3.1260 \(\int \frac {(a+b \tan (e+f x))^3}{(c+d \tan (e+f x))^{5/2}} \, dx\)
Optimal. Leaf size=219 \[ -\frac {4 (b c-a d)^2 \left (3 a c d+b \left (c^2+4 d^2\right )\right )}{3 d^2 f \left (c^2+d^2\right )^2 \sqrt {c+d \tan (e+f x)}}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}-\frac {(-b+i a)^3 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (c+i d)^{5/2}}+\frac {(b+i a)^3 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (c-i d)^{5/2}} \]
[Out]
(I*a+b)^3*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/(c-I*d)^(5/2)/f-(I*a-b)^3*arctanh((c+d*tan(f*x+e))^(1/
2)/(c+I*d)^(1/2))/(c+I*d)^(5/2)/f-4/3*(-a*d+b*c)^2*(3*a*c*d+b*(c^2+4*d^2))/d^2/(c^2+d^2)^2/f/(c+d*tan(f*x+e))^
(1/2)-2/3*(-a*d+b*c)^2*(a+b*tan(f*x+e))/d/(c^2+d^2)/f/(c+d*tan(f*x+e))^(3/2)
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Rubi [A] time = 0.65, antiderivative size = 219, normalized size of antiderivative = 1.00,
number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used =
{3565, 3628, 3539, 3537, 63, 208} \[ -\frac {4 (b c-a d)^2 \left (3 a c d+b \left (c^2+4 d^2\right )\right )}{3 d^2 f \left (c^2+d^2\right )^2 \sqrt {c+d \tan (e+f x)}}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}-\frac {(-b+i a)^3 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (c+i d)^{5/2}}+\frac {(b+i a)^3 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (c-i d)^{5/2}} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*Tan[e + f*x])^3/(c + d*Tan[e + f*x])^(5/2),x]
[Out]
((I*a + b)^3*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(5/2)*f) - ((I*a - b)^3*ArcTanh[Sqrt[
c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((c + I*d)^(5/2)*f) - (2*(b*c - a*d)^2*(a + b*Tan[e + f*x]))/(3*d*(c^2 + d
^2)*f*(c + d*Tan[e + f*x])^(3/2)) - (4*(b*c - a*d)^2*(3*a*c*d + b*(c^2 + 4*d^2)))/(3*d^2*(c^2 + d^2)^2*f*Sqrt[
c + d*Tan[e + f*x]])
Rule 63
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 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rule 3537
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c*
d)/f, Subst[Int[(a + (b*x)/d)^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 3539
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 3565
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[((b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 + d^2)), x] - D
ist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(
m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3
*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]
Rule 3628
