} \tan ^{-1}\left(\frac{-b \sqrt{a^2-2 a c+b^2+c^2} \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}","-\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,"-((Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e)) - (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*Sqrt[c]*e) + (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])])/(16*c^(5/2)*e) + (Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTanh[(b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) + b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) - Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]/e - (b*(b + 2*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(8*c^2*e) + (a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2)/(3*c*e)","A",16,12,33,0.3636,1,"{3700, 6725, 640, 612, 621, 206, 1021, 1078, 1036, 1030, 208, 205}"
4,1,676,0,26.5713319,"\int \tan ^2(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx","Int[Tan[d + e*x]^2*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","\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}","\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,"(Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b*Sqrt[a^2 + b^2 - 2*a*c + c^2] - (a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) - ((b^2 - 4*(a - 2*c)*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])])/(8*c^(3/2)*e) + (Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTanh[(b*Sqrt[a^2 + b^2 - 2*a*c + c^2] + (a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) + ((b + 2*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(4*c*e)","A",10,9,33,0.2727,1,"{3700, 1071, 1078, 621, 206, 1036, 1030, 208, 205}"
5,1,601,0,23.2876201,"\int \tan (d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx","Int[Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","\frac{\sqrt{-a \left(\sqrt{a^2-2 a c+b^2+c^2}+2 c\right)+c \left(\sqrt{a^2-2 a c+b^2+c^2}+c\right)+a^2+b^2} \tan ^{-1}\left(\frac{-b \sqrt{a^2-2 a c+b^2+c^2} \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}","\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^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) + (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*Sqrt[c]*e) - (Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTanh[(b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) + b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) + Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]/e","A",10,9,31,0.2903,1,"{3700, 1021, 1078, 621, 206, 1036, 1030, 208, 205}"
6,1,574,0,23.2685593,"\int \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx","Int[Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","-\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}","-\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,"-((Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b*Sqrt[a^2 + b^2 - 2*a*c + c^2] - (b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e)) + (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])])/e - (Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTanh[(b*Sqrt[a^2 + b^2 - 2*a*c + c^2] + (b^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e)","A",9,7,24,0.2917,1,"{990, 621, 206, 1036, 1030, 208, 205}"
7,1,571,0,23.582838,"\int \cot (d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx","Int[Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","-\frac{\sqrt{-a \left(\sqrt{a^2-2 a c+b^2+c^2}+2 c\right)+c \left(\sqrt{a^2-2 a c+b^2+c^2}+c\right)+a^2+b^2} \tan ^{-1}\left(\frac{-b \sqrt{a^2-2 a c+b^2+c^2} \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{\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,"-((Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e)) - (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])])/e + (Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTanh[(b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) + b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e)","A",18,13,31,0.4194,1,"{3700, 6725, 734, 843, 621, 206, 724, 1021, 1078, 1036, 1030, 208, 205}"
8,1,612,0,23.7443176,"\int \cot ^2(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx","Int[Cot[d + e*x]^2*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","\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}","\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,"(Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b*Sqrt[a^2 + b^2 - 2*a*c + c^2] - (b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) - (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])])/(2*Sqrt[a]*e) + (Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTanh[(b*Sqrt[a^2 + b^2 - 2*a*c + c^2] + (b^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) - (Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/e","A",17,12,33,0.3636,1,"{3700, 6725, 732, 843, 621, 206, 724, 990, 1036, 1030, 205, 208}"
