3.13.0 ∫ (a + b tan(e + fx))2(c + d tan(e + fx))5∕2 dx [1242]

Integrand size = 27, antiderivative size = 231 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \, dx=-\frac {i (a-i b)^2 (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {i (a+i b)^2 (c+i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {4 (b c+a d) (a c-b d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (2 a b c+a^2 d-b^2 d\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {4 a b (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {2 b^2 (c+d \tan (e+f x))^{7/2}}{7 d f} \]

-I*(a-I*b)^2*(c-I*d)^(5/2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/f+I*(a+I*b)^2*(c+I*d)^(5/2)*arctanh((

c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/f+4*(a*d+b*c)*(a*c-b*d)*(c+d*tan(f*x+e))^(1/2)/f+2/3*(a^2*d+2*a*b*c-b^2*d

)*(c+d*tan(f*x+e))^(3/2)/f+4/5*a*b*(c+d*tan(f*x+e))^(5/2)/f+2/7*b^2*(c+d*tan(f*x+e))^(7/2)/d/f

Rubi [A] (veriﬁed)

Time = 0.69 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3624, 3609, 3620, 3618, 65, 214} \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \, dx=\frac {2 \left (a^2 d+2 a b c-b^2 d\right ) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {i (a-i b)^2 (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {i (a+i b)^2 (c+i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {4 a b (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {4 (a d+b c) (a c-b d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 b^2 (c+d \tan (e+f x))^{7/2}}{7 d f} \]

Int[(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5/2),x]

((-I)*(a - I*b)^2*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f + (I*(a + I*b)^2*(c + I*d

)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (4*(b*c + a*d)*(a*c - b*d)*Sqrt[c + d*Tan[e + f*x

]])/f + (2*(2*a*b*c + a^2*d - b^2*d)*(c + d*Tan[e + f*x])^(3/2))/(3*f) + (4*a*b*(c + d*Tan[e + f*x])^(5/2))/(5

*f) + (2*b^2*(c + d*Tan[e + f*x])^(7/2))/(7*d*f)

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub

st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &

& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*

((a + b*Tan[e + f*x])^m/(f*m)), x] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x

], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c*(

d/f), Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &&

NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c

+ I*d)/2, Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x], x] + Dist[(c - I*d)/2, Int[(a + b*Tan[e + f*x]

)^m*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0]

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[

d^2*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Simp[c^2 - d^2 + 2*c*d*Tan[e

+ f*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && !LeQ[m, -1] && !(EqQ[m, 2] && EqQ

Rubi steps \begin{align*} \text {integral}& = \frac {2 b^2 (c+d \tan (e+f x))^{7/2}}{7 d f}+\int \left (a^2-b^2+2 a b \tan (e+f x)\right ) (c+d \tan (e+f x))^{5/2} \, dx \\ & = \frac {4 a b (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {2 b^2 (c+d \tan (e+f x))^{7/2}}{7 d f}+\int (c+d \tan (e+f x))^{3/2} \left (a^2 c-b^2 c-2 a b d+\left (2 a b c+a^2 d-b^2 d\right ) \tan (e+f x)\right ) \, dx \\ & = \frac {2 \left (2 a b c+a^2 d-b^2 d\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {4 a b (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {2 b^2 (c+d \tan (e+f x))^{7/2}}{7 d f}+\int \sqrt {c+d \tan (e+f x)} ((a c-b c-a d-b d) (a c+b c+a d-b d)+2 (b c+a d) (a c-b d) \tan (e+f x)) \, dx \\ & = \frac {4 (b c+a d) (a c-b d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (2 a b c+a^2 d-b^2 d\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {4 a b (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {2 b^2 (c+d \tan (e+f x))^{7/2}}{7 d f}+\int \frac {-b^2 c \left (c^2-3 d^2\right )-2 a b d \left (3 c^2-d^2\right )+a^2 \left (c^3-3 c d^2\right )+\left (2 a b c \left (c^2-3 d^2\right )-b^2 d \left (3 c^2-d^2\right )+a^2 \left (3 c^2 d-d^3\right )\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx \\ & = \frac {4 (b c+a d) (a c-b d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (2 a b c+a^2 d-b^2 d\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {4 a b (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {2 b^2 (c+d \tan (e+f x))^{7/2}}{7 d f}+\frac {1}{2} \left ((a-i b)^2 (c-i d)^3\right ) \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx+\frac {1}{2} \left ((a+i b)^2 (c+i d)^3\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx \\ & = \frac {4 (b c+a d) (a c-b d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (2 a b c+a^2 d-b^2 d\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {4 a b (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {2 b^2 (c+d \tan (e+f x))^{7/2}}{7 d f}+\frac {\left ((a+i b)^2 (i c-d)^3\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 f}-\frac {\left ((a-i b)^2 (i c+d)^3\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 f} \\ & = \frac {4 (b c+a d) (a c-b d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (2 a b c+a^2 d-b^2 d\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {4 a b (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {2 b^2 (c+d \tan (e+f x))^{7/2}}{7 d f}-\frac {\left ((a-i b)^2 (c-i d)^3\right ) \text {Subst}\left (\int \frac {1}{-1-\frac {i c}{d}+\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{d f}-\frac {\left ((a+i b)^2 (c+i d)^3\right ) \text {Subst}\left (\int \frac {1}{-1+\frac {i c}{d}-\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{d f} \\ & = -\frac {i (a-i b)^2 (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {i (a+i b)^2 (c+i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {4 (b c+a d) (a c-b d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (2 a b c+a^2 d-b^2 d\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {4 a b (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {2 b^2 (c+d \tan (e+f x))^{7/2}}{7 d f} \\ \end{align*}

Mathematica [A] (veriﬁed)

Time = 2.20 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.13 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \, dx=\frac {\frac {4 b^2 (c+d \tan (e+f x))^{7/2}}{d}+7 i (a-i b)^2 \left (\frac {2}{5} (c+d \tan (e+f x))^{5/2}+\frac {2}{3} (c-i d) \left (-3 (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )+\sqrt {c+d \tan (e+f x)} (4 c-3 i d+d \tan (e+f x))\right )\right )-7 i (a+i b)^2 \left (\frac {2}{5} (c+d \tan (e+f x))^{5/2}+\frac {2}{3} (c+i d) \left (-3 (c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )+\sqrt {c+d \tan (e+f x)} (4 c+3 i d+d \tan (e+f x))\right )\right )}{14 f} \]

Integrate[(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5/2),x]

((4*b^2*(c + d*Tan[e + f*x])^(7/2))/d + (7*I)*(a - I*b)^2*((2*(c + d*Tan[e + f*x])^(5/2))/5 + (2*(c - I*d)*(-3

*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]] + Sqrt[c + d*Tan[e + f*x]]*(4*c - (3*I)*d + d

*Tan[e + f*x])))/3) - (7*I)*(a + I*b)^2*((2*(c + d*Tan[e + f*x])^(5/2))/5 + (2*(c + I*d)*(-3*(c + I*d)^(3/2)*A

rcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]] + Sqrt[c + d*Tan[e + f*x]]*(4*c + (3*I)*d + d*Tan[e + f*x])))/3

Maple [B] (veriﬁed)

Leaf count of result is larger than twice the leaf count of optimal. \(3586\) vs. \(2(197)=394\).

Time = 0.94 (sec) , antiderivative size = 3587, normalized size of antiderivative = 15.53


method	result	size

parts	\(\text {Expression too large to display}\)	\(3587\)
derivativedivides	\(\text {Expression too large to display}\)	\(3700\)
default	\(\text {Expression too large to display}\)	\(3700\)

int((a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^(5/2),x,method=_RETURNVERBOSE)

a^2*(2/3/f*d*(c+d*tan(f*x+e))^(3/2)+4/f*d*(c+d*tan(f*x+e))^(1/2)*c-1/4/f/d*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^

(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*c^2+1/4/f*d

*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*

c)^(1/2)*(c^2+d^2)^(1/2)+1/4/f/d*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d

^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^3-3/4/f*d*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/

2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c+3/f*d/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(

c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*c^2-1/f*d^3/(2*(c^2+d^2)^(

1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))

-2/f*d/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d

^2)^(1/2)-2*c)^(1/2))*(c^2+d^2)^(1/2)*c+1/4/f/d*ln(d*tan(f*x+e)+c-(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*

c)^(1/2)+(c^2+d^2)^(1/2))*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2-1/4/f*d*ln(d*tan(f*x+e)+c-(c+d*tan

(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)-1/

4/f/d*ln(d*tan(f*x+e)+c-(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/

2)+2*c)^(1/2)*c^3+3/4/f*d*ln(d*tan(f*x+e)+c-(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/

2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c+3/f*d/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)-(2*(c

^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*c^2-1/f*d^3/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(

c+d*tan(f*x+e))^(1/2)-(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))-2/f*d/(2*(c^2+d^2)^(1/2)-2

*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)-(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*(c^2+

d^2)^(1/2)*c)+b^2*(2/7/d/f*(c+d*tan(f*x+e))^(7/2)-2/3/f*d*(c+d*tan(f*x+e))^(3/2)-4/f*d*(c+d*tan(f*x+e))^(1/2)*

c+1/4/f/d*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(2*(c^2+d^2)

^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*c^2-1/4/f*d*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)

^(1/2)+(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)-1/4/f/d*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e

))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^3+3/4/f*d*ln(d*tan(f*x

+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c-3/

f*d/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)

^(1/2)-2*c)^(1/2))*c^2+1/f*d^3/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/

2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))+2/f*d/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(

1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*(c^2+d^2)^(1/2)*c-1/4/f/d*ln(d*tan(f*x+e)+c

-(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)

^(1/2)*c^2+1/4/f*d*ln(d*tan(f*x+e)+c-(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(c^

2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+1/4/f/d*ln(d*tan(f*x+e)+c-(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)

+2*c)^(1/2)+(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^3-3/4/f*d*ln(d*tan(f*x+e)+c-(c+d*tan(f*x+e))^(1/2

)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c-3/f*d/(2*(c^2+d^2)^(1/2)-2*c)

^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)-(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*c^2+1/f*

d^3/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)-(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)

^(1/2)-2*c)^(1/2))+2/f*d/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)-(2*(c^2+d^2)^(1/2)+2*c

)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*(c^2+d^2)^(1/2)*c)-2*a*b/f/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+

d*tan(f*x+e))^(1/2)-(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*(c^2+d^2)^(1/2)*c^2+2*a*b/f/

(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)-(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/

2)-2*c)^(1/2))*(c^2+d^2)^(1/2)*d^2-6*a*b/f/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)-(2*(

c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*c*d^2-3/2/f*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)

*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^2+1/2/f*d^2*ln(d*tan(f*x+e

)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b+4/

5*a*b*(c+d*tan(f*x+e))^(5/2)/f+3/2*a*b/f*ln(d*tan(f*x+e)+c-(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2

)+(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2-1/2*a*b/f*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*ln(d*tan(f*x+e)+c

-(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*d^2-2/f/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*a

rctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*(c^2+d^2)^(1/2)*

a*b*c^2+2/f*d^2/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/

(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*(c^2+d^2)^(1/2)*a*b-6/f*d^2/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*

x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*a*b*c+2/f/(2*(c^2+d^2)^(1/2)-2*c)^(1

/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*a*b*c^3+4/3

/f*a*b*c*(c+d*tan(f*x+e))^(3/2)+4/f*a*b*c^2*(c+d*tan(f*x+e))^(1/2)-4/f*d^2*a*b*(c+d*tan(f*x+e))^(1/2)+1/f*ln(d

*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1

/2)*(c^2+d^2)^(1/2)*a*b*c+2*a*b/f/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)-(2*(c^2+d^2)^

(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*c^3-a*b/f*ln(d*tan(f*x+e)+c-(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^

2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*c

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8466 vs. \(2 (189) = 378\).