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> Simp[((A*b^2 - a*b*B + a^2*C)*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)*(a^2 + b^2
)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[b*B + a*(A - C) - (A*b - a*B - b*C)*Tan[e +
f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && LtQ[m, -1] && NeQ[a^2
+ b^2, 0]
Rubi steps
\begin {align*} \int \frac {(a+b \tan (e+f x))^3}{(c+d \tan (e+f x))^{5/2}} \, dx &=-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 \int \frac {\frac {1}{2} \left (2 b^3 c^2+3 a^3 c d-7 a b^2 c d+8 a^2 b d^2\right )+\frac {3}{2} d \left (3 a^2 b c-b^3 c-a^3 d+3 a b^2 d\right ) \tan (e+f x)+\frac {1}{2} b \left (2 a b c d-a^2 d^2+b^2 \left (2 c^2+3 d^2\right )\right ) \tan ^2(e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx}{3 d \left (c^2+d^2\right )}\\ &=-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {4 (b c-a d)^2 \left (3 a c d+b \left (c^2+4 d^2\right )\right )}{3 d^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {2 \int \frac {\frac {3}{2} d \left (6 a^2 b c d-2 b^3 c d+a^3 \left (c^2-d^2\right )-3 a b^2 \left (c^2-d^2\right )\right )-\frac {3}{2} d \left (2 a^3 c d-6 a b^2 c d-3 a^2 b \left (c^2-d^2\right )+b^3 \left (c^2-d^2\right )\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{3 d \left (c^2+d^2\right )^2}\\ &=-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {4 (b c-a d)^2 \left (3 a c d+b \left (c^2+4 d^2\right )\right )}{3 d^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {(a-i b)^3 \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (c-i d)^2}+\frac {(a+i b)^3 \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (c+i d)^2}\\ &=-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {4 (b c-a d)^2 \left (3 a c d+b \left (c^2+4 d^2\right )\right )}{3 d^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}-\frac {(i a+b)^3 \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 (c-i d)^2 f}+\frac {(i a-b)^3 \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 (c+i d)^2 f}\\ &=-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {4 (b c-a d)^2 \left (3 a c d+b \left (c^2+4 d^2\right )\right )}{3 d^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}-\frac {(a-i b)^3 \operatorname {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 )}{(c-i d)^2 d f}-\frac {(a+i b)^3 \operatorname {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 )}{(c+i d)^2 d f}\\ &=\frac {(i a+b)^3 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{5/2} f}-\frac {(i a-b)^3 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(c+i d)^{5/2} f}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {4 (b c-a d)^2 \left (3 a c d+b \left (c^2+4 d^2\right )\right )}{3 d^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}\\ \end {align*}