9,1,690,0,23.7333439,"\int \cot ^3(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx","Int[Cot[d + e*x]^3*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","\frac{\sqrt{-a \left(\sqrt{a^2-2 a c+b^2+c^2}+2 c\right)+c \left(\sqrt{a^2-2 a c+b^2+c^2}+c\right)+a^2+b^2} \tan ^{-1}\left(\frac{-b \sqrt{a^2-2 a c+b^2+c^2} \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{\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} \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}","\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{\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} \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,"(Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) + (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])])/e + ((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])])/(8*a^(3/2)*e) - (Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTanh[(b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) + b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) - (Cot[d + e*x]^2*(2*a + b*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(4*a*e)","A",21,14,33,0.4242,1,"{3700, 6725, 720, 724, 206, 734, 843, 621, 1021, 1078, 1036, 1030, 208, 205}"
10,1,548,0,1.0854246,"\int \frac{\tan ^5(d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx","Int[Tan[d + e*x]^5/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","\frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left(\frac{-\sqrt{a^2-2 a c+b^2+c^2}+a+b \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{\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 \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{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}","\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{\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 \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{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,"(Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) - (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) + (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)*e) - (b*(5*b^2 - 12*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])])/(16*c^(7/2)*e) - Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]/(c*e) + (Tan[d + e*x]^2*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(3*c*e) + ((15*b^2 - 16*a*c - 10*b*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(24*c^3*e)","A",15,10,33,0.3030,1,"{3700, 6725, 640, 621, 206, 742, 779, 1036, 1030, 208}"
11,1,495,0,0.8200109,"\int \frac{\tan ^4(d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx","Int[Tan[d + e*x]^4/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","\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}","\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,"(Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b - (a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) - (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b - (a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) - 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) + ((3*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])])/(8*c^(5/2)*e) - (3*b*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(4*c^2*e) + (Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(2*c*e)","A",14,9,33,0.2727,1,"{3700, 6725, 621, 206, 742, 640, 987, 1030, 205}"
12,1,383,0,0.7307361,"\int \frac{\tan ^3(d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx","Int[Tan[d + e*x]^3/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","-\frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left(\frac{-\sqrt{a^2-2 a c+b^2+c^2}+a+b \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}","-\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,"-((Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e)) + (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) - (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)*e) + Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]/(c*e)","A",11,8,33,0.2424,1,"{3700, 6725, 640, 621, 206, 1036, 1030, 208}"
13,1,352,0,0.3729331,"\int \frac{\tan ^2(d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx","Int[Tan[d + e*x]^2/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","-\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}","-\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,"-((Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b - (a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e)) + (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b - (a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) + 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)","A",9,7,33,0.2121,1,"{3700, 1079, 621, 206, 987, 1030, 205}"
14,1,294,0,0.2858365,"\int \frac{\tan (d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx","Int[Tan[d + e*x]/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","\frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left(\frac{-\sqrt{a^2-2 a c+b^2+c^2}+a+b \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{\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,"(Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) - (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e)","A",6,4,31,0.1290,1,"{3700, 1036, 1030, 208}"
15,1,298,0,0.3111567,"\int \frac{1}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx","Int[1/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","\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{\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,"(Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b - (a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) - (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b - (a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e)","A",6,3,24,0.1250,1,"{987, 1030, 205}"