Time = 2.77 (sec) , antiderivative size = 8466, normalized size of antiderivative = 36.65 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \, dx=\text {Too large to display} \]

integrate((a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^(5/2),x, algorithm="fricas")

Sympy [F]

\[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \, dx=\int \left (a + b \tan {\left (e + f x \right )}\right )^{2} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}\, dx \]

integrate((a+b*tan(f*x+e))**2*(c+d*tan(f*x+e))**(5/2),x)

Integral((a + b*tan(e + f*x))**2*(c + d*tan(e + f*x))**(5/2), x)

Maxima [F]

\[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \, dx=\int { {\left (b \tan \left (f x + e\right ) + a\right )}^{2} {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]

integrate((a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^(5/2),x, algorithm="maxima")

integrate((b*tan(f*x + e) + a)^2*(d*tan(f*x + e) + c)^(5/2), x)

Giac [F(-1)]

Timed out. \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \, dx=\text {Timed out} \]

integrate((a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^(5/2),x, algorithm="giac")

Mupad [B] (veriﬁcation not implemented)

Time = 78.78 (sec) , antiderivative size = 23642, normalized size of antiderivative = 102.35 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \, dx=\text {Too large to display} \]

int((a + b*tan(e + f*x))^2*(c + d*tan(e + f*x))^(5/2),x)

(((4*b^2*c - 4*a*b*d)/(d*f) - (4*b^2*c)/(d*f))*(c^2 + d^2) - 2*c*(2*c*((4*b^2*c - 4*a*b*d)/(d*f) - (4*b^2*c)/(

d*f)) - (2*(a*d - b*c)^2)/(d*f) + (2*b^2*(c^2 + d^2))/(d*f)))*(c + d*tan(e + f*x))^(1/2) - (c + d*tan(e + f*x)

)^(3/2)*((2*c*((4*b^2*c - 4*a*b*d)/(d*f) - (4*b^2*c)/(d*f)))/3 - (2*(a*d - b*c)^2)/(3*d*f) + (2*b^2*(c^2 + d^2

))/(3*d*f)) - ((4*b^2*c - 4*a*b*d)/(5*d*f) - (4*b^2*c)/(5*d*f))*(c + d*tan(e + f*x))^(5/2) - atan(((((8*(8*a^2

*c*d^5*f^2 - 8*b^2*c*d^5*f^2 + 8*a^2*c^3*d^3*f^2 - 8*b^2*c^3*d^3*f^2 - 8*a*b*d^6*f^2 + 8*a*b*c^4*d^2*f^2))/f^3

- 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(((8*a^4*c^5*f^2 + 8*b^4*c^5*f^2 + 32*a*b^3*d^5*f^2 - 32*a^3*b*d^5*f^

2 + 40*a^4*c*d^4*f^2 + 40*b^4*c*d^4*f^2 - 48*a^2*b^2*c^5*f^2 - 80*a^4*c^3*d^2*f^2 - 80*b^4*c^3*d^2*f^2 + 480*a

^2*b^2*c^3*d^2*f^2 + 160*a*b^3*c^4*d*f^2 - 160*a^3*b*c^4*d*f^2 - 320*a*b^3*c^2*d^3*f^2 - 240*a^2*b^2*c*d^4*f^2

+ 320*a^3*b*c^2*d^3*f^2)^2/64 - f^4*(a^8*c^10 + a^8*d^10 + b^8*c^10 + b^8*d^10 + 4*a^2*b^6*c^10 + 6*a^4*b^4*c

^10 + 4*a^6*b^2*c^10 + 4*a^2*b^6*d^10 + 6*a^4*b^4*d^10 + 4*a^6*b^2*d^10 + 5*a^8*c^2*d^8 + 10*a^8*c^4*d^6 + 10*

a^8*c^6*d^4 + 5*a^8*c^8*d^2 + 5*b^8*c^2*d^8 + 10*b^8*c^4*d^6 + 10*b^8*c^6*d^4 + 5*b^8*c^8*d^2 + 20*a^2*b^6*c^2

*d^8 + 40*a^2*b^6*c^4*d^6 + 40*a^2*b^6*c^6*d^4 + 20*a^2*b^6*c^8*d^2 + 30*a^4*b^4*c^2*d^8 + 60*a^4*b^4*c^4*d^6

+ 60*a^4*b^4*c^6*d^4 + 30*a^4*b^4*c^8*d^2 + 20*a^6*b^2*c^2*d^8 + 40*a^6*b^2*c^4*d^6 + 40*a^6*b^2*c^6*d^4 + 20*

a^6*b^2*c^8*d^2))^(1/2) + a^4*c^5*f^2 + b^4*c^5*f^2 + 4*a*b^3*d^5*f^2 - 4*a^3*b*d^5*f^2 + 5*a^4*c*d^4*f^2 + 5*

b^4*c*d^4*f^2 - 6*a^2*b^2*c^5*f^2 - 10*a^4*c^3*d^2*f^2 - 10*b^4*c^3*d^2*f^2 + 60*a^2*b^2*c^3*d^2*f^2 + 20*a*b^

3*c^4*d*f^2 - 20*a^3*b*c^4*d*f^2 - 40*a*b^3*c^2*d^3*f^2 - 30*a^2*b^2*c*d^4*f^2 + 40*a^3*b*c^2*d^3*f^2)/(4*f^4)

)^(1/2))*(-(((8*a^4*c^5*f^2 + 8*b^4*c^5*f^2 + 32*a*b^3*d^5*f^2 - 32*a^3*b*d^5*f^2 + 40*a^4*c*d^4*f^2 + 40*b^4*

c*d^4*f^2 - 48*a^2*b^2*c^5*f^2 - 80*a^4*c^3*d^2*f^2 - 80*b^4*c^3*d^2*f^2 + 480*a^2*b^2*c^3*d^2*f^2 + 160*a*b^3

*c^4*d*f^2 - 160*a^3*b*c^4*d*f^2 - 320*a*b^3*c^2*d^3*f^2 - 240*a^2*b^2*c*d^4*f^2 + 320*a^3*b*c^2*d^3*f^2)^2/64

- f^4*(a^8*c^10 + a^8*d^10 + b^8*c^10 + b^8*d^10 + 4*a^2*b^6*c^10 + 6*a^4*b^4*c^10 + 4*a^6*b^2*c^10 + 4*a^2*b

^6*d^10 + 6*a^4*b^4*d^10 + 4*a^6*b^2*d^10 + 5*a^8*c^2*d^8 + 10*a^8*c^4*d^6 + 10*a^8*c^6*d^4 + 5*a^8*c^8*d^2 +

5*b^8*c^2*d^8 + 10*b^8*c^4*d^6 + 10*b^8*c^6*d^4 + 5*b^8*c^8*d^2 + 20*a^2*b^6*c^2*d^8 + 40*a^2*b^6*c^4*d^6 + 40

*a^2*b^6*c^6*d^4 + 20*a^2*b^6*c^8*d^2 + 30*a^4*b^4*c^2*d^8 + 60*a^4*b^4*c^4*d^6 + 60*a^4*b^4*c^6*d^4 + 30*a^4*

b^4*c^8*d^2 + 20*a^6*b^2*c^2*d^8 + 40*a^6*b^2*c^4*d^6 + 40*a^6*b^2*c^6*d^4 + 20*a^6*b^2*c^8*d^2))^(1/2) + a^4*

c^5*f^2 + b^4*c^5*f^2 + 4*a*b^3*d^5*f^2 - 4*a^3*b*d^5*f^2 + 5*a^4*c*d^4*f^2 + 5*b^4*c*d^4*f^2 - 6*a^2*b^2*c^5*

f^2 - 10*a^4*c^3*d^2*f^2 - 10*b^4*c^3*d^2*f^2 + 60*a^2*b^2*c^3*d^2*f^2 + 20*a*b^3*c^4*d*f^2 - 20*a^3*b*c^4*d*f

^2 - 40*a*b^3*c^2*d^3*f^2 - 30*a^2*b^2*c*d^4*f^2 + 40*a^3*b*c^2*d^3*f^2)/(4*f^4))^(1/2) + (16*(c + d*tan(e + f

*x))^(1/2)*(a^4*d^8 + b^4*d^8 - 6*a^2*b^2*d^8 - 15*a^4*c^2*d^6 + 15*a^4*c^4*d^4 - a^4*c^6*d^2 - 15*b^4*c^2*d^6

+ 15*b^4*c^4*d^4 - b^4*c^6*d^2 + 80*a*b^3*c^3*d^5 - 24*a*b^3*c^5*d^3 - 80*a^3*b*c^3*d^5 + 24*a^3*b*c^5*d^3 +

90*a^2*b^2*c^2*d^6 - 90*a^2*b^2*c^4*d^4 + 6*a^2*b^2*c^6*d^2 - 24*a*b^3*c*d^7 + 24*a^3*b*c*d^7))/f^2)*(-(((8*a^

4*c^5*f^2 + 8*b^4*c^5*f^2 + 32*a*b^3*d^5*f^2 - 32*a^3*b*d^5*f^2 + 40*a^4*c*d^4*f^2 + 40*b^4*c*d^4*f^2 - 48*a^2

*b^2*c^5*f^2 - 80*a^4*c^3*d^2*f^2 - 80*b^4*c^3*d^2*f^2 + 480*a^2*b^2*c^3*d^2*f^2 + 160*a*b^3*c^4*d*f^2 - 160*a

^3*b*c^4*d*f^2 - 320*a*b^3*c^2*d^3*f^2 - 240*a^2*b^2*c*d^4*f^2 + 320*a^3*b*c^2*d^3*f^2)^2/64 - f^4*(a^8*c^10 +

a^8*d^10 + b^8*c^10 + b^8*d^10 + 4*a^2*b^6*c^10 + 6*a^4*b^4*c^10 + 4*a^6*b^2*c^10 + 4*a^2*b^6*d^10 + 6*a^4*b^

4*d^10 + 4*a^6*b^2*d^10 + 5*a^8*c^2*d^8 + 10*a^8*c^4*d^6 + 10*a^8*c^6*d^4 + 5*a^8*c^8*d^2 + 5*b^8*c^2*d^8 + 10

*b^8*c^4*d^6 + 10*b^8*c^6*d^4 + 5*b^8*c^8*d^2 + 20*a^2*b^6*c^2*d^8 + 40*a^2*b^6*c^4*d^6 + 40*a^2*b^6*c^6*d^4 +

20*a^2*b^6*c^8*d^2 + 30*a^4*b^4*c^2*d^8 + 60*a^4*b^4*c^4*d^6 + 60*a^4*b^4*c^6*d^4 + 30*a^4*b^4*c^8*d^2 + 20*a

^6*b^2*c^2*d^8 + 40*a^6*b^2*c^4*d^6 + 40*a^6*b^2*c^6*d^4 + 20*a^6*b^2*c^8*d^2))^(1/2) + a^4*c^5*f^2 + b^4*c^5*

f^2 + 4*a*b^3*d^5*f^2 - 4*a^3*b*d^5*f^2 + 5*a^4*c*d^4*f^2 + 5*b^4*c*d^4*f^2 - 6*a^2*b^2*c^5*f^2 - 10*a^4*c^3*d

^2*f^2 - 10*b^4*c^3*d^2*f^2 + 60*a^2*b^2*c^3*d^2*f^2 + 20*a*b^3*c^4*d*f^2 - 20*a^3*b*c^4*d*f^2 - 40*a*b^3*c^2*

d^3*f^2 - 30*a^2*b^2*c*d^4*f^2 + 40*a^3*b*c^2*d^3*f^2)/(4*f^4))^(1/2)*1i - (((8*(8*a^2*c*d^5*f^2 - 8*b^2*c*d^5