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Mathematica [C] time = 1.56, size = 284, normalized size = 1.30 \[ -\frac {-3 b d \left (3 a^2-b^2\right ) (c+d \tan (e+f x)) \left (i (c+i d) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c+d \tan (e+f x)}{c-i d}\right )-(d+i c) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c+d \tan (e+f x)}{c+i d}\right )\right )-d \left (a^3 d-3 a^2 b c-3 a b^2 d+b^3 c\right ) \left (i (c+i d) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {c+d \tan (e+f x)}{c-i d}\right )-(d+i c) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {c+d \tan (e+f x)}{c+i d}\right )\right )+6 b^2 d (c-i d) (c+i d) (a+b \tan (e+f x))+4 b^3 c \left (c^2+d^2\right )}{3 d^2 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*Tan[e + f*x])^3/(c + d*Tan[e + f*x])^(5/2),x]
[Out]
-1/3*(4*b^3*c*(c^2 + d^2) - d*(-3*a^2*b*c + b^3*c + a^3*d - 3*a*b^2*d)*(I*(c + I*d)*Hypergeometric2F1[-3/2, 1,
-1/2, (c + d*Tan[e + f*x])/(c - I*d)] - (I*c + d)*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c +
I*d)]) + 6*b^2*(c - I*d)*(c + I*d)*d*(a + b*Tan[e + f*x]) - 3*b*(3*a^2 - b^2)*d*(I*(c + I*d)*Hypergeometric2F1
[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c - I*d)] - (I*c + d)*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x]
)/(c + I*d)])*(c + d*Tan[e + f*x]))/(d^2*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2))
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*tan(f*x+e))^3/(c+d*tan(f*x+e))^(5/2),x, algorithm="fricas")
[Out]
Timed out
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*tan(f*x+e))^3/(c+d*tan(f*x+e))^(5/2),x, algorithm="giac")
[Out]
Timed out
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maple [B] time = 0.36, size = 26303, normalized size = 120.11 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*tan(f*x+e))^3/(c+d*tan(f*x+e))^(5/2),x)
[Out]
result too large to display
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*tan(f*x+e))^3/(c+d*tan(f*x+e))^(5/2),x, algorithm="maxima")
[Out]
Timed out
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mupad [B] time = 24.50, size = 34142, normalized size = 155.90 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*tan(e + f*x))^3/(c + d*tan(e + f*x))^(5/2),x)
[Out]