16,1,350,0,0.7010566,"\int \frac{\cot (d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx","Int[Cot[d + e*x]/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","-\frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left(\frac{-\sqrt{a^2-2 a c+b^2+c^2}+a+b \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}","-\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,"-(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]*e)) - (Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) + (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e)","A",10,7,31,0.2258,1,"{3700, 6725, 724, 206, 1036, 1030, 208}"
17,1,395,0,0.7645393,"\int \frac{\cot ^2(d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx","Int[Cot[d + e*x]^2/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","-\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{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{\cot (d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{a 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{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{\cot (d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{a e}",1,"-((Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b - (a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e)) + (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b - (a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) + (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])])/(2*a^(3/2)*e) - (Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(a*e)","A",11,8,33,0.2424,1,"{3700, 6725, 730, 724, 206, 987, 1030, 205}"
18,1,500,0,0.7999174,"\int \frac{\cot ^3(d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx","Int[Cot[d + e*x]^3/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]","\frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left(\frac{-\sqrt{a^2-2 a c+b^2+c^2}+a+b \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{\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{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}","\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{\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{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,"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]*e) - ((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])])/(8*a^(5/2)*e) + (Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) - (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) + (3*b*Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(4*a^2*e) - (Cot[d + e*x]^2*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(2*a*e)","A",14,9,33,0.2727,1,"{3700, 6725, 744, 806, 724, 206, 1036, 1030, 208}"
19,1,1190,0,6.507454,"\int \frac{\tan ^7(d+e x)}{\left(a+b \tan (d+e x)+c \tan ^2(d+e x)\right)^{3/2}} \, dx","Int[Tan[d + e*x]^7/(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2),x]","\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}}","\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,"(3*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^(5/2)*e) - (5*b*(7*b^2 - 12*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])])/(16*c^(9/2)*e) - (Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) + (Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) + (2*(2*a + b*Tan[d + e*x]))/((b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) - (2*Tan[d + e*x]^2*(2*a + b*Tan[d + e*x]))/((b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) + (2*Tan[d + e*x]^4*(2*a + b*Tan[d + e*x]))/((b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) - (2*(a*(b^2 - 2*(a - c)*c) + b*c*(a + c)*Tan[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) + ((7*b^2 - 16*a*c)*Tan[d + e*x]^2*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(3*c^2*(b^2 - 4*a*c)*e) - (2*b*Tan[d + e*x]^3*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(c*(b^2 - 4*a*c)*e) - ((3*b^2 - 8*a*c - 2*b*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(c^2*(b^2 - 4*a*c)*e) + ((105*b^4 - 460*a*b^2*c + 256*a^2*c^2 - 2*b*c*(35*b^2 - 116*a*c)*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(24*c^4*(b^2 - 4*a*c)*e)","A",20,12,33,0.3636,1,"{3700, 6725, 636, 738, 779, 621, 206, 832, 1018, 1036, 1030, 208}"
20,1,864,0,4.7903415,"\int \frac{\tan ^5(d+e x)}{\left(a+b \tan (d+e x)+c \tan ^2(d+e x)\right)^{3/2}} \, dx","Int[Tan[d + e*x]^5/(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2),x]","\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}}","\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,"(-3*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^(5/2)*e) + (Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) - (Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) - (2*(2*a + b*Tan[d + e*x]))/((b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) + (2*Tan[d + e*x]^2*(2*a + b*Tan[d + e*x]))/((b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) + (2*(a*(b^2 - 2*(a - c)*c) + b*c*(a + c)*Tan[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) + ((3*b^2 - 8*a*c - 2*b*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(c^2*(b^2 - 4*a*c)*e)","A",14,11,33,0.3333,1,"{3700, 6725, 636, 738, 779, 621, 206, 1018, 1036, 1030, 208}"