*f^2 + 8*a^2*c^3*d^3*f^2 - 8*b^2*c^3*d^3*f^2 - 8*a*b*d^6*f^2 + 8*a*b*c^4*d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e

+ f*x))^(1/2)*(-(((8*a^4*c^5*f^2 + 8*b^4*c^5*f^2 + 32*a*b^3*d^5*f^2 - 32*a^3*b*d^5*f^2 + 40*a^4*c*d^4*f^2 + 4

0*b^4*c*d^4*f^2 - 48*a^2*b^2*c^5*f^2 - 80*a^4*c^3*d^2*f^2 - 80*b^4*c^3*d^2*f^2 + 480*a^2*b^2*c^3*d^2*f^2 + 160

*a*b^3*c^4*d*f^2 - 160*a^3*b*c^4*d*f^2 - 320*a*b^3*c^2*d^3*f^2 - 240*a^2*b^2*c*d^4*f^2 + 320*a^3*b*c^2*d^3*f^2

)^2/64 - f^4*(a^8*c^10 + a^8*d^10 + b^8*c^10 + b^8*d^10 + 4*a^2*b^6*c^10 + 6*a^4*b^4*c^10 + 4*a^6*b^2*c^10 + 4

*a^2*b^6*d^10 + 6*a^4*b^4*d^10 + 4*a^6*b^2*d^10 + 5*a^8*c^2*d^8 + 10*a^8*c^4*d^6 + 10*a^8*c^6*d^4 + 5*a^8*c^8*

d^2 + 5*b^8*c^2*d^8 + 10*b^8*c^4*d^6 + 10*b^8*c^6*d^4 + 5*b^8*c^8*d^2 + 20*a^2*b^6*c^2*d^8 + 40*a^2*b^6*c^4*d^

6 + 40*a^2*b^6*c^6*d^4 + 20*a^2*b^6*c^8*d^2 + 30*a^4*b^4*c^2*d^8 + 60*a^4*b^4*c^4*d^6 + 60*a^4*b^4*c^6*d^4 + 3

0*a^4*b^4*c^8*d^2 + 20*a^6*b^2*c^2*d^8 + 40*a^6*b^2*c^4*d^6 + 40*a^6*b^2*c^6*d^4 + 20*a^6*b^2*c^8*d^2))^(1/2)

+ a^4*c^5*f^2 + b^4*c^5*f^2 + 4*a*b^3*d^5*f^2 - 4*a^3*b*d^5*f^2 + 5*a^4*c*d^4*f^2 + 5*b^4*c*d^4*f^2 - 6*a^2*b^

2*c^5*f^2 - 10*a^4*c^3*d^2*f^2 - 10*b^4*c^3*d^2*f^2 + 60*a^2*b^2*c^3*d^2*f^2 + 20*a*b^3*c^4*d*f^2 - 20*a^3*b*c

^4*d*f^2 - 40*a*b^3*c^2*d^3*f^2 - 30*a^2*b^2*c*d^4*f^2 + 40*a^3*b*c^2*d^3*f^2)/(4*f^4))^(1/2))*(-(((8*a^4*c^5*

f^2 + 8*b^4*c^5*f^2 + 32*a*b^3*d^5*f^2 - 32*a^3*b*d^5*f^2 + 40*a^4*c*d^4*f^2 + 40*b^4*c*d^4*f^2 - 48*a^2*b^2*c

^5*f^2 - 80*a^4*c^3*d^2*f^2 - 80*b^4*c^3*d^2*f^2 + 480*a^2*b^2*c^3*d^2*f^2 + 160*a*b^3*c^4*d*f^2 - 160*a^3*b*c

^4*d*f^2 - 320*a*b^3*c^2*d^3*f^2 - 240*a^2*b^2*c*d^4*f^2 + 320*a^3*b*c^2*d^3*f^2)^2/64 - f^4*(a^8*c^10 + a^8*d

^10 + b^8*c^10 + b^8*d^10 + 4*a^2*b^6*c^10 + 6*a^4*b^4*c^10 + 4*a^6*b^2*c^10 + 4*a^2*b^6*d^10 + 6*a^4*b^4*d^10

+ 4*a^6*b^2*d^10 + 5*a^8*c^2*d^8 + 10*a^8*c^4*d^6 + 10*a^8*c^6*d^4 + 5*a^8*c^8*d^2 + 5*b^8*c^2*d^8 + 10*b^8*c

^4*d^6 + 10*b^8*c^6*d^4 + 5*b^8*c^8*d^2 + 20*a^2*b^6*c^2*d^8 + 40*a^2*b^6*c^4*d^6 + 40*a^2*b^6*c^6*d^4 + 20*a^

2*b^6*c^8*d^2 + 30*a^4*b^4*c^2*d^8 + 60*a^4*b^4*c^4*d^6 + 60*a^4*b^4*c^6*d^4 + 30*a^4*b^4*c^8*d^2 + 20*a^6*b^2

*c^2*d^8 + 40*a^6*b^2*c^4*d^6 + 40*a^6*b^2*c^6*d^4 + 20*a^6*b^2*c^8*d^2))^(1/2) + a^4*c^5*f^2 + b^4*c^5*f^2 +

4*a*b^3*d^5*f^2 - 4*a^3*b*d^5*f^2 + 5*a^4*c*d^4*f^2 + 5*b^4*c*d^4*f^2 - 6*a^2*b^2*c^5*f^2 - 10*a^4*c^3*d^2*f^2

- 10*b^4*c^3*d^2*f^2 + 60*a^2*b^2*c^3*d^2*f^2 + 20*a*b^3*c^4*d*f^2 - 20*a^3*b*c^4*d*f^2 - 40*a*b^3*c^2*d^3*f^

2 - 30*a^2*b^2*c*d^4*f^2 + 40*a^3*b*c^2*d^3*f^2)/(4*f^4))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(a^4*d^8 + b^

4*d^8 - 6*a^2*b^2*d^8 - 15*a^4*c^2*d^6 + 15*a^4*c^4*d^4 - a^4*c^6*d^2 - 15*b^4*c^2*d^6 + 15*b^4*c^4*d^4 - b^4*

c^6*d^2 + 80*a*b^3*c^3*d^5 - 24*a*b^3*c^5*d^3 - 80*a^3*b*c^3*d^5 + 24*a^3*b*c^5*d^3 + 90*a^2*b^2*c^2*d^6 - 90*

a^2*b^2*c^4*d^4 + 6*a^2*b^2*c^6*d^2 - 24*a*b^3*c*d^7 + 24*a^3*b*c*d^7))/f^2)*(-(((8*a^4*c^5*f^2 + 8*b^4*c^5*f^

2 + 32*a*b^3*d^5*f^2 - 32*a^3*b*d^5*f^2 + 40*a^4*c*d^4*f^2 + 40*b^4*c*d^4*f^2 - 48*a^2*b^2*c^5*f^2 - 80*a^4*c^

3*d^2*f^2 - 80*b^4*c^3*d^2*f^2 + 480*a^2*b^2*c^3*d^2*f^2 + 160*a*b^3*c^4*d*f^2 - 160*a^3*b*c^4*d*f^2 - 320*a*b

^3*c^2*d^3*f^2 - 240*a^2*b^2*c*d^4*f^2 + 320*a^3*b*c^2*d^3*f^2)^2/64 - f^4*(a^8*c^10 + a^8*d^10 + b^8*c^10 + b

^8*d^10 + 4*a^2*b^6*c^10 + 6*a^4*b^4*c^10 + 4*a^6*b^2*c^10 + 4*a^2*b^6*d^10 + 6*a^4*b^4*d^10 + 4*a^6*b^2*d^10

+ 5*a^8*c^2*d^8 + 10*a^8*c^4*d^6 + 10*a^8*c^6*d^4 + 5*a^8*c^8*d^2 + 5*b^8*c^2*d^8 + 10*b^8*c^4*d^6 + 10*b^8*c^

6*d^4 + 5*b^8*c^8*d^2 + 20*a^2*b^6*c^2*d^8 + 40*a^2*b^6*c^4*d^6 + 40*a^2*b^6*c^6*d^4 + 20*a^2*b^6*c^8*d^2 + 30

*a^4*b^4*c^2*d^8 + 60*a^4*b^4*c^4*d^6 + 60*a^4*b^4*c^6*d^4 + 30*a^4*b^4*c^8*d^2 + 20*a^6*b^2*c^2*d^8 + 40*a^6*

b^2*c^4*d^6 + 40*a^6*b^2*c^6*d^4 + 20*a^6*b^2*c^8*d^2))^(1/2) + a^4*c^5*f^2 + b^4*c^5*f^2 + 4*a*b^3*d^5*f^2 -

4*a^3*b*d^5*f^2 + 5*a^4*c*d^4*f^2 + 5*b^4*c*d^4*f^2 - 6*a^2*b^2*c^5*f^2 - 10*a^4*c^3*d^2*f^2 - 10*b^4*c^3*d^2*

f^2 + 60*a^2*b^2*c^3*d^2*f^2 + 20*a*b^3*c^4*d*f^2 - 20*a^3*b*c^4*d*f^2 - 40*a*b^3*c^2*d^3*f^2 - 30*a^2*b^2*c*d

^4*f^2 + 40*a^3*b*c^2*d^3*f^2)/(4*f^4))^(1/2)*1i)/((16*(a^6*d^11 - b^6*d^11 - a^2*b^4*d^11 + a^4*b^2*d^11 - 6*

a^6*c^4*d^7 - 8*a^6*c^6*d^5 - 3*a^6*c^8*d^3 + 6*b^6*c^4*d^7 + 8*b^6*c^6*d^5 + 3*b^6*c^8*d^3 + 16*a*b^5*c^3*d^8

+ 12*a*b^5*c^5*d^6 - 2*a*b^5*c^9*d^2 + 12*a^3*b^3*c*d^10 + 16*a^5*b*c^3*d^8 + 12*a^5*b*c^5*d^6 - 2*a^5*b*c^9*

d^2 + 6*a^2*b^4*c^4*d^7 + 8*a^2*b^4*c^6*d^5 + 3*a^2*b^4*c^8*d^3 + 32*a^3*b^3*c^3*d^8 + 24*a^3*b^3*c^5*d^6 - 4*

a^3*b^3*c^9*d^2 - 6*a^4*b^2*c^4*d^7 - 8*a^4*b^2*c^6*d^5 - 3*a^4*b^2*c^8*d^3 + 6*a*b^5*c*d^10 + 6*a^5*b*c*d^10)

)/f^3 + (((8*(8*a^2*c*d^5*f^2 - 8*b^2*c*d^5*f^2 + 8*a^2*c^3*d^3*f^2 - 8*b^2*c^3*d^3*f^2 - 8*a*b*d^6*f^2 + 8*a*

b*c^4*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(((8*a^4*c^5*f^2 + 8*b^4*c^5*f^2 + 32*a*b^3*d^5*f^

2 - 32*a^3*b*d^5*f^2 + 40*a^4*c*d^4*f^2 + 40*b^4*c*d^4*f^2 - 48*a^2*b^2*c^5*f^2 - 80*a^4*c^3*d^2*f^2 - 80*b^4*

c^3*d^2*f^2 + 480*a^2*b^2*c^3*d^2*f^2 + 160*a*b^3*c^4*d*f^2 - 160*a^3*b*c^4*d*f^2 - 320*a*b^3*c^2*d^3*f^2 - 24

0*a^2*b^2*c*d^4*f^2 + 320*a^3*b*c^2*d^3*f^2)^2/64 - f^4*(a^8*c^10 + a^8*d^10 + b^8*c^10 + b^8*d^10 + 4*a^2*b^6

*c^10 + 6*a^4*b^4*c^10 + 4*a^6*b^2*c^10 + 4*a^2*b^6*d^10 + 6*a^4*b^4*d^10 + 4*a^6*b^2*d^10 + 5*a^8*c^2*d^8 + 1