- atan(((((((8*a^6*c^5*f^2 - 8*b^6*c^5*f^2 + 48*a*b^5*d^5*f^2 + 48*a^5*b*d^5*f^2 + 40*a^6*c*d^4*f^2 - 40*b^6*c
*d^4*f^2 + 120*a^2*b^4*c^5*f^2 - 120*a^4*b^2*c^5*f^2 - 160*a^3*b^3*d^5*f^2 - 80*a^6*c^3*d^2*f^2 + 80*b^6*c^3*d
^2*f^2 - 1200*a^2*b^4*c^3*d^2*f^2 + 1600*a^3*b^3*c^2*d^3*f^2 + 1200*a^4*b^2*c^3*d^2*f^2 + 240*a*b^5*c^4*d*f^2
+ 240*a^5*b*c^4*d*f^2 - 480*a*b^5*c^2*d^3*f^2 + 600*a^2*b^4*c*d^4*f^2 - 800*a^3*b^3*c^4*d*f^2 - 600*a^4*b^2*c*
d^4*f^2 - 480*a^5*b*c^2*d^3*f^2)^2/4 - (16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6
*d^4*f^4 + 80*c^8*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/
2) - 4*a^6*c^5*f^2 + 4*b^6*c^5*f^2 - 24*a*b^5*d^5*f^2 - 24*a^5*b*d^5*f^2 - 20*a^6*c*d^4*f^2 + 20*b^6*c*d^4*f^2
- 60*a^2*b^4*c^5*f^2 + 60*a^4*b^2*c^5*f^2 + 80*a^3*b^3*d^5*f^2 + 40*a^6*c^3*d^2*f^2 - 40*b^6*c^3*d^2*f^2 + 60
0*a^2*b^4*c^3*d^2*f^2 - 800*a^3*b^3*c^2*d^3*f^2 - 600*a^4*b^2*c^3*d^2*f^2 - 120*a*b^5*c^4*d*f^2 - 120*a^5*b*c^
4*d*f^2 + 240*a*b^5*c^2*d^3*f^2 - 300*a^2*b^4*c*d^4*f^2 + 400*a^3*b^3*c^4*d*f^2 + 300*a^4*b^2*c*d^4*f^2 + 240*
a^5*b*c^2*d^3*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4)
))^(1/2)*((c + d*tan(e + f*x))^(1/2)*((((8*a^6*c^5*f^2 - 8*b^6*c^5*f^2 + 48*a*b^5*d^5*f^2 + 48*a^5*b*d^5*f^2 +
40*a^6*c*d^4*f^2 - 40*b^6*c*d^4*f^2 + 120*a^2*b^4*c^5*f^2 - 120*a^4*b^2*c^5*f^2 - 160*a^3*b^3*d^5*f^2 - 80*a^
6*c^3*d^2*f^2 + 80*b^6*c^3*d^2*f^2 - 1200*a^2*b^4*c^3*d^2*f^2 + 1600*a^3*b^3*c^2*d^3*f^2 + 1200*a^4*b^2*c^3*d^
2*f^2 + 240*a*b^5*c^4*d*f^2 + 240*a^5*b*c^4*d*f^2 - 480*a*b^5*c^2*d^3*f^2 + 600*a^2*b^4*c*d^4*f^2 - 800*a^3*b^
3*c^4*d*f^2 - 600*a^4*b^2*c*d^4*f^2 - 480*a^5*b*c^2*d^3*f^2)^2/4 - (16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4
+ 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 1
5*a^8*b^4 + 6*a^10*b^2))^(1/2) - 4*a^6*c^5*f^2 + 4*b^6*c^5*f^2 - 24*a*b^5*d^5*f^2 - 24*a^5*b*d^5*f^2 - 20*a^6*
c*d^4*f^2 + 20*b^6*c*d^4*f^2 - 60*a^2*b^4*c^5*f^2 + 60*a^4*b^2*c^5*f^2 + 80*a^3*b^3*d^5*f^2 + 40*a^6*c^3*d^2*f
^2 - 40*b^6*c^3*d^2*f^2 + 600*a^2*b^4*c^3*d^2*f^2 - 800*a^3*b^3*c^2*d^3*f^2 - 600*a^4*b^2*c^3*d^2*f^2 - 120*a*
b^5*c^4*d*f^2 - 120*a^5*b*c^4*d*f^2 + 240*a*b^5*c^2*d^3*f^2 - 300*a^2*b^4*c*d^4*f^2 + 400*a^3*b^3*c^4*d*f^2 +
300*a^4*b^2*c*d^4*f^2 + 240*a^5*b*c^2*d^3*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*
c^6*d^4*f^4 + 5*c^8*d^2*f^4)))^(1/2)*(64*c*d^22*f^5 + 640*c^3*d^20*f^5 + 2880*c^5*d^18*f^5 + 7680*c^7*d^16*f^5
+ 13440*c^9*d^14*f^5 + 16128*c^11*d^12*f^5 + 13440*c^13*d^10*f^5 + 7680*c^15*d^8*f^5 + 2880*c^17*d^6*f^5 + 64
