\(\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \, dx\) [1242]
Optimal result
Integrand size = 27, antiderivative size = 231 \[
\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \, dx=-\frac {i (a-i b)^2 (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {i (a+i b)^2 (c+i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {4 (b c+a d) (a c-b d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (2 a b c+a^2 d-b^2 d\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {4 a b (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {2 b^2 (c+d \tan (e+f x))^{7/2}}{7 d f}
\]
[Out]
-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} \[
\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \, dx=\frac {2 \left (a^2 d+2 a b c-b^2 d\right ) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {i (a-i b)^2 (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {i (a+i b)^2 (c+i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {4 a b (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {4 (a d+b c) (a c-b d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 b^2 (c+d \tan (e+f x))^{7/2}}{7 d f}
\]
[In]
Int[(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5/2),x]
[Out]
((-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)
Rule 65
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,
b, c, d, m, n, x]
Rule 214
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]
Rule 3609
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]
Rule 3618
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]
Rule 3620
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]
&& NeQ[c^2 + d^2, 0] && !IntegerQ[m]
Rule 3624
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
[a, 0])
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
\[
\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \, dx=\frac {\frac {4 b^2 (c+d \tan (e+f x))^{7/2}}{d}+7 i (a-i b)^2 \left (\frac {2}{5} (c+d \tan (e+f x))^{5/2}+\frac {2}{3} (c-i d) \left (-3 (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )+\sqrt {c+d \tan (e+f x)} (4 c-3 i d+d \tan (e+f x))\right )\right )-7 i (a+i b)^2 \left (\frac {2}{5} (c+d \tan (e+f x))^{5/2}+\frac {2}{3} (c+i d) \left (-3 (c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )+\sqrt {c+d \tan (e+f x)} (4 c+3 i d+d \tan (e+f x))\right )\right )}{14 f}
\]
[In]
Integrate[(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5/2),x]
[Out]
((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
))/(14*f)
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 |
\(\text {Expression too large to display}\) |
\(3700\) |
| | |
|
|
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[In]
int((a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^(5/2),x,method=_RETURNVERBOSE)
[Out]
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
\[
\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \, dx=\text {Too large to display}
\]
[In]
integrate((a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^(5/2),x, algorithm="fricas")
[Out]
Too large to include
Sympy [F]
\[
\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \, dx=\int \left (a + b \tan {\left (e + f x \right )}\right )^{2} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}\, dx
\]
[In]
integrate((a+b*tan(f*x+e))**2*(c+d*tan(f*x+e))**(5/2),x)
[Out]
Integral((a + b*tan(e + f*x))**2*(c + d*tan(e + f*x))**(5/2), x)
Maxima [F]
\[
\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \, dx=\int { {\left (b \tan \left (f x + e\right ) + a\right )}^{2} {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x }
\]
[In]
integrate((a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^(5/2),x, algorithm="maxima")
[Out]
integrate((b*tan(f*x + e) + a)^2*(d*tan(f*x + e) + c)^(5/2), x)
Giac [F(-1)]
Timed out. \[
\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \, dx=\text {Timed out}
\]
[In]
integrate((a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^(5/2),x, algorithm="giac")
[Out]
Timed out
Mupad [B] (verification not implemented)
Time = 78.78 (sec) , antiderivative size = 23642, normalized size of antiderivative = 102.35
\[
\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \, dx=\text {Too large to display}
\]
[In]
int((a + b*tan(e + f*x))^2*(c + d*tan(e + f*x))^(5/2),x)
[Out]
(((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)))*(((
(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)*2i + (2*b^2*(c + d*tan(e + f*x))^(7
/2))/(7*d*f)