0*a^8*c^4*d^6 + 10*a^8*c^6*d^4 + 5*a^8*c^8*d^2 + 5*b^8*c^2*d^8 + 10*b^8*c^4*d^6 + 10*b^8*c^6*d^4 + 5*b^8*c^8*d

^2 + 20*a^2*b^6*c^2*d^8 + 40*a^2*b^6*c^4*d^6 + 40*a^2*b^6*c^6*d^4 + 20*a^2*b^6*c^8*d^2 + 30*a^4*b^4*c^2*d^8 +

60*a^4*b^4*c^4*d^6 + 60*a^4*b^4*c^6*d^4 + 30*a^4*b^4*c^8*d^2 + 20*a^6*b^2*c^2*d^8 + 40*a^6*b^2*c^4*d^6 + 40*a^

6*b^2*c^6*d^4 + 20*a^6*b^2*c^8*d^2))^(1/2) + a^4*c^5*f^2 + b^4*c^5*f^2 + 4*a*b^3*d^5*f^2 - 4*a^3*b*d^5*f^2 + 5

*a^4*c*d^4*f^2 + 5*b^4*c*d^4*f^2 - 6*a^2*b^2*c^5*f^2 - 10*a^4*c^3*d^2*f^2 - 10*b^4*c^3*d^2*f^2 + 60*a^2*b^2*c^

3*d^2*f^2 + 20*a*b^3*c^4*d*f^2 - 20*a^3*b*c^4*d*f^2 - 40*a*b^3*c^2*d^3*f^2 - 30*a^2*b^2*c*d^4*f^2 + 40*a^3*b*c

^2*d^3*f^2)/(4*f^4))^(1/2))*(-(((8*a^4*c^5*f^2 + 8*b^4*c^5*f^2 + 32*a*b^3*d^5*f^2 - 32*a^3*b*d^5*f^2 + 40*a^4*

c*d^4*f^2 + 40*b^4*c*d^4*f^2 - 48*a^2*b^2*c^5*f^2 - 80*a^4*c^3*d^2*f^2 - 80*b^4*c^3*d^2*f^2 + 480*a^2*b^2*c^3*

d^2*f^2 + 160*a*b^3*c^4*d*f^2 - 160*a^3*b*c^4*d*f^2 - 320*a*b^3*c^2*d^3*f^2 - 240*a^2*b^2*c*d^4*f^2 + 320*a^3*

b*c^2*d^3*f^2)^2/64 - f^4*(a^8*c^10 + a^8*d^10 + b^8*c^10 + b^8*d^10 + 4*a^2*b^6*c^10 + 6*a^4*b^4*c^10 + 4*a^6

*b^2*c^10 + 4*a^2*b^6*d^10 + 6*a^4*b^4*d^10 + 4*a^6*b^2*d^10 + 5*a^8*c^2*d^8 + 10*a^8*c^4*d^6 + 10*a^8*c^6*d^4

+ 5*a^8*c^8*d^2 + 5*b^8*c^2*d^8 + 10*b^8*c^4*d^6 + 10*b^8*c^6*d^4 + 5*b^8*c^8*d^2 + 20*a^2*b^6*c^2*d^8 + 40*a

^2*b^6*c^4*d^6 + 40*a^2*b^6*c^6*d^4 + 20*a^2*b^6*c^8*d^2 + 30*a^4*b^4*c^2*d^8 + 60*a^4*b^4*c^4*d^6 + 60*a^4*b^

4*c^6*d^4 + 30*a^4*b^4*c^8*d^2 + 20*a^6*b^2*c^2*d^8 + 40*a^6*b^2*c^4*d^6 + 40*a^6*b^2*c^6*d^4 + 20*a^6*b^2*c^8

*d^2))^(1/2) + a^4*c^5*f^2 + b^4*c^5*f^2 + 4*a*b^3*d^5*f^2 - 4*a^3*b*d^5*f^2 + 5*a^4*c*d^4*f^2 + 5*b^4*c*d^4*f

^2 - 6*a^2*b^2*c^5*f^2 - 10*a^4*c^3*d^2*f^2 - 10*b^4*c^3*d^2*f^2 + 60*a^2*b^2*c^3*d^2*f^2 + 20*a*b^3*c^4*d*f^2

- 20*a^3*b*c^4*d*f^2 - 40*a*b^3*c^2*d^3*f^2 - 30*a^2*b^2*c*d^4*f^2 + 40*a^3*b*c^2*d^3*f^2)/(4*f^4))^(1/2) + (

16*(c + d*tan(e + f*x))^(1/2)*(a^4*d^8 + b^4*d^8 - 6*a^2*b^2*d^8 - 15*a^4*c^2*d^6 + 15*a^4*c^4*d^4 - a^4*c^6*d

^2 - 15*b^4*c^2*d^6 + 15*b^4*c^4*d^4 - b^4*c^6*d^2 + 80*a*b^3*c^3*d^5 - 24*a*b^3*c^5*d^3 - 80*a^3*b*c^3*d^5 +

24*a^3*b*c^5*d^3 + 90*a^2*b^2*c^2*d^6 - 90*a^2*b^2*c^4*d^4 + 6*a^2*b^2*c^6*d^2 - 24*a*b^3*c*d^7 + 24*a^3*b*c*d

^7))/f^2)*(-(((8*a^4*c^5*f^2 + 8*b^4*c^5*f^2 + 32*a*b^3*d^5*f^2 - 32*a^3*b*d^5*f^2 + 40*a^4*c*d^4*f^2 + 40*b^4

*c*d^4*f^2 - 48*a^2*b^2*c^5*f^2 - 80*a^4*c^3*d^2*f^2 - 80*b^4*c^3*d^2*f^2 + 480*a^2*b^2*c^3*d^2*f^2 + 160*a*b^

3*c^4*d*f^2 - 160*a^3*b*c^4*d*f^2 - 320*a*b^3*c^2*d^3*f^2 - 240*a^2*b^2*c*d^4*f^2 + 320*a^3*b*c^2*d^3*f^2)^2/6

4 - f^4*(a^8*c^10 + a^8*d^10 + b^8*c^10 + b^8*d^10 + 4*a^2*b^6*c^10 + 6*a^4*b^4*c^10 + 4*a^6*b^2*c^10 + 4*a^2*

b^6*d^10 + 6*a^4*b^4*d^10 + 4*a^6*b^2*d^10 + 5*a^8*c^2*d^8 + 10*a^8*c^4*d^6 + 10*a^8*c^6*d^4 + 5*a^8*c^8*d^2 +

5*b^8*c^2*d^8 + 10*b^8*c^4*d^6 + 10*b^8*c^6*d^4 + 5*b^8*c^8*d^2 + 20*a^2*b^6*c^2*d^8 + 40*a^2*b^6*c^4*d^6 + 4

0*a^2*b^6*c^6*d^4 + 20*a^2*b^6*c^8*d^2 + 30*a^4*b^4*c^2*d^8 + 60*a^4*b^4*c^4*d^6 + 60*a^4*b^4*c^6*d^4 + 30*a^4

*b^4*c^8*d^2 + 20*a^6*b^2*c^2*d^8 + 40*a^6*b^2*c^4*d^6 + 40*a^6*b^2*c^6*d^4 + 20*a^6*b^2*c^8*d^2))^(1/2) + a^4

*c^5*f^2 + b^4*c^5*f^2 + 4*a*b^3*d^5*f^2 - 4*a^3*b*d^5*f^2 + 5*a^4*c*d^4*f^2 + 5*b^4*c*d^4*f^2 - 6*a^2*b^2*c^5

*f^2 - 10*a^4*c^3*d^2*f^2 - 10*b^4*c^3*d^2*f^2 + 60*a^2*b^2*c^3*d^2*f^2 + 20*a*b^3*c^4*d*f^2 - 20*a^3*b*c^4*d*

f^2 - 40*a*b^3*c^2*d^3*f^2 - 30*a^2*b^2*c*d^4*f^2 + 40*a^3*b*c^2*d^3*f^2)/(4*f^4))^(1/2) + (((8*(8*a^2*c*d^5*f

^2 - 8*b^2*c*d^5*f^2 + 8*a^2*c^3*d^3*f^2 - 8*b^2*c^3*d^3*f^2 - 8*a*b*d^6*f^2 + 8*a*b*c^4*d^2*f^2))/f^3 + 64*c*

d^2*(c + d*tan(e + f*x))^(1/2)*(-(((8*a^4*c^5*f^2 + 8*b^4*c^5*f^2 + 32*a*b^3*d^5*f^2 - 32*a^3*b*d^5*f^2 + 40*a

^4*c*d^4*f^2 + 40*b^4*c*d^4*f^2 - 48*a^2*b^2*c^5*f^2 - 80*a^4*c^3*d^2*f^2 - 80*b^4*c^3*d^2*f^2 + 480*a^2*b^2*c

^3*d^2*f^2 + 160*a*b^3*c^4*d*f^2 - 160*a^3*b*c^4*d*f^2 - 320*a*b^3*c^2*d^3*f^2 - 240*a^2*b^2*c*d^4*f^2 + 320*a

^3*b*c^2*d^3*f^2)^2/64 - f^4*(a^8*c^10 + a^8*d^10 + b^8*c^10 + b^8*d^10 + 4*a^2*b^6*c^10 + 6*a^4*b^4*c^10 + 4*

a^6*b^2*c^10 + 4*a^2*b^6*d^10 + 6*a^4*b^4*d^10 + 4*a^6*b^2*d^10 + 5*a^8*c^2*d^8 + 10*a^8*c^4*d^6 + 10*a^8*c^6*

d^4 + 5*a^8*c^8*d^2 + 5*b^8*c^2*d^8 + 10*b^8*c^4*d^6 + 10*b^8*c^6*d^4 + 5*b^8*c^8*d^2 + 20*a^2*b^6*c^2*d^8 + 4

0*a^2*b^6*c^4*d^6 + 40*a^2*b^6*c^6*d^4 + 20*a^2*b^6*c^8*d^2 + 30*a^4*b^4*c^2*d^8 + 60*a^4*b^4*c^4*d^6 + 60*a^4

*b^4*c^6*d^4 + 30*a^4*b^4*c^8*d^2 + 20*a^6*b^2*c^2*d^8 + 40*a^6*b^2*c^4*d^6 + 40*a^6*b^2*c^6*d^4 + 20*a^6*b^2*

c^8*d^2))^(1/2) + a^4*c^5*f^2 + b^4*c^5*f^2 + 4*a*b^3*d^5*f^2 - 4*a^3*b*d^5*f^2 + 5*a^4*c*d^4*f^2 + 5*b^4*c*d^

4*f^2 - 6*a^2*b^2*c^5*f^2 - 10*a^4*c^3*d^2*f^2 - 10*b^4*c^3*d^2*f^2 + 60*a^2*b^2*c^3*d^2*f^2 + 20*a*b^3*c^4*d*

f^2 - 20*a^3*b*c^4*d*f^2 - 40*a*b^3*c^2*d^3*f^2 - 30*a^2*b^2*c*d^4*f^2 + 40*a^3*b*c^2*d^3*f^2)/(4*f^4))^(1/2))

*(-(((8*a^4*c^5*f^2 + 8*b^4*c^5*f^2 + 32*a*b^3*d^5*f^2 - 32*a^3*b*d^5*f^2 + 40*a^4*c*d^4*f^2 + 40*b^4*c*d^4*f^

2 - 48*a^2*b^2*c^5*f^2 - 80*a^4*c^3*d^2*f^2 - 80*b^4*c^3*d^2*f^2 + 480*a^2*b^2*c^3*d^2*f^2 + 160*a*b^3*c^4*d*f