0*c^19*d^4*f^5 + 64*c^21*d^2*f^5) - 32*a^3*d^21*f^4 + 96*a*b^2*d^21*f^4 - 96*b^3*c*d^20*f^4 - 160*a^3*c^2*d^19
*f^4 - 128*a^3*c^4*d^17*f^4 + 896*a^3*c^6*d^15*f^4 + 3136*a^3*c^8*d^13*f^4 + 4928*a^3*c^10*d^11*f^4 + 4480*a^3
*c^12*d^9*f^4 + 2432*a^3*c^14*d^7*f^4 + 736*a^3*c^16*d^5*f^4 + 96*a^3*c^18*d^3*f^4 - 736*b^3*c^3*d^18*f^4 - 24
32*b^3*c^5*d^16*f^4 - 4480*b^3*c^7*d^14*f^4 - 4928*b^3*c^9*d^12*f^4 - 3136*b^3*c^11*d^10*f^4 - 896*b^3*c^13*d^
8*f^4 + 128*b^3*c^15*d^6*f^4 + 160*b^3*c^17*d^4*f^4 + 32*b^3*c^19*d^2*f^4 + 288*a^2*b*c*d^20*f^4 + 480*a*b^2*c
^2*d^19*f^4 + 384*a*b^2*c^4*d^17*f^4 - 2688*a*b^2*c^6*d^15*f^4 - 9408*a*b^2*c^8*d^13*f^4 - 14784*a*b^2*c^10*d^
11*f^4 - 13440*a*b^2*c^12*d^9*f^4 - 7296*a*b^2*c^14*d^7*f^4 - 2208*a*b^2*c^16*d^5*f^4 - 288*a*b^2*c^18*d^3*f^4
+ 2208*a^2*b*c^3*d^18*f^4 + 7296*a^2*b*c^5*d^16*f^4 + 13440*a^2*b*c^7*d^14*f^4 + 14784*a^2*b*c^9*d^12*f^4 + 9
408*a^2*b*c^11*d^10*f^4 + 2688*a^2*b*c^13*d^8*f^4 - 384*a^2*b*c^15*d^6*f^4 - 480*a^2*b*c^17*d^4*f^4 - 96*a^2*b
*c^19*d^2*f^4) + (c + d*tan(e + f*x))^(1/2)*(16*a^6*d^18*f^3 - 16*b^6*d^18*f^3 + 240*a^2*b^4*d^18*f^3 - 240*a^
4*b^2*d^18*f^3 - 320*a^6*c^4*d^14*f^3 - 1024*a^6*c^6*d^12*f^3 - 1440*a^6*c^8*d^10*f^3 - 1024*a^6*c^10*d^8*f^3
- 320*a^6*c^12*d^6*f^3 + 16*a^6*c^16*d^2*f^3 + 320*b^6*c^4*d^14*f^3 + 1024*b^6*c^6*d^12*f^3 + 1440*b^6*c^8*d^1
0*f^3 + 1024*b^6*c^10*d^8*f^3 + 320*b^6*c^12*d^6*f^3 - 16*b^6*c^16*d^2*f^3 - 4800*a^2*b^4*c^4*d^14*f^3 - 15360
*a^2*b^4*c^6*d^12*f^3 - 21600*a^2*b^4*c^8*d^10*f^3 - 15360*a^2*b^4*c^10*d^8*f^3 - 4800*a^2*b^4*c^12*d^6*f^3 +
240*a^2*b^4*c^16*d^2*f^3 + 6400*a^3*b^3*c^3*d^15*f^3 + 11520*a^3*b^3*c^5*d^13*f^3 + 6400*a^3*b^3*c^7*d^11*f^3
- 6400*a^3*b^3*c^9*d^9*f^3 - 11520*a^3*b^3*c^11*d^7*f^3 - 6400*a^3*b^3*c^13*d^5*f^3 - 1280*a^3*b^3*c^15*d^3*f^
3 + 4800*a^4*b^2*c^4*d^14*f^3 + 15360*a^4*b^2*c^6*d^12*f^3 + 21600*a^4*b^2*c^8*d^10*f^3 + 15360*a^4*b^2*c^10*d
^8*f^3 + 4800*a^4*b^2*c^12*d^6*f^3 - 240*a^4*b^2*c^16*d^2*f^3 - 384*a*b^5*c*d^17*f^3 - 384*a^5*b*c*d^17*f^3 -
1920*a*b^5*c^3*d^15*f^3 - 3456*a*b^5*c^5*d^13*f^3 - 1920*a*b^5*c^7*d^11*f^3 + 1920*a*b^5*c^9*d^9*f^3 + 3456*a*
b^5*c^11*d^7*f^3 + 1920*a*b^5*c^13*d^5*f^3 + 384*a*b^5*c^15*d^3*f^3 + 1280*a^3*b^3*c*d^17*f^3 - 1920*a^5*b*c^3
*d^15*f^3 - 3456*a^5*b*c^5*d^13*f^3 - 1920*a^5*b*c^7*d^11*f^3 + 1920*a^5*b*c^9*d^9*f^3 + 3456*a^5*b*c^11*d^7*f
^3 + 1920*a^5*b*c^13*d^5*f^3 + 384*a^5*b*c^15*d^3*f^3))*((((8*a^6*c^5*f^2 - 8*b^6*c^5*f^2 + 48*a*b^5*d^5*f^2 +
48*a^5*b*d^5*f^2 + 40*a^6*c*d^4*f^2 - 40*b^6*c*d^4*f^2 + 120*a^2*b^4*c^5*f^2 - 120*a^4*b^2*c^5*f^2 - 160*a^3*