21,1,686,0,4.5136327,"\int \frac{\tan ^3(d+e x)}{\left(a+b \tan (d+e x)+c \tan ^2(d+e x)\right)^{3/2}} \, dx","Int[Tan[d + e*x]^3/(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2),x]","-\frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \tanh ^{-1}\left(\frac{-b \left(-\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c\right) \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)}}","-\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,"-((Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e)) + (Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) + (2*(2*a + b*Tan[d + e*x]))/((b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) - (2*(a*(b^2 - 2*(a - c)*c) + b*c*(a + c)*Tan[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])","A",10,7,33,0.2121,1,"{3700, 6725, 636, 1018, 1036, 1030, 208}"
22,1,638,0,3.9502634,"\int \frac{\tan ^2(d+e x)}{\left(a+b \tan (d+e x)+c \tan ^2(d+e x)\right)^{3/2}} \, dx","Int[Tan[d + e*x]^2/(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2),x]","-\frac{\sqrt{\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{-(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \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)}}","-\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,"-((Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) + (b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e)) + (Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) + (b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) - (2*(a*b*(a + c) + c*(2*a^2 + b^2 - 2*a*c)*Tan[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])","A",7,5,33,0.1515,1,"{3700, 1065, 1036, 1030, 205}"
23,1,635,0,3.7914709,"\int \frac{\tan (d+e x)}{\left(a+b \tan (d+e x)+c \tan ^2(d+e x)\right)^{3/2}} \, dx","Int[Tan[d + e*x]/(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2),x]","\frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \tanh ^{-1}\left(\frac{-b \left(-\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c\right) \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)}}","\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,"(Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) - (Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) + (2*(a*(b^2 - 2*(a - c)*c) + b*c*(a + c)*Tan[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])","A",7,5,31,0.1613,1,"{3700, 1018, 1036, 1030, 208}"
24,1,750,0,4.5775047,"\int \frac{\cot (d+e x)}{\left(a+b \tan (d+e x)+c \tan ^2(d+e x)\right)^{3/2}} \, dx","Int[Cot[d + e*x]/(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2),x]","-\frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \tanh ^{-1}\left(\frac{-b \left(-\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c\right) \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{\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{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)}}","-\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{\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{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,"-(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)*e)) - (Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) + (Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) + (2*(b^2 - 2*a*c + b*c*Tan[d + e*x]))/(a*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) - (2*(a*(b^2 - 2*(a - c)*c) + b*c*(a + c)*Tan[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])","A",13,10,31,0.3226,1,"{3700, 6725, 740, 12, 724, 206, 1018, 1036, 1030, 208}"
25,1,829,0,4.9387572,"\int \frac{\cot ^2(d+e x)}{\left(a+b \tan (d+e x)+c \tan ^2(d+e x)\right)^{3/2}} \, dx","Int[Cot[d + e*x]^2/(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2),x]","-\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}}","-\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,"-((Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) + (b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e)) + (Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) + (b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) + (3*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])])/(2*a^(5/2)*e) + (2*Cot[d + e*x]*(b^2 - 2*a*c + b*c*Tan[d + e*x]))/(a*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) + (2*(b*(b^2 - (3*a - c)*c) + c*(b^2 - 2*(a - c)*c)*Tan[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) - ((3*b^2 - 8*a*c)*Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(a^2*(b^2 - 4*a*c)*e)","A",13,10,33,0.3030,1,"{3700, 6725, 740, 806, 724, 206, 975, 1036, 1030, 205}"
26,1,1007,0,4.927684,"\int \frac{\cot ^3(d+e x)}{\left(a+b \tan (d+e x)+c \tan ^2(d+e x)\right)^{3/2}} \, dx","Int[Cot[d + e*x]^3/(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2),x]","-\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}}","-\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,"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)*e) - (3*(5*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])])/(8*a^(7/2)*e) + (Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) - (Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) - (2*(b^2 - 2*a*c + b*c*Tan[d + e*x]))/(a*(b^2 - 4*a*c)*e*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)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) + (2*(a*(b^2 - 2*(a - c)*c) + b*c*(a + c)*Tan[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^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^3*(b^2 - 4*a*c)*e) - ((5*b^2 - 12*a*c)*Cot[d + e*x]^2*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(2*a^2*(b^2 - 4*a*c)*e)","A",18,12,33,0.3636,1,"{3700, 6725, 740, 834, 806, 724, 206, 12, 1018, 1036, 1030, 208}"