^2 - 160*a^3*b*c^4*d*f^2 - 320*a*b^3*c^2*d^3*f^2 - 240*a^2*b^2*c*d^4*f^2 + 320*a^3*b*c^2*d^3*f^2)^2/64 - f^4*(

a^8*c^10 + a^8*d^10 + b^8*c^10 + b^8*d^10 + 4*a^2*b^6*c^10 + 6*a^4*b^4*c^10 + 4*a^6*b^2*c^10 + 4*a^2*b^6*d^10

+ 6*a^4*b^4*d^10 + 4*a^6*b^2*d^10 + 5*a^8*c^2*d^8 + 10*a^8*c^4*d^6 + 10*a^8*c^6*d^4 + 5*a^8*c^8*d^2 + 5*b^8*c^

2*d^8 + 10*b^8*c^4*d^6 + 10*b^8*c^6*d^4 + 5*b^8*c^8*d^2 + 20*a^2*b^6*c^2*d^8 + 40*a^2*b^6*c^4*d^6 + 40*a^2*b^6

*c^6*d^4 + 20*a^2*b^6*c^8*d^2 + 30*a^4*b^4*c^2*d^8 + 60*a^4*b^4*c^4*d^6 + 60*a^4*b^4*c^6*d^4 + 30*a^4*b^4*c^8*

d^2 + 20*a^6*b^2*c^2*d^8 + 40*a^6*b^2*c^4*d^6 + 40*a^6*b^2*c^6*d^4 + 20*a^6*b^2*c^8*d^2))^(1/2) + a^4*c^5*f^2

+ b^4*c^5*f^2 + 4*a*b^3*d^5*f^2 - 4*a^3*b*d^5*f^2 + 5*a^4*c*d^4*f^2 + 5*b^4*c*d^4*f^2 - 6*a^2*b^2*c^5*f^2 - 10

*a^4*c^3*d^2*f^2 - 10*b^4*c^3*d^2*f^2 + 60*a^2*b^2*c^3*d^2*f^2 + 20*a*b^3*c^4*d*f^2 - 20*a^3*b*c^4*d*f^2 - 40*

a*b^3*c^2*d^3*f^2 - 30*a^2*b^2*c*d^4*f^2 + 40*a^3*b*c^2*d^3*f^2)/(4*f^4))^(1/2) - (16*(c + d*tan(e + f*x))^(1/

2)*(a^4*d^8 + b^4*d^8 - 6*a^2*b^2*d^8 - 15*a^4*c^2*d^6 + 15*a^4*c^4*d^4 - a^4*c^6*d^2 - 15*b^4*c^2*d^6 + 15*b^

4*c^4*d^4 - b^4*c^6*d^2 + 80*a*b^3*c^3*d^5 - 24*a*b^3*c^5*d^3 - 80*a^3*b*c^3*d^5 + 24*a^3*b*c^5*d^3 + 90*a^2*b

^2*c^2*d^6 - 90*a^2*b^2*c^4*d^4 + 6*a^2*b^2*c^6*d^2 - 24*a*b^3*c*d^7 + 24*a^3*b*c*d^7))/f^2)*(-(((8*a^4*c^5*f^

2 + 8*b^4*c^5*f^2 + 32*a*b^3*d^5*f^2 - 32*a^3*b*d^5*f^2 + 40*a^4*c*d^4*f^2 + 40*b^4*c*d^4*f^2 - 48*a^2*b^2*c^5

*f^2 - 80*a^4*c^3*d^2*f^2 - 80*b^4*c^3*d^2*f^2 + 480*a^2*b^2*c^3*d^2*f^2 + 160*a*b^3*c^4*d*f^2 - 160*a^3*b*c^4

*d*f^2 - 320*a*b^3*c^2*d^3*f^2 - 240*a^2*b^2*c*d^4*f^2 + 320*a^3*b*c^2*d^3*f^2)^2/64 - f^4*(a^8*c^10 + a^8*d^1

0 + b^8*c^10 + b^8*d^10 + 4*a^2*b^6*c^10 + 6*a^4*b^4*c^10 + 4*a^6*b^2*c^10 + 4*a^2*b^6*d^10 + 6*a^4*b^4*d^10 +

4*a^6*b^2*d^10 + 5*a^8*c^2*d^8 + 10*a^8*c^4*d^6 + 10*a^8*c^6*d^4 + 5*a^8*c^8*d^2 + 5*b^8*c^2*d^8 + 10*b^8*c^4

*d^6 + 10*b^8*c^6*d^4 + 5*b^8*c^8*d^2 + 20*a^2*b^6*c^2*d^8 + 40*a^2*b^6*c^4*d^6 + 40*a^2*b^6*c^6*d^4 + 20*a^2*

b^6*c^8*d^2 + 30*a^4*b^4*c^2*d^8 + 60*a^4*b^4*c^4*d^6 + 60*a^4*b^4*c^6*d^4 + 30*a^4*b^4*c^8*d^2 + 20*a^6*b^2*c

^2*d^8 + 40*a^6*b^2*c^4*d^6 + 40*a^6*b^2*c^6*d^4 + 20*a^6*b^2*c^8*d^2))^(1/2) + a^4*c^5*f^2 + b^4*c^5*f^2 + 4*

a*b^3*d^5*f^2 - 4*a^3*b*d^5*f^2 + 5*a^4*c*d^4*f^2 + 5*b^4*c*d^4*f^2 - 6*a^2*b^2*c^5*f^2 - 10*a^4*c^3*d^2*f^2 -

10*b^4*c^3*d^2*f^2 + 60*a^2*b^2*c^3*d^2*f^2 + 20*a*b^3*c^4*d*f^2 - 20*a^3*b*c^4*d*f^2 - 40*a*b^3*c^2*d^3*f^2

- 30*a^2*b^2*c*d^4*f^2 + 40*a^3*b*c^2*d^3*f^2)/(4*f^4))^(1/2)))*(-(((8*a^4*c^5*f^2 + 8*b^4*c^5*f^2 + 32*a*b^3*

d^5*f^2 - 32*a^3*b*d^5*f^2 + 40*a^4*c*d^4*f^2 + 40*b^4*c*d^4*f^2 - 48*a^2*b^2*c^5*f^2 - 80*a^4*c^3*d^2*f^2 - 8

0*b^4*c^3*d^2*f^2 + 480*a^2*b^2*c^3*d^2*f^2 + 160*a*b^3*c^4*d*f^2 - 160*a^3*b*c^4*d*f^2 - 320*a*b^3*c^2*d^3*f^

2 - 240*a^2*b^2*c*d^4*f^2 + 320*a^3*b*c^2*d^3*f^2)^2/64 - f^4*(a^8*c^10 + a^8*d^10 + b^8*c^10 + b^8*d^10 + 4*a

^2*b^6*c^10 + 6*a^4*b^4*c^10 + 4*a^6*b^2*c^10 + 4*a^2*b^6*d^10 + 6*a^4*b^4*d^10 + 4*a^6*b^2*d^10 + 5*a^8*c^2*d

^8 + 10*a^8*c^4*d^6 + 10*a^8*c^6*d^4 + 5*a^8*c^8*d^2 + 5*b^8*c^2*d^8 + 10*b^8*c^4*d^6 + 10*b^8*c^6*d^4 + 5*b^8

*c^8*d^2 + 20*a^2*b^6*c^2*d^8 + 40*a^2*b^6*c^4*d^6 + 40*a^2*b^6*c^6*d^4 + 20*a^2*b^6*c^8*d^2 + 30*a^4*b^4*c^2*

d^8 + 60*a^4*b^4*c^4*d^6 + 60*a^4*b^4*c^6*d^4 + 30*a^4*b^4*c^8*d^2 + 20*a^6*b^2*c^2*d^8 + 40*a^6*b^2*c^4*d^6 +

40*a^6*b^2*c^6*d^4 + 20*a^6*b^2*c^8*d^2))^(1/2) + a^4*c^5*f^2 + b^4*c^5*f^2 + 4*a*b^3*d^5*f^2 - 4*a^3*b*d^5*f

^2 + 5*a^4*c*d^4*f^2 + 5*b^4*c*d^4*f^2 - 6*a^2*b^2*c^5*f^2 - 10*a^4*c^3*d^2*f^2 - 10*b^4*c^3*d^2*f^2 + 60*a^2*

b^2*c^3*d^2*f^2 + 20*a*b^3*c^4*d*f^2 - 20*a^3*b*c^4*d*f^2 - 40*a*b^3*c^2*d^3*f^2 - 30*a^2*b^2*c*d^4*f^2 + 40*a

^3*b*c^2*d^3*f^2)/(4*f^4))^(1/2)*2i - atan(((((8*(8*a^2*c*d^5*f^2 - 8*b^2*c*d^5*f^2 + 8*a^2*c^3*d^3*f^2 - 8*b^

2*c^3*d^3*f^2 - 8*a*b*d^6*f^2 + 8*a*b*c^4*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((((8*a^4*c^5*f^

2 + 8*b^4*c^5*f^2 + 32*a*b^3*d^5*f^2 - 32*a^3*b*d^5*f^2 + 40*a^4*c*d^4*f^2 + 40*b^4*c*d^4*f^2 - 48*a^2*b^2*c^5

*f^2 - 80*a^4*c^3*d^2*f^2 - 80*b^4*c^3*d^2*f^2 + 480*a^2*b^2*c^3*d^2*f^2 + 160*a*b^3*c^4*d*f^2 - 160*a^3*b*c^4

*d*f^2 - 320*a*b^3*c^2*d^3*f^2 - 240*a^2*b^2*c*d^4*f^2 + 320*a^3*b*c^2*d^3*f^2)^2/64 - f^4*(a^8*c^10 + a^8*d^1

0 + b^8*c^10 + b^8*d^10 + 4*a^2*b^6*c^10 + 6*a^4*b^4*c^10 + 4*a^6*b^2*c^10 + 4*a^2*b^6*d^10 + 6*a^4*b^4*d^10 +

4*a^6*b^2*d^10 + 5*a^8*c^2*d^8 + 10*a^8*c^4*d^6 + 10*a^8*c^6*d^4 + 5*a^8*c^8*d^2 + 5*b^8*c^2*d^8 + 10*b^8*c^4

*d^6 + 10*b^8*c^6*d^4 + 5*b^8*c^8*d^2 + 20*a^2*b^6*c^2*d^8 + 40*a^2*b^6*c^4*d^6 + 40*a^2*b^6*c^6*d^4 + 20*a^2*

b^6*c^8*d^2 + 30*a^4*b^4*c^2*d^8 + 60*a^4*b^4*c^4*d^6 + 60*a^4*b^4*c^6*d^4 + 30*a^4*b^4*c^8*d^2 + 20*a^6*b^2*c

^2*d^8 + 40*a^6*b^2*c^4*d^6 + 40*a^6*b^2*c^6*d^4 + 20*a^6*b^2*c^8*d^2))^(1/2) - a^4*c^5*f^2 - b^4*c^5*f^2 - 4*

a*b^3*d^5*f^2 + 4*a^3*b*d^5*f^2 - 5*a^4*c*d^4*f^2 - 5*b^4*c*d^4*f^2 + 6*a^2*b^2*c^5*f^2 + 10*a^4*c^3*d^2*f^2 +

10*b^4*c^3*d^2*f^2 - 60*a^2*b^2*c^3*d^2*f^2 - 20*a*b^3*c^4*d*f^2 + 20*a^3*b*c^4*d*f^2 + 40*a*b^3*c^2*d^3*f^2

+ 30*a^2*b^2*c*d^4*f^2 - 40*a^3*b*c^2*d^3*f^2)/(4*f^4))^(1/2))*((((8*a^4*c^5*f^2 + 8*b^4*c^5*f^2 + 32*a*b^3*d^