b^3*d^5*f^2 - 80*a^6*c^3*d^2*f^2 + 80*b^6*c^3*d^2*f^2 - 1200*a^2*b^4*c^3*d^2*f^2 + 1600*a^3*b^3*c^2*d^3*f^2 +
1200*a^4*b^2*c^3*d^2*f^2 + 240*a*b^5*c^4*d*f^2 + 240*a^5*b*c^4*d*f^2 - 480*a*b^5*c^2*d^3*f^2 + 600*a^2*b^4*c*d
^4*f^2 - 800*a^3*b^3*c^4*d*f^2 - 600*a^4*b^2*c*d^4*f^2 - 480*a^5*b*c^2*d^3*f^2)^2/4 - (16*c^10*f^4 + 16*d^10*f
^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b
^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) - 4*a^6*c^5*f^2 + 4*b^6*c^5*f^2 - 24*a*b^5*d^5*f^2 - 24*a^5*
b*d^5*f^2 - 20*a^6*c*d^4*f^2 + 20*b^6*c*d^4*f^2 - 60*a^2*b^4*c^5*f^2 + 60*a^4*b^2*c^5*f^2 + 80*a^3*b^3*d^5*f^2
+ 40*a^6*c^3*d^2*f^2 - 40*b^6*c^3*d^2*f^2 + 600*a^2*b^4*c^3*d^2*f^2 - 800*a^3*b^3*c^2*d^3*f^2 - 600*a^4*b^2*c
^3*d^2*f^2 - 120*a*b^5*c^4*d*f^2 - 120*a^5*b*c^4*d*f^2 + 240*a*b^5*c^2*d^3*f^2 - 300*a^2*b^4*c*d^4*f^2 + 400*a
^3*b^3*c^4*d*f^2 + 300*a^4*b^2*c*d^4*f^2 + 240*a^5*b*c^2*d^3*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 1
0*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4)))^(1/2)*1i + (((((8*a^6*c^5*f^2 - 8*b^6*c^5*f^2 + 48*a*b^5*d^5
*f^2 + 48*a^5*b*d^5*f^2 + 40*a^6*c*d^4*f^2 - 40*b^6*c*d^4*f^2 + 120*a^2*b^4*c^5*f^2 - 120*a^4*b^2*c^5*f^2 - 16
0*a^3*b^3*d^5*f^2 - 80*a^6*c^3*d^2*f^2 + 80*b^6*c^3*d^2*f^2 - 1200*a^2*b^4*c^3*d^2*f^2 + 1600*a^3*b^3*c^2*d^3*
f^2 + 1200*a^4*b^2*c^3*d^2*f^2 + 240*a*b^5*c^4*d*f^2 + 240*a^5*b*c^4*d*f^2 - 480*a*b^5*c^2*d^3*f^2 + 600*a^2*b
^4*c*d^4*f^2 - 800*a^3*b^3*c^4*d*f^2 - 600*a^4*b^2*c*d^4*f^2 - 480*a^5*b*c^2*d^3*f^2)^2/4 - (16*c^10*f^4 + 16*
d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15
*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) - 4*a^6*c^5*f^2 + 4*b^6*c^5*f^2 - 24*a*b^5*d^5*f^2 - 2
4*a^5*b*d^5*f^2 - 20*a^6*c*d^4*f^2 + 20*b^6*c*d^4*f^2 - 60*a^2*b^4*c^5*f^2 + 60*a^4*b^2*c^5*f^2 + 80*a^3*b^3*d
^5*f^2 + 40*a^6*c^3*d^2*f^2 - 40*b^6*c^3*d^2*f^2 + 600*a^2*b^4*c^3*d^2*f^2 - 800*a^3*b^3*c^2*d^3*f^2 - 600*a^4
*b^2*c^3*d^2*f^2 - 120*a*b^5*c^4*d*f^2 - 120*a^5*b*c^4*d*f^2 + 240*a*b^5*c^2*d^3*f^2 - 300*a^2*b^4*c*d^4*f^2 +
400*a^3*b^3*c^4*d*f^2 + 300*a^4*b^2*c*d^4*f^2 + 240*a^5*b*c^2*d^3*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f
^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4)))^(1/2)*(32*a^3*d^21*f^4 + (c + d*tan(e + f*x))^(1/2)*((
((8*a^6*c^5*f^2 - 8*b^6*c^5*f^2 + 48*a*b^5*d^5*f^2 + 48*a^5*b*d^5*f^2 + 40*a^6*c*d^4*f^2 - 40*b^6*c*d^4*f^2 +
120*a^2*b^4*c^5*f^2 - 120*a^4*b^2*c^5*f^2 - 160*a^3*b^3*d^5*f^2 - 80*a^6*c^3*d^2*f^2 + 80*b^6*c^3*d^2*f^2 - 12