27,1,270,0,0.6125871,"\int \tan ^5(d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx","Int[Tan[d + e*x]^5*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],x]","\frac{\left(-4 b c (a-2 c)-8 c^2 (a+2 c)+2 b^2 c+b^3\right) \tanh ^{-1}\left(\frac{b+2 c \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}","\frac{\left(-4 b c (a-2 c)-8 c^2 (a+2 c)+2 b^2 c+b^3\right) \tanh ^{-1}\left(\frac{b+2 c \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,"-(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])])/(2*e) + ((b^3 + 2*b^2*c - 4*b*(a - 2*c)*c - 8*c^2*(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])])/(32*c^(5/2)*e) - (((b - 2*c)*(b + 4*c) + 2*c*(b + 2*c)*Tan[d + e*x]^2)*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(16*c^2*e) + (a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)^(3/2)/(6*c*e)","A",9,8,35,0.2286,1,"{3700, 1251, 1653, 814, 843, 621, 206, 724}"
28,1,209,0,0.3492797,"\int \tan ^3(d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx","Int[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)}{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}","-\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,"(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])])/(2*e) - ((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])])/(16*c^(3/2)*e) + ((b - 4*c + 2*c*Tan[d + e*x]^2)*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(8*c*e)","A",8,7,35,0.2000,1,"{3700, 1251, 814, 843, 621, 206, 724}"
29,1,179,0,0.2242276,"\int \tan (d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx","Int[Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],x]","\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}","\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,"-(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])])/(2*e) + ((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])])/(4*Sqrt[c]*e) + Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]/(2*e)","A",8,7,33,0.2121,1,"{3700, 1247, 734, 843, 621, 206, 724}"
30,1,203,0,0.2924685,"\int \cot (d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx","Int[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)}{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{\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,"-(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])])/(2*e) + (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])])/(2*e) + (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])])/(2*e)","A",10,7,33,0.2121,1,"{3700, 1251, 895, 724, 206, 843, 621}"
31,1,435,0,0.5441831,"\int \cot ^3(d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx","Int[Cot[d + e*x]^3*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],x]","-\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}","-\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,"(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])])/(2*e) - (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])])/(4*Sqrt[a]*e) - (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])])/(2*e) - (b*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])])/(4*Sqrt[c]*e) + ((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])])/(4*Sqrt[c]*e) + (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])])/(2*e) - (Cot[d + e*x]^2*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(2*e)","A",22,9,35,0.2571,1,"{3700, 1251, 960, 732, 843, 621, 206, 724, 734}"
32,1,1254,0,0.9932531,"\int \tan ^2(d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx","Int[Tan[d + e*x]^2*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],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}}","-\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[a - b + c]*ArcTan[(Sqrt[a - b + c]*Tan[d + e*x])/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]])/(2*e) + (Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(3*e) + (b*Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(3*Sqrt[c]*e*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)) - (Sqrt[c]*Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(e*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)) - (a^(1/4)*b*EllipticE[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(3*c^(3/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + (a^(1/4)*c^(1/4)*EllipticE[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + (a^(1/4)*(b + 2*Sqrt[a]*Sqrt[c])*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(6*c^(3/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - ((b + Sqrt[a]*Sqrt[c] - c)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*a^(1/4)*c^(1/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + (c^(1/4)*(a - b + c)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*a^(1/4)*(Sqrt[a] - Sqrt[c])*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - ((Sqrt[a] + Sqrt[c])*(a - b + c)*EllipticPi[-(Sqrt[a] - Sqrt[c])^2/(4*Sqrt[a]*Sqrt[c]), 2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(4*a^(1/4)*(Sqrt[a] - Sqrt[c])*c^(1/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])","A",14,9,35,0.2571,1,"{3700, 1335, 1091, 1197, 1103, 1195, 1208, 1216, 1706}"