5*f^2 - 32*a^3*b*d^5*f^2 + 40*a^4*c*d^4*f^2 + 40*b^4*c*d^4*f^2 - 48*a^2*b^2*c^5*f^2 - 80*a^4*c^3*d^2*f^2 - 80*

b^4*c^3*d^2*f^2 + 480*a^2*b^2*c^3*d^2*f^2 + 160*a*b^3*c^4*d*f^2 - 160*a^3*b*c^4*d*f^2 - 320*a*b^3*c^2*d^3*f^2

- 240*a^2*b^2*c*d^4*f^2 + 320*a^3*b*c^2*d^3*f^2)^2/64 - f^4*(a^8*c^10 + a^8*d^10 + b^8*c^10 + b^8*d^10 + 4*a^2

*b^6*c^10 + 6*a^4*b^4*c^10 + 4*a^6*b^2*c^10 + 4*a^2*b^6*d^10 + 6*a^4*b^4*d^10 + 4*a^6*b^2*d^10 + 5*a^8*c^2*d^8

+ 10*a^8*c^4*d^6 + 10*a^8*c^6*d^4 + 5*a^8*c^8*d^2 + 5*b^8*c^2*d^8 + 10*b^8*c^4*d^6 + 10*b^8*c^6*d^4 + 5*b^8*c

^8*d^2 + 20*a^2*b^6*c^2*d^8 + 40*a^2*b^6*c^4*d^6 + 40*a^2*b^6*c^6*d^4 + 20*a^2*b^6*c^8*d^2 + 30*a^4*b^4*c^2*d^

8 + 60*a^4*b^4*c^4*d^6 + 60*a^4*b^4*c^6*d^4 + 30*a^4*b^4*c^8*d^2 + 20*a^6*b^2*c^2*d^8 + 40*a^6*b^2*c^4*d^6 + 4

0*a^6*b^2*c^6*d^4 + 20*a^6*b^2*c^8*d^2))^(1/2) - a^4*c^5*f^2 - b^4*c^5*f^2 - 4*a*b^3*d^5*f^2 + 4*a^3*b*d^5*f^2

- 5*a^4*c*d^4*f^2 - 5*b^4*c*d^4*f^2 + 6*a^2*b^2*c^5*f^2 + 10*a^4*c^3*d^2*f^2 + 10*b^4*c^3*d^2*f^2 - 60*a^2*b^

2*c^3*d^2*f^2 - 20*a*b^3*c^4*d*f^2 + 20*a^3*b*c^4*d*f^2 + 40*a*b^3*c^2*d^3*f^2 + 30*a^2*b^2*c*d^4*f^2 - 40*a^3

*b*c^2*d^3*f^2)/(4*f^4))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(a^4*d^8 + b^4*d^8 - 6*a^2*b^2*d^8 - 15*a^4*c^

2*d^6 + 15*a^4*c^4*d^4 - a^4*c^6*d^2 - 15*b^4*c^2*d^6 + 15*b^4*c^4*d^4 - b^4*c^6*d^2 + 80*a*b^3*c^3*d^5 - 24*a

*b^3*c^5*d^3 - 80*a^3*b*c^3*d^5 + 24*a^3*b*c^5*d^3 + 90*a^2*b^2*c^2*d^6 - 90*a^2*b^2*c^4*d^4 + 6*a^2*b^2*c^6*d

^2 - 24*a*b^3*c*d^7 + 24*a^3*b*c*d^7))/f^2)*((((8*a^4*c^5*f^2 + 8*b^4*c^5*f^2 + 32*a*b^3*d^5*f^2 - 32*a^3*b*d^

5*f^2 + 40*a^4*c*d^4*f^2 + 40*b^4*c*d^4*f^2 - 48*a^2*b^2*c^5*f^2 - 80*a^4*c^3*d^2*f^2 - 80*b^4*c^3*d^2*f^2 + 4

80*a^2*b^2*c^3*d^2*f^2 + 160*a*b^3*c^4*d*f^2 - 160*a^3*b*c^4*d*f^2 - 320*a*b^3*c^2*d^3*f^2 - 240*a^2*b^2*c*d^4

*f^2 + 320*a^3*b*c^2*d^3*f^2)^2/64 - f^4*(a^8*c^10 + a^8*d^10 + b^8*c^10 + b^8*d^10 + 4*a^2*b^6*c^10 + 6*a^4*b

^4*c^10 + 4*a^6*b^2*c^10 + 4*a^2*b^6*d^10 + 6*a^4*b^4*d^10 + 4*a^6*b^2*d^10 + 5*a^8*c^2*d^8 + 10*a^8*c^4*d^6 +

10*a^8*c^6*d^4 + 5*a^8*c^8*d^2 + 5*b^8*c^2*d^8 + 10*b^8*c^4*d^6 + 10*b^8*c^6*d^4 + 5*b^8*c^8*d^2 + 20*a^2*b^6

*c^2*d^8 + 40*a^2*b^6*c^4*d^6 + 40*a^2*b^6*c^6*d^4 + 20*a^2*b^6*c^8*d^2 + 30*a^4*b^4*c^2*d^8 + 60*a^4*b^4*c^4*

d^6 + 60*a^4*b^4*c^6*d^4 + 30*a^4*b^4*c^8*d^2 + 20*a^6*b^2*c^2*d^8 + 40*a^6*b^2*c^4*d^6 + 40*a^6*b^2*c^6*d^4 +

20*a^6*b^2*c^8*d^2))^(1/2) - a^4*c^5*f^2 - b^4*c^5*f^2 - 4*a*b^3*d^5*f^2 + 4*a^3*b*d^5*f^2 - 5*a^4*c*d^4*f^2

- 5*b^4*c*d^4*f^2 + 6*a^2*b^2*c^5*f^2 + 10*a^4*c^3*d^2*f^2 + 10*b^4*c^3*d^2*f^2 - 60*a^2*b^2*c^3*d^2*f^2 - 20*

a*b^3*c^4*d*f^2 + 20*a^3*b*c^4*d*f^2 + 40*a*b^3*c^2*d^3*f^2 + 30*a^2*b^2*c*d^4*f^2 - 40*a^3*b*c^2*d^3*f^2)/(4*

f^4))^(1/2)*1i - (((8*(8*a^2*c*d^5*f^2 - 8*b^2*c*d^5*f^2 + 8*a^2*c^3*d^3*f^2 - 8*b^2*c^3*d^3*f^2 - 8*a*b*d^6*f

^2 + 8*a*b*c^4*d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((((8*a^4*c^5*f^2 + 8*b^4*c^5*f^2 + 32*a*b^

3*d^5*f^2 - 32*a^3*b*d^5*f^2 + 40*a^4*c*d^4*f^2 + 40*b^4*c*d^4*f^2 - 48*a^2*b^2*c^5*f^2 - 80*a^4*c^3*d^2*f^2 -

80*b^4*c^3*d^2*f^2 + 480*a^2*b^2*c^3*d^2*f^2 + 160*a*b^3*c^4*d*f^2 - 160*a^3*b*c^4*d*f^2 - 320*a*b^3*c^2*d^3*

f^2 - 240*a^2*b^2*c*d^4*f^2 + 320*a^3*b*c^2*d^3*f^2)^2/64 - f^4*(a^8*c^10 + a^8*d^10 + b^8*c^10 + b^8*d^10 + 4

*a^2*b^6*c^10 + 6*a^4*b^4*c^10 + 4*a^6*b^2*c^10 + 4*a^2*b^6*d^10 + 6*a^4*b^4*d^10 + 4*a^6*b^2*d^10 + 5*a^8*c^2

*d^8 + 10*a^8*c^4*d^6 + 10*a^8*c^6*d^4 + 5*a^8*c^8*d^2 + 5*b^8*c^2*d^8 + 10*b^8*c^4*d^6 + 10*b^8*c^6*d^4 + 5*b

^8*c^8*d^2 + 20*a^2*b^6*c^2*d^8 + 40*a^2*b^6*c^4*d^6 + 40*a^2*b^6*c^6*d^4 + 20*a^2*b^6*c^8*d^2 + 30*a^4*b^4*c^

2*d^8 + 60*a^4*b^4*c^4*d^6 + 60*a^4*b^4*c^6*d^4 + 30*a^4*b^4*c^8*d^2 + 20*a^6*b^2*c^2*d^8 + 40*a^6*b^2*c^4*d^6

+ 40*a^6*b^2*c^6*d^4 + 20*a^6*b^2*c^8*d^2))^(1/2) - a^4*c^5*f^2 - b^4*c^5*f^2 - 4*a*b^3*d^5*f^2 + 4*a^3*b*d^5

*f^2 - 5*a^4*c*d^4*f^2 - 5*b^4*c*d^4*f^2 + 6*a^2*b^2*c^5*f^2 + 10*a^4*c^3*d^2*f^2 + 10*b^4*c^3*d^2*f^2 - 60*a^

2*b^2*c^3*d^2*f^2 - 20*a*b^3*c^4*d*f^2 + 20*a^3*b*c^4*d*f^2 + 40*a*b^3*c^2*d^3*f^2 + 30*a^2*b^2*c*d^4*f^2 - 40

*a^3*b*c^2*d^3*f^2)/(4*f^4))^(1/2))*((((8*a^4*c^5*f^2 + 8*b^4*c^5*f^2 + 32*a*b^3*d^5*f^2 - 32*a^3*b*d^5*f^2 +

40*a^4*c*d^4*f^2 + 40*b^4*c*d^4*f^2 - 48*a^2*b^2*c^5*f^2 - 80*a^4*c^3*d^2*f^2 - 80*b^4*c^3*d^2*f^2 + 480*a^2*b

^2*c^3*d^2*f^2 + 160*a*b^3*c^4*d*f^2 - 160*a^3*b*c^4*d*f^2 - 320*a*b^3*c^2*d^3*f^2 - 240*a^2*b^2*c*d^4*f^2 + 3

20*a^3*b*c^2*d^3*f^2)^2/64 - f^4*(a^8*c^10 + a^8*d^10 + b^8*c^10 + b^8*d^10 + 4*a^2*b^6*c^10 + 6*a^4*b^4*c^10

+ 4*a^6*b^2*c^10 + 4*a^2*b^6*d^10 + 6*a^4*b^4*d^10 + 4*a^6*b^2*d^10 + 5*a^8*c^2*d^8 + 10*a^8*c^4*d^6 + 10*a^8*

c^6*d^4 + 5*a^8*c^8*d^2 + 5*b^8*c^2*d^8 + 10*b^8*c^4*d^6 + 10*b^8*c^6*d^4 + 5*b^8*c^8*d^2 + 20*a^2*b^6*c^2*d^8

+ 40*a^2*b^6*c^4*d^6 + 40*a^2*b^6*c^6*d^4 + 20*a^2*b^6*c^8*d^2 + 30*a^4*b^4*c^2*d^8 + 60*a^4*b^4*c^4*d^6 + 60

*a^4*b^4*c^6*d^4 + 30*a^4*b^4*c^8*d^2 + 20*a^6*b^2*c^2*d^8 + 40*a^6*b^2*c^4*d^6 + 40*a^6*b^2*c^6*d^4 + 20*a^6*

b^2*c^8*d^2))^(1/2) - a^4*c^5*f^2 - b^4*c^5*f^2 - 4*a*b^3*d^5*f^2 + 4*a^3*b*d^5*f^2 - 5*a^4*c*d^4*f^2 - 5*b^4*

c*d^4*f^2 + 6*a^2*b^2*c^5*f^2 + 10*a^4*c^3*d^2*f^2 + 10*b^4*c^3*d^2*f^2 - 60*a^2*b^2*c^3*d^2*f^2 - 20*a*b^3*c^