00*a^2*b^4*c^3*d^2*f^2 + 1600*a^3*b^3*c^2*d^3*f^2 + 1200*a^4*b^2*c^3*d^2*f^2 + 240*a*b^5*c^4*d*f^2 + 240*a^5*b
*c^4*d*f^2 - 480*a*b^5*c^2*d^3*f^2 + 600*a^2*b^4*c*d^4*f^2 - 800*a^3*b^3*c^4*d*f^2 - 600*a^4*b^2*c*d^4*f^2 - 4
80*a^5*b*c^2*d^3*f^2)^2/4 - (16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 +
80*c^8*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) - 4*a^6*
c^5*f^2 + 4*b^6*c^5*f^2 - 24*a*b^5*d^5*f^2 - 24*a^5*b*d^5*f^2 - 20*a^6*c*d^4*f^2 + 20*b^6*c*d^4*f^2 - 60*a^2*b
^4*c^5*f^2 + 60*a^4*b^2*c^5*f^2 + 80*a^3*b^3*d^5*f^2 + 40*a^6*c^3*d^2*f^2 - 40*b^6*c^3*d^2*f^2 + 600*a^2*b^4*c
^3*d^2*f^2 - 800*a^3*b^3*c^2*d^3*f^2 - 600*a^4*b^2*c^3*d^2*f^2 - 120*a*b^5*c^4*d*f^2 - 120*a^5*b*c^4*d*f^2 + 2
40*a*b^5*c^2*d^3*f^2 - 300*a^2*b^4*c*d^4*f^2 + 400*a^3*b^3*c^4*d*f^2 + 300*a^4*b^2*c*d^4*f^2 + 240*a^5*b*c^2*d
^3*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4)))^(1/2)*(6
4*c*d^22*f^5 + 640*c^3*d^20*f^5 + 2880*c^5*d^18*f^5 + 7680*c^7*d^16*f^5 + 13440*c^9*d^14*f^5 + 16128*c^11*d^12
*f^5 + 13440*c^13*d^10*f^5 + 7680*c^15*d^8*f^5 + 2880*c^17*d^6*f^5 + 640*c^19*d^4*f^5 + 64*c^21*d^2*f^5) - 96*
a*b^2*d^21*f^4 + 96*b^3*c*d^20*f^4 + 160*a^3*c^2*d^19*f^4 + 128*a^3*c^4*d^17*f^4 - 896*a^3*c^6*d^15*f^4 - 3136
*a^3*c^8*d^13*f^4 - 4928*a^3*c^10*d^11*f^4 - 4480*a^3*c^12*d^9*f^4 - 2432*a^3*c^14*d^7*f^4 - 736*a^3*c^16*d^5*
f^4 - 96*a^3*c^18*d^3*f^4 + 736*b^3*c^3*d^18*f^4 + 2432*b^3*c^5*d^16*f^4 + 4480*b^3*c^7*d^14*f^4 + 4928*b^3*c^
9*d^12*f^4 + 3136*b^3*c^11*d^10*f^4 + 896*b^3*c^13*d^8*f^4 - 128*b^3*c^15*d^6*f^4 - 160*b^3*c^17*d^4*f^4 - 32*
b^3*c^19*d^2*f^4 - 288*a^2*b*c*d^20*f^4 - 480*a*b^2*c^2*d^19*f^4 - 384*a*b^2*c^4*d^17*f^4 + 2688*a*b^2*c^6*d^1
5*f^4 + 9408*a*b^2*c^8*d^13*f^4 + 14784*a*b^2*c^10*d^11*f^4 + 13440*a*b^2*c^12*d^9*f^4 + 7296*a*b^2*c^14*d^7*f
^4 + 2208*a*b^2*c^16*d^5*f^4 + 288*a*b^2*c^18*d^3*f^4 - 2208*a^2*b*c^3*d^18*f^4 - 7296*a^2*b*c^5*d^16*f^4 - 13
440*a^2*b*c^7*d^14*f^4 - 14784*a^2*b*c^9*d^12*f^4 - 9408*a^2*b*c^11*d^10*f^4 - 2688*a^2*b*c^13*d^8*f^4 + 384*a
^2*b*c^15*d^6*f^4 + 480*a^2*b*c^17*d^4*f^4 + 96*a^2*b*c^19*d^2*f^4) + (c + d*tan(e + f*x))^(1/2)*(16*a^6*d^18*
f^3 - 16*b^6*d^18*f^3 + 240*a^2*b^4*d^18*f^3 - 240*a^4*b^2*d^18*f^3 - 320*a^6*c^4*d^14*f^3 - 1024*a^6*c^6*d^12
*f^3 - 1440*a^6*c^8*d^10*f^3 - 1024*a^6*c^10*d^8*f^3 - 320*a^6*c^12*d^6*f^3 + 16*a^6*c^16*d^2*f^3 + 320*b^6*c^
4*d^14*f^3 + 1024*b^6*c^6*d^12*f^3 + 1440*b^6*c^8*d^10*f^3 + 1024*b^6*c^10*d^8*f^3 + 320*b^6*c^12*d^6*f^3 - 16