33,1,829,0,0.5516543,"\int \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx","Int[Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],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}}","\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,"(Sqrt[a - b + c]*ArcTan[(Sqrt[a - b + c]*Tan[d + e*x])/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]])/(2*e) + (Sqrt[c]*Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(e*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)) - (a^(1/4)*c^(1/4)*EllipticE[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + ((b + Sqrt[a]*Sqrt[c] - c)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*a^(1/4)*c^(1/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - (c^(1/4)*(a - b + c)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*a^(1/4)*(Sqrt[a] - Sqrt[c])*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + ((Sqrt[a] + Sqrt[c])*(a - b + c)*EllipticPi[-(Sqrt[a] - Sqrt[c])^2/(4*Sqrt[a]*Sqrt[c]), 2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(4*a^(1/4)*(Sqrt[a] - Sqrt[c])*c^(1/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])","A",8,6,26,0.2308,1,"{1208, 1197, 1103, 1195, 1216, 1706}"
34,1,861,0,0.5850234,"\int \cot ^2(d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx","Int[Cot[d + e*x]^2*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],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{\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}}","-\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[a - b + c]*ArcTan[(Sqrt[a - b + c]*Tan[d + e*x])/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]])/(2*e) - (Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/e + (Sqrt[c]*Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(e*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)) - (a^(1/4)*c^(1/4)*EllipticE[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + ((Sqrt[a] + Sqrt[c])*c^(1/4)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*a^(1/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + (c^(1/4)*(a - b + c)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*a^(1/4)*(Sqrt[a] - Sqrt[c])*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - ((Sqrt[a] + Sqrt[c])*(a - b + c)*EllipticPi[-(Sqrt[a] - Sqrt[c])^2/(4*Sqrt[a]*Sqrt[c]), 2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(4*a^(1/4)*(Sqrt[a] - Sqrt[c])*c^(1/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])","A",9,8,35,0.2286,1,"{3700, 1311, 1281, 1197, 1103, 1195, 1216, 1706}"
35,1,943,0,0.7170997,"\int \cot ^4(d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx","Int[Cot[d + e*x]^4*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],x]","-\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}}","-\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[a - b + c]*ArcTan[(Sqrt[a - b + c]*Tan[d + e*x])/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]])/(2*e) + ((3*a - b)*Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(3*a*e) - (Cot[d + e*x]^3*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(3*e) - ((3*a - b)*Sqrt[c]*Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(3*a*e*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)) + ((3*a - b)*c^(1/4)*EllipticE[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(3*a^(3/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - ((3*a - b + Sqrt[a]*Sqrt[c])*c^(1/4)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(6*a^(3/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - (c^(1/4)*(a - b + c)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*a^(1/4)*(Sqrt[a] - Sqrt[c])*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + ((Sqrt[a] + Sqrt[c])*(a - b + c)*EllipticPi[-(Sqrt[a] - Sqrt[c])^2/(4*Sqrt[a]*Sqrt[c]), 2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(4*a^(1/4)*(Sqrt[a] - Sqrt[c])*c^(1/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])","A",10,8,35,0.2286,1,"{3700, 1309, 1281, 1197, 1103, 1195, 1216, 1706}"
36,1,182,0,0.353049,"\int \frac{\tan ^5(d+e x)}{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx","Int[Tan[d + e*x]^5/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],x]","-\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}}","-\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,"-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*Sqrt[a - b + c]*e) - ((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])])/(4*c^(3/2)*e) + Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]/(2*c*e)","A",8,7,35,0.2000,1,"{3700, 1251, 1653, 843, 621, 206, 724}"
37,1,141,0,0.21029,"\int \frac{\tan ^3(d+e x)}{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx","Int[Tan[d + e*x]^3/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{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}","\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])]/(2*Sqrt[a - b + c]*e) + 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]*e)","A",7,6,35,0.1714,1,"{3700, 1251, 843, 621, 206, 724}"
38,1,79,0,0.1138367,"\int \frac{\tan (d+e x)}{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx","Int[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,"-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*Sqrt[a - b + c]*e)","A",4,4,33,0.1212,1,"{3700, 1247, 724, 206}"
39,1,142,0,0.2275648,"\int \frac{\cot (d+e x)}{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx","Int[Cot[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 \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{\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,"-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]*e) + 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*Sqrt[a - b + c]*e)","A",8,5,33,0.1515,1,"{3700, 1251, 960, 724, 206}"