4*d*f^2 + 20*a^3*b*c^4*d*f^2 + 40*a*b^3*c^2*d^3*f^2 + 30*a^2*b^2*c*d^4*f^2 - 40*a^3*b*c^2*d^3*f^2)/(4*f^4))^(1

/2) - (16*(c + d*tan(e + f*x))^(1/2)*(a^4*d^8 + b^4*d^8 - 6*a^2*b^2*d^8 - 15*a^4*c^2*d^6 + 15*a^4*c^4*d^4 - a^

4*c^6*d^2 - 15*b^4*c^2*d^6 + 15*b^4*c^4*d^4 - b^4*c^6*d^2 + 80*a*b^3*c^3*d^5 - 24*a*b^3*c^5*d^3 - 80*a^3*b*c^3

*d^5 + 24*a^3*b*c^5*d^3 + 90*a^2*b^2*c^2*d^6 - 90*a^2*b^2*c^4*d^4 + 6*a^2*b^2*c^6*d^2 - 24*a*b^3*c*d^7 + 24*a^

3*b*c*d^7))/f^2)*((((8*a^4*c^5*f^2 + 8*b^4*c^5*f^2 + 32*a*b^3*d^5*f^2 - 32*a^3*b*d^5*f^2 + 40*a^4*c*d^4*f^2 +

40*b^4*c*d^4*f^2 - 48*a^2*b^2*c^5*f^2 - 80*a^4*c^3*d^2*f^2 - 80*b^4*c^3*d^2*f^2 + 480*a^2*b^2*c^3*d^2*f^2 + 16

0*a*b^3*c^4*d*f^2 - 160*a^3*b*c^4*d*f^2 - 320*a*b^3*c^2*d^3*f^2 - 240*a^2*b^2*c*d^4*f^2 + 320*a^3*b*c^2*d^3*f^

2)^2/64 - f^4*(a^8*c^10 + a^8*d^10 + b^8*c^10 + b^8*d^10 + 4*a^2*b^6*c^10 + 6*a^4*b^4*c^10 + 4*a^6*b^2*c^10 +

4*a^2*b^6*d^10 + 6*a^4*b^4*d^10 + 4*a^6*b^2*d^10 + 5*a^8*c^2*d^8 + 10*a^8*c^4*d^6 + 10*a^8*c^6*d^4 + 5*a^8*c^8

*d^2 + 5*b^8*c^2*d^8 + 10*b^8*c^4*d^6 + 10*b^8*c^6*d^4 + 5*b^8*c^8*d^2 + 20*a^2*b^6*c^2*d^8 + 40*a^2*b^6*c^4*d

^6 + 40*a^2*b^6*c^6*d^4 + 20*a^2*b^6*c^8*d^2 + 30*a^4*b^4*c^2*d^8 + 60*a^4*b^4*c^4*d^6 + 60*a^4*b^4*c^6*d^4 +

30*a^4*b^4*c^8*d^2 + 20*a^6*b^2*c^2*d^8 + 40*a^6*b^2*c^4*d^6 + 40*a^6*b^2*c^6*d^4 + 20*a^6*b^2*c^8*d^2))^(1/2)

- a^4*c^5*f^2 - b^4*c^5*f^2 - 4*a*b^3*d^5*f^2 + 4*a^3*b*d^5*f^2 - 5*a^4*c*d^4*f^2 - 5*b^4*c*d^4*f^2 + 6*a^2*b

^2*c^5*f^2 + 10*a^4*c^3*d^2*f^2 + 10*b^4*c^3*d^2*f^2 - 60*a^2*b^2*c^3*d^2*f^2 - 20*a*b^3*c^4*d*f^2 + 20*a^3*b*

c^4*d*f^2 + 40*a*b^3*c^2*d^3*f^2 + 30*a^2*b^2*c*d^4*f^2 - 40*a^3*b*c^2*d^3*f^2)/(4*f^4))^(1/2)*1i)/((16*(a^6*d

^11 - b^6*d^11 - a^2*b^4*d^11 + a^4*b^2*d^11 - 6*a^6*c^4*d^7 - 8*a^6*c^6*d^5 - 3*a^6*c^8*d^3 + 6*b^6*c^4*d^7 +

8*b^6*c^6*d^5 + 3*b^6*c^8*d^3 + 16*a*b^5*c^3*d^8 + 12*a*b^5*c^5*d^6 - 2*a*b^5*c^9*d^2 + 12*a^3*b^3*c*d^10 + 1

6*a^5*b*c^3*d^8 + 12*a^5*b*c^5*d^6 - 2*a^5*b*c^9*d^2 + 6*a^2*b^4*c^4*d^7 + 8*a^2*b^4*c^6*d^5 + 3*a^2*b^4*c^8*d

^3 + 32*a^3*b^3*c^3*d^8 + 24*a^3*b^3*c^5*d^6 - 4*a^3*b^3*c^9*d^2 - 6*a^4*b^2*c^4*d^7 - 8*a^4*b^2*c^6*d^5 - 3*a

^4*b^2*c^8*d^3 + 6*a*b^5*c*d^10 + 6*a^5*b*c*d^10))/f^3 + (((8*(8*a^2*c*d^5*f^2 - 8*b^2*c*d^5*f^2 + 8*a^2*c^3*d

^3*f^2 - 8*b^2*c^3*d^3*f^2 - 8*a*b*d^6*f^2 + 8*a*b*c^4*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(((

(8*a^4*c^5*f^2 + 8*b^4*c^5*f^2 + 32*a*b^3*d^5*f^2 - 32*a^3*b*d^5*f^2 + 40*a^4*c*d^4*f^2 + 40*b^4*c*d^4*f^2 - 4

8*a^2*b^2*c^5*f^2 - 80*a^4*c^3*d^2*f^2 - 80*b^4*c^3*d^2*f^2 + 480*a^2*b^2*c^3*d^2*f^2 + 160*a*b^3*c^4*d*f^2 -

160*a^3*b*c^4*d*f^2 - 320*a*b^3*c^2*d^3*f^2 - 240*a^2*b^2*c*d^4*f^2 + 320*a^3*b*c^2*d^3*f^2)^2/64 - f^4*(a^8*c

^10 + a^8*d^10 + b^8*c^10 + b^8*d^10 + 4*a^2*b^6*c^10 + 6*a^4*b^4*c^10 + 4*a^6*b^2*c^10 + 4*a^2*b^6*d^10 + 6*a

^4*b^4*d^10 + 4*a^6*b^2*d^10 + 5*a^8*c^2*d^8 + 10*a^8*c^4*d^6 + 10*a^8*c^6*d^4 + 5*a^8*c^8*d^2 + 5*b^8*c^2*d^8

+ 10*b^8*c^4*d^6 + 10*b^8*c^6*d^4 + 5*b^8*c^8*d^2 + 20*a^2*b^6*c^2*d^8 + 40*a^2*b^6*c^4*d^6 + 40*a^2*b^6*c^6*

d^4 + 20*a^2*b^6*c^8*d^2 + 30*a^4*b^4*c^2*d^8 + 60*a^4*b^4*c^4*d^6 + 60*a^4*b^4*c^6*d^4 + 30*a^4*b^4*c^8*d^2 +

20*a^6*b^2*c^2*d^8 + 40*a^6*b^2*c^4*d^6 + 40*a^6*b^2*c^6*d^4 + 20*a^6*b^2*c^8*d^2))^(1/2) - a^4*c^5*f^2 - b^4

*c^5*f^2 - 4*a*b^3*d^5*f^2 + 4*a^3*b*d^5*f^2 - 5*a^4*c*d^4*f^2 - 5*b^4*c*d^4*f^2 + 6*a^2*b^2*c^5*f^2 + 10*a^4*

c^3*d^2*f^2 + 10*b^4*c^3*d^2*f^2 - 60*a^2*b^2*c^3*d^2*f^2 - 20*a*b^3*c^4*d*f^2 + 20*a^3*b*c^4*d*f^2 + 40*a*b^3

*c^2*d^3*f^2 + 30*a^2*b^2*c*d^4*f^2 - 40*a^3*b*c^2*d^3*f^2)/(4*f^4))^(1/2))*((((8*a^4*c^5*f^2 + 8*b^4*c^5*f^2

+ 32*a*b^3*d^5*f^2 - 32*a^3*b*d^5*f^2 + 40*a^4*c*d^4*f^2 + 40*b^4*c*d^4*f^2 - 48*a^2*b^2*c^5*f^2 - 80*a^4*c^3*

d^2*f^2 - 80*b^4*c^3*d^2*f^2 + 480*a^2*b^2*c^3*d^2*f^2 + 160*a*b^3*c^4*d*f^2 - 160*a^3*b*c^4*d*f^2 - 320*a*b^3

*c^2*d^3*f^2 - 240*a^2*b^2*c*d^4*f^2 + 320*a^3*b*c^2*d^3*f^2)^2/64 - f^4*(a^8*c^10 + a^8*d^10 + b^8*c^10 + b^8

*d^10 + 4*a^2*b^6*c^10 + 6*a^4*b^4*c^10 + 4*a^6*b^2*c^10 + 4*a^2*b^6*d^10 + 6*a^4*b^4*d^10 + 4*a^6*b^2*d^10 +

5*a^8*c^2*d^8 + 10*a^8*c^4*d^6 + 10*a^8*c^6*d^4 + 5*a^8*c^8*d^2 + 5*b^8*c^2*d^8 + 10*b^8*c^4*d^6 + 10*b^8*c^6*

d^4 + 5*b^8*c^8*d^2 + 20*a^2*b^6*c^2*d^8 + 40*a^2*b^6*c^4*d^6 + 40*a^2*b^6*c^6*d^4 + 20*a^2*b^6*c^8*d^2 + 30*a

^4*b^4*c^2*d^8 + 60*a^4*b^4*c^4*d^6 + 60*a^4*b^4*c^6*d^4 + 30*a^4*b^4*c^8*d^2 + 20*a^6*b^2*c^2*d^8 + 40*a^6*b^

2*c^4*d^6 + 40*a^6*b^2*c^6*d^4 + 20*a^6*b^2*c^8*d^2))^(1/2) - a^4*c^5*f^2 - b^4*c^5*f^2 - 4*a*b^3*d^5*f^2 + 4*

a^3*b*d^5*f^2 - 5*a^4*c*d^4*f^2 - 5*b^4*c*d^4*f^2 + 6*a^2*b^2*c^5*f^2 + 10*a^4*c^3*d^2*f^2 + 10*b^4*c^3*d^2*f^

2 - 60*a^2*b^2*c^3*d^2*f^2 - 20*a*b^3*c^4*d*f^2 + 20*a^3*b*c^4*d*f^2 + 40*a*b^3*c^2*d^3*f^2 + 30*a^2*b^2*c*d^4

*f^2 - 40*a^3*b*c^2*d^3*f^2)/(4*f^4))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(a^4*d^8 + b^4*d^8 - 6*a^2*b^2*d^

8 - 15*a^4*c^2*d^6 + 15*a^4*c^4*d^4 - a^4*c^6*d^2 - 15*b^4*c^2*d^6 + 15*b^4*c^4*d^4 - b^4*c^6*d^2 + 80*a*b^3*c

^3*d^5 - 24*a*b^3*c^5*d^3 - 80*a^3*b*c^3*d^5 + 24*a^3*b*c^5*d^3 + 90*a^2*b^2*c^2*d^6 - 90*a^2*b^2*c^4*d^4 + 6*

a^2*b^2*c^6*d^2 - 24*a*b^3*c*d^7 + 24*a^3*b*c*d^7))/f^2)*((((8*a^4*c^5*f^2 + 8*b^4*c^5*f^2 + 32*a*b^3*d^5*f^2

- 32*a^3*b*d^5*f^2 + 40*a^4*c*d^4*f^2 + 40*b^4*c*d^4*f^2 - 48*a^2*b^2*c^5*f^2 - 80*a^4*c^3*d^2*f^2 - 80*b^4*c^

3*d^2*f^2 + 480*a^2*b^2*c^3*d^2*f^2 + 160*a*b^3*c^4*d*f^2 - 160*a^3*b*c^4*d*f^2 - 320*a*b^3*c^2*d^3*f^2 - 240*

a^2*b^2*c*d^4*f^2 + 320*a^3*b*c^2*d^3*f^2)^2/64 - f^4*(a^8*c^10 + a^8*d^10 + b^8*c^10 + b^8*d^10 + 4*a^2*b^6*c

^10 + 6*a^4*b^4*c^10 + 4*a^6*b^2*c^10 + 4*a^2*b^6*d^10 + 6*a^4*b^4*d^10 + 4*a^6*b^2*d^10 + 5*a^8*c^2*d^8 + 10*

a^8*c^4*d^6 + 10*a^8*c^6*d^4 + 5*a^8*c^8*d^2 + 5*b^8*c^2*d^8 + 10*b^8*c^4*d^6 + 10*b^8*c^6*d^4 + 5*b^8*c^8*d^2

+ 20*a^2*b^6*c^2*d^8 + 40*a^2*b^6*c^4*d^6 + 40*a^2*b^6*c^6*d^4 + 20*a^2*b^6*c^8*d^2 + 30*a^4*b^4*c^2*d^8 + 60

*a^4*b^4*c^4*d^6 + 60*a^4*b^4*c^6*d^4 + 30*a^4*b^4*c^8*d^2 + 20*a^6*b^2*c^2*d^8 + 40*a^6*b^2*c^4*d^6 + 40*a^6*

b^2*c^6*d^4 + 20*a^6*b^2*c^8*d^2))^(1/2) - a^4*c^5*f^2 - b^4*c^5*f^2 - 4*a*b^3*d^5*f^2 + 4*a^3*b*d^5*f^2 - 5*a

^4*c*d^4*f^2 - 5*b^4*c*d^4*f^2 + 6*a^2*b^2*c^5*f^2 + 10*a^4*c^3*d^2*f^2 + 10*b^4*c^3*d^2*f^2 - 60*a^2*b^2*c^3*

d^2*f^2 - 20*a*b^3*c^4*d*f^2 + 20*a^3*b*c^4*d*f^2 + 40*a*b^3*c^2*d^3*f^2 + 30*a^2*b^2*c*d^4*f^2 - 40*a^3*b*c^2

*d^3*f^2)/(4*f^4))^(1/2) + (((8*(8*a^2*c*d^5*f^2 - 8*b^2*c*d^5*f^2 + 8*a^2*c^3*d^3*f^2 - 8*b^2*c^3*d^3*f^2 - 8

*a*b*d^6*f^2 + 8*a*b*c^4*d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((((8*a^4*c^5*f^2 + 8*b^4*c^5*f^2

+ 32*a*b^3*d^5*f^2 - 32*a^3*b*d^5*f^2 + 40*a^4*c*d^4*f^2 + 40*b^4*c*d^4*f^2 - 48*a^2*b^2*c^5*f^2 - 80*a^4*c^3

*d^2*f^2 - 80*b^4*c^3*d^2*f^2 + 480*a^2*b^2*c^3*d^2*f^2 + 160*a*b^3*c^4*d*f^2 - 160*a^3*b*c^4*d*f^2 - 320*a*b^

3*c^2*d^3*f^2 - 240*a^2*b^2*c*d^4*f^2 + 320*a^3*b*c^2*d^3*f^2)^2/64 - f^4*(a^8*c^10 + a^8*d^10 + b^8*c^10 + b^

8*d^10 + 4*a^2*b^6*c^10 + 6*a^4*b^4*c^10 + 4*a^6*b^2*c^10 + 4*a^2*b^6*d^10 + 6*a^4*b^4*d^10 + 4*a^6*b^2*d^10 +

5*a^8*c^2*d^8 + 10*a^8*c^4*d^6 + 10*a^8*c^6*d^4 + 5*a^8*c^8*d^2 + 5*b^8*c^2*d^8 + 10*b^8*c^4*d^6 + 10*b^8*c^6

*d^4 + 5*b^8*c^8*d^2 + 20*a^2*b^6*c^2*d^8 + 40*a^2*b^6*c^4*d^6 + 40*a^2*b^6*c^6*d^4 + 20*a^2*b^6*c^8*d^2 + 30*

a^4*b^4*c^2*d^8 + 60*a^4*b^4*c^4*d^6 + 60*a^4*b^4*c^6*d^4 + 30*a^4*b^4*c^8*d^2 + 20*a^6*b^2*c^2*d^8 + 40*a^6*b

^2*c^4*d^6 + 40*a^6*b^2*c^6*d^4 + 20*a^6*b^2*c^8*d^2))^(1/2) - a^4*c^5*f^2 - b^4*c^5*f^2 - 4*a*b^3*d^5*f^2 + 4

*a^3*b*d^5*f^2 - 5*a^4*c*d^4*f^2 - 5*b^4*c*d^4*f^2 + 6*a^2*b^2*c^5*f^2 + 10*a^4*c^3*d^2*f^2 + 10*b^4*c^3*d^2*f

^2 - 60*a^2*b^2*c^3*d^2*f^2 - 20*a*b^3*c^4*d*f^2 + 20*a^3*b*c^4*d*f^2 + 40*a*b^3*c^2*d^3*f^2 + 30*a^2*b^2*c*d^

4*f^2 - 40*a^3*b*c^2*d^3*f^2)/(4*f^4))^(1/2))*((((8*a^4*c^5*f^2 + 8*b^4*c^5*f^2 + 32*a*b^3*d^5*f^2 - 32*a^3*b*

d^5*f^2 + 40*a^4*c*d^4*f^2 + 40*b^4*c*d^4*f^2 - 48*a^2*b^2*c^5*f^2 - 80*a^4*c^3*d^2*f^2 - 80*b^4*c^3*d^2*f^2 +

480*a^2*b^2*c^3*d^2*f^2 + 160*a*b^3*c^4*d*f^2 - 160*a^3*b*c^4*d*f^2 - 320*a*b^3*c^2*d^3*f^2 - 240*a^2*b^2*c*d

^4*f^2 + 320*a^3*b*c^2*d^3*f^2)^2/64 - f^4*(a^8*c^10 + a^8*d^10 + b^8*c^10 + b^8*d^10 + 4*a^2*b^6*c^10 + 6*a^4

*b^4*c^10 + 4*a^6*b^2*c^10 + 4*a^2*b^6*d^10 + 6*a^4*b^4*d^10 + 4*a^6*b^2*d^10 + 5*a^8*c^2*d^8 + 10*a^8*c^4*d^6

+ 10*a^8*c^6*d^4 + 5*a^8*c^8*d^2 + 5*b^8*c^2*d^8 + 10*b^8*c^4*d^6 + 10*b^8*c^6*d^4 + 5*b^8*c^8*d^2 + 20*a^2*b

^6*c^2*d^8 + 40*a^2*b^6*c^4*d^6 + 40*a^2*b^6*c^6*d^4 + 20*a^2*b^6*c^8*d^2 + 30*a^4*b^4*c^2*d^8 + 60*a^4*b^4*c^

4*d^6 + 60*a^4*b^4*c^6*d^4 + 30*a^4*b^4*c^8*d^2 + 20*a^6*b^2*c^2*d^8 + 40*a^6*b^2*c^4*d^6 + 40*a^6*b^2*c^6*d^4

+ 20*a^6*b^2*c^8*d^2))^(1/2) - a^4*c^5*f^2 - b^4*c^5*f^2 - 4*a*b^3*d^5*f^2 + 4*a^3*b*d^5*f^2 - 5*a^4*c*d^4*f^

2 - 5*b^4*c*d^4*f^2 + 6*a^2*b^2*c^5*f^2 + 10*a^4*c^3*d^2*f^2 + 10*b^4*c^3*d^2*f^2 - 60*a^2*b^2*c^3*d^2*f^2 - 2

0*a*b^3*c^4*d*f^2 + 20*a^3*b*c^4*d*f^2 + 40*a*b^3*c^2*d^3*f^2 + 30*a^2*b^2*c*d^4*f^2 - 40*a^3*b*c^2*d^3*f^2)/(

4*f^4))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(a^4*d^8 + b^4*d^8 - 6*a^2*b^2*d^8 - 15*a^4*c^2*d^6 + 15*a^4*c^

4*d^4 - a^4*c^6*d^2 - 15*b^4*c^2*d^6 + 15*b^4*c^4*d^4 - b^4*c^6*d^2 + 80*a*b^3*c^3*d^5 - 24*a*b^3*c^5*d^3 - 80

*a^3*b*c^3*d^5 + 24*a^3*b*c^5*d^3 + 90*a^2*b^2*c^2*d^6 - 90*a^2*b^2*c^4*d^4 + 6*a^2*b^2*c^6*d^2 - 24*a*b^3*c*d

^7 + 24*a^3*b*c*d^7))/f^2)*((((8*a^4*c^5*f^2 + 8*b^4*c^5*f^2 + 32*a*b^3*d^5*f^2 - 32*a^3*b*d^5*f^2 + 40*a^4*c*

d^4*f^2 + 40*b^4*c*d^4*f^2 - 48*a^2*b^2*c^5*f^2 - 80*a^4*c^3*d^2*f^2 - 80*b^4*c^3*d^2*f^2 + 480*a^2*b^2*c^3*d^

2*f^2 + 160*a*b^3*c^4*d*f^2 - 160*a^3*b*c^4*d*f^2 - 320*a*b^3*c^2*d^3*f^2 - 240*a^2*b^2*c*d^4*f^2 + 320*a^3*b*

c^2*d^3*f^2)^2/64 - f^4*(a^8*c^10 + a^8*d^10 + b^8*c^10 + b^8*d^10 + 4*a^2*b^6*c^10 + 6*a^4*b^4*c^10 + 4*a^6*b

^2*c^10 + 4*a^2*b^6*d^10 + 6*a^4*b^4*d^10 + 4*a^6*b^2*d^10 + 5*a^8*c^2*d^8 + 10*a^8*c^4*d^6 + 10*a^8*c^6*d^4 +

5*a^8*c^8*d^2 + 5*b^8*c^2*d^8 + 10*b^8*c^4*d^6 + 10*b^8*c^6*d^4 + 5*b^8*c^8*d^2 + 20*a^2*b^6*c^2*d^8 + 40*a^2

*b^6*c^4*d^6 + 40*a^2*b^6*c^6*d^4 + 20*a^2*b^6*c^8*d^2 + 30*a^4*b^4*c^2*d^8 + 60*a^4*b^4*c^4*d^6 + 60*a^4*b^4*

c^6*d^4 + 30*a^4*b^4*c^8*d^2 + 20*a^6*b^2*c^2*d^8 + 40*a^6*b^2*c^4*d^6 + 40*a^6*b^2*c^6*d^4 + 20*a^6*b^2*c^8*d

^2))^(1/2) - a^4*c^5*f^2 - b^4*c^5*f^2 - 4*a*b^3*d^5*f^2 + 4*a^3*b*d^5*f^2 - 5*a^4*c*d^4*f^2 - 5*b^4*c*d^4*f^2

+ 6*a^2*b^2*c^5*f^2 + 10*a^4*c^3*d^2*f^2 + 10*b^4*c^3*d^2*f^2 - 60*a^2*b^2*c^3*d^2*f^2 - 20*a*b^3*c^4*d*f^2 +

20*a^3*b*c^4*d*f^2 + 40*a*b^3*c^2*d^3*f^2 + 30*a^2*b^2*c*d^4*f^2 - 40*a^3*b*c^2*d^3*f^2)/(4*f^4))^(1/2)))*(((