40,1,249,0,0.3177192,"\int \frac{\cot ^3(d+e x)}{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx","Int[Cot[d + e*x]^3/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],x]","\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}","\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,"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]*e) + (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])])/(4*a^(3/2)*e) - 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*Sqrt[a - b + c]*e) - (Cot[d + e*x]^2*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(2*a*e)","A",11,6,35,0.1714,1,"{3700, 1251, 960, 730, 724, 206}"
41,1,662,0,0.4389262,"\int \frac{\tan ^4(d+e x)}{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx","Int[Tan[d + e*x]^4/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],x]","\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 ^{-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{\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{\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)}}","\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 ^{-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{\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{\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,"ArcTan[(Sqrt[a - b + c]*Tan[d + e*x])/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]]/(2*Sqrt[a - b + c]*e) + (Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(Sqrt[c]*e*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)) - (a^(1/4)*EllipticE[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(c^(3/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + (a^(1/4)*(Sqrt[a] - 2*Sqrt[c])*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*(Sqrt[a] - Sqrt[c])*c^(3/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + ((Sqrt[a] + Sqrt[c])*EllipticPi[-(Sqrt[a] - Sqrt[c])^2/(4*Sqrt[a]*Sqrt[c]), 2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(4*a^(1/4)*(Sqrt[a] - Sqrt[c])*c^(1/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])","A",5,5,35,0.1429,1,"{3700, 1325, 1103, 1195, 1706}"
42,1,436,0,0.3406996,"\int \frac{\tan ^2(d+e x)}{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx","Int[Tan[d + e*x]^2/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],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)}}","-\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,"-ArcTan[(Sqrt[a - b + c]*Tan[d + e*x])/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]]/(2*Sqrt[a - b + c]*e) + (a^(1/4)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*(Sqrt[a] - Sqrt[c])*c^(1/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - ((Sqrt[a] + Sqrt[c])*EllipticPi[-(Sqrt[a] - Sqrt[c])^2/(4*Sqrt[a]*Sqrt[c]), 2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(4*a^(1/4)*(Sqrt[a] - Sqrt[c])*c^(1/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])","A",4,4,35,0.1143,1,"{3700, 1319, 1103, 1706}"
43,1,436,0,0.3063344,"\int \frac{1}{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx","Int[1/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],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)}}","\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,"ArcTan[(Sqrt[a - b + c]*Tan[d + e*x])/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]]/(2*Sqrt[a - b + c]*e) - (c^(1/4)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*a^(1/4)*(Sqrt[a] - Sqrt[c])*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + ((Sqrt[a] + Sqrt[c])*EllipticPi[-(Sqrt[a] - Sqrt[c])^2/(4*Sqrt[a]*Sqrt[c]), 2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(4*a^(1/4)*(Sqrt[a] - Sqrt[c])*c^(1/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])","A",4,3,26,0.1154,1,"{1216, 1103, 1706}"
44,1,707,0,0.7275193,"\int \frac{\cot ^2(d+e x)}{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx","Int[Cot[d + e*x]^2/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],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{\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{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{\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)}}","\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{\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{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{\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,"-ArcTan[(Sqrt[a - b + c]*Tan[d + e*x])/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]]/(2*Sqrt[a - b + c]*e) - (Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(a*e) + (Sqrt[c]*Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(a*e*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)) - (c^(1/4)*EllipticE[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(a^(3/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + ((2*Sqrt[a] - Sqrt[c])*c^(1/4)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*a^(3/4)*(Sqrt[a] - Sqrt[c])*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - ((Sqrt[a] + Sqrt[c])*EllipticPi[-(Sqrt[a] - Sqrt[c])^2/(4*Sqrt[a]*Sqrt[c]), 2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(4*a^(1/4)*(Sqrt[a] - Sqrt[c])*c^(1/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])","A",7,7,35,0.2000,1,"{3700, 1329, 1714, 1195, 1708, 1103, 1706}"
45,1,235,0,0.5492134,"\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","Int[Tan[d + e*x]^7/(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)^(3/2),x]","\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}}","\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,"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)*e) + 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*c^(3/2)*e) + (a*(b^2 - a*(b + 2*c)) + (b^3 + 2*a^2*c - a*b*(b + 3*c))*Tan[d + e*x]^2)/(c*(a - b + c)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])","A",8,7,35,0.2000,1,"{3700, 1251, 1646, 843, 621, 206, 724}"
46,1,159,0,0.3936957,"\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","Int[Tan[d + e*x]^5/(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)^(3/2),x]","\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}}","\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,"-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)*e) + (a*(2*a - b) + ((a - b)*b + 2*a*c)*Tan[d + e*x]^2)/((a - b + c)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])","A",6,6,35,0.1714,1,"{3700, 1251, 1646, 12, 724, 206}"
47,1,154,0,0.2847761,"\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","Int[Tan[d + e*x]^3/(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)^(3/2),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}}-\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)}}","\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])]/(2*(a - b + c)^(3/2)*e) - (a*(b - 2*c) + (2*a - b)*c*Tan[d + e*x]^2)/((a - b + c)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])","A",6,6,35,0.1714,1,"{3700, 1251, 822, 12, 724, 206}"
48,1,155,0,0.2181652,"\int \frac{\tan (d+e x)}{\left(a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right)^{3/2}} \, dx","Int[Tan[d + e*x]/(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)^(3/2),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}}","\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,"-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)*e) + (b^2 - 2*a*c - b*c + (b - 2*c)*c*Tan[d + e*x]^2)/((a - b + c)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])","A",6,6,33,0.1818,1,"{3700, 1247, 740, 12, 724, 206}"
49,1,280,0,0.3866459,"\int \frac{\cot (d+e x)}{\left(a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right)^{3/2}} \, dx","Int[Cot[d + e*x]/(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)^(3/2),x]","-\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}}","-\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,"-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)*e) + 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)*e) + (b^2 - 2*a*c + b*c*Tan[d + e*x]^2)/(a*(b^2 - 4*a*c)*e*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)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])","A",12,7,33,0.2121,1,"{3700, 1251, 960, 740, 12, 724, 206}"
50,1,477,0,0.5518491,"\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","Int[Cot[d + e*x]^3/(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)^(3/2),x]","-\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{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{-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}}","-\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{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{-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,"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)*e) + (3*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])])/(4*a^(5/2)*e) - 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)*e) - (b^2 - 2*a*c + b*c*Tan[d + e*x]^2)/(a*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + (Cot[d + e*x]^2*(b^2 - 2*a*c + b*c*Tan[d + e*x]^2))/(a*(b^2 - 4*a*c)*e*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)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - ((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^2*(b^2 - 4*a*c)*e)","A",16,8,35,0.2286,1,"{3700, 1251, 960, 740, 806, 724, 206, 12}"
51,1,981,0,0.8755096,"\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","Int[Tan[d + e*x]^2/(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)^(3/2),x]","-\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}}","-\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,"-ArcTan[(Sqrt[a - b + c]*Tan[d + e*x])/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]]/(2*(a - b + c)^(3/2)*e) + (Tan[d + e*x]*(b^2 - 2*a*c - b*c + (b - 2*c)*c*Tan[d + e*x]^2))/((a - b + c)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - ((b - 2*c)*Sqrt[c]*Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/((a - b + c)*(b^2 - 4*a*c)*e*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)) + (a^(1/4)*(b - 2*c)*c^(1/4)*EllipticE[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/((a - b + c)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + (c^(1/4)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*a^(1/4)*(Sqrt[a] - Sqrt[c])*(a - b + c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - ((Sqrt[a] - Sqrt[c])*c^(1/4)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*a^(1/4)*(b - 2*Sqrt[a]*Sqrt[c])*(a - b + c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - ((Sqrt[a] + Sqrt[c])*EllipticPi[-(Sqrt[a] - Sqrt[c])^2/(4*Sqrt[a]*Sqrt[c]), 2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(4*a^(1/4)*(Sqrt[a] - Sqrt[c])*c^(1/4)*(a - b + c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])","A",9,8,35,0.2286,1,"{3700, 1315, 1178, 1197, 1103, 1195, 1216, 1706}"