*b^6*c^16*d^2*f^3 - 4800*a^2*b^4*c^4*d^14*f^3 - 15360*a^2*b^4*c^6*d^12*f^3 - 21600*a^2*b^4*c^8*d^10*f^3 - 1536
0*a^2*b^4*c^10*d^8*f^3 - 4800*a^2*b^4*c^12*d^6*f^3 + 240*a^2*b^4*c^16*d^2*f^3 + 6400*a^3*b^3*c^3*d^15*f^3 + 11
520*a^3*b^3*c^5*d^13*f^3 + 6400*a^3*b^3*c^7*d^11*f^3 - 6400*a^3*b^3*c^9*d^9*f^3 - 11520*a^3*b^3*c^11*d^7*f^3 -
6400*a^3*b^3*c^13*d^5*f^3 - 1280*a^3*b^3*c^15*d^3*f^3 + 4800*a^4*b^2*c^4*d^14*f^3 + 15360*a^4*b^2*c^6*d^12*f^
3 + 21600*a^4*b^2*c^8*d^10*f^3 + 15360*a^4*b^2*c^10*d^8*f^3 + 4800*a^4*b^2*c^12*d^6*f^3 - 240*a^4*b^2*c^16*d^2
*f^3 - 384*a*b^5*c*d^17*f^3 - 384*a^5*b*c*d^17*f^3 - 1920*a*b^5*c^3*d^15*f^3 - 3456*a*b^5*c^5*d^13*f^3 - 1920*
a*b^5*c^7*d^11*f^3 + 1920*a*b^5*c^9*d^9*f^3 + 3456*a*b^5*c^11*d^7*f^3 + 1920*a*b^5*c^13*d^5*f^3 + 384*a*b^5*c^
15*d^3*f^3 + 1280*a^3*b^3*c*d^17*f^3 - 1920*a^5*b*c^3*d^15*f^3 - 3456*a^5*b*c^5*d^13*f^3 - 1920*a^5*b*c^7*d^11
*f^3 + 1920*a^5*b*c^9*d^9*f^3 + 3456*a^5*b*c^11*d^7*f^3 + 1920*a^5*b*c^13*d^5*f^3 + 384*a^5*b*c^15*d^3*f^3))*(
(((8*a^6*c^5*f^2 - 8*b^6*c^5*f^2 + 48*a*b^5*d^5*f^2 + 48*a^5*b*d^5*f^2 + 40*a^6*c*d^4*f^2 - 40*b^6*c*d^4*f^2 +
120*a^2*b^4*c^5*f^2 - 120*a^4*b^2*c^5*f^2 - 160*a^3*b^3*d^5*f^2 - 80*a^6*c^3*d^2*f^2 + 80*b^6*c^3*d^2*f^2 - 1
200*a^2*b^4*c^3*d^2*f^2 + 1600*a^3*b^3*c^2*d^3*f^2 + 1200*a^4*b^2*c^3*d^2*f^2 + 240*a*b^5*c^4*d*f^2 + 240*a^5*
b*c^4*d*f^2 - 480*a*b^5*c^2*d^3*f^2 + 600*a^2*b^4*c*d^4*f^2 - 800*a^3*b^3*c^4*d*f^2 - 600*a^4*b^2*c*d^4*f^2 -
480*a^5*b*c^2*d^3*f^2)^2/4 - (16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 +
80*c^8*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) - 4*a^6
*c^5*f^2 + 4*b^6*c^5*f^2 - 24*a*b^5*d^5*f^2 - 24*a^5*b*d^5*f^2 - 20*a^6*c*d^4*f^2 + 20*b^6*c*d^4*f^2 - 60*a^2*
b^4*c^5*f^2 + 60*a^4*b^2*c^5*f^2 + 80*a^3*b^3*d^5*f^2 + 40*a^6*c^3*d^2*f^2 - 40*b^6*c^3*d^2*f^2 + 600*a^2*b^4*
c^3*d^2*f^2 - 800*a^3*b^3*c^2*d^3*f^2 - 600*a^4*b^2*c^3*d^2*f^2 - 120*a*b^5*c^4*d*f^2 - 120*a^5*b*c^4*d*f^2 +
240*a*b^5*c^2*d^3*f^2 - 300*a^2*b^4*c*d^4*f^2 + 400*a^3*b^3*c^4*d*f^2 + 300*a^4*b^2*c*d^4*f^2 + 240*a^5*b*c^2*
d^3*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4)))^(1/2)*1
i)/((((((8*a^6*c^5*f^2 - 8*b^6*c^5*f^2 + 48*a*b^5*d^5*f^2 + 48*a^5*b*d^5*f^2 + 40*a^6*c*d^4*f^2 - 40*b^6*c*d^4
*f^2 + 120*a^2*b^4*c^5*f^2 - 120*a^4*b^2*c^5*f^2 - 160*a^3*b^3*d^5*f^2 - 80*a^6*c^3*d^2*f^2 + 80*b^6*c^3*d^2*f
^2 - 1200*a^2*b^4*c^3*d^2*f^2 + 1600*a^3*b^3*c^2*d^3*f^2 + 1200*a^4*b^2*c^3*d^2*f^2 + 240*a*b^5*c^4*d*f^2 + 24
0*a^5*b*c^4*d*f^2 - 480*a*b^5*c^2*d^3*f^2 + 600*a^2*b^4*c*d^4*f^2 - 800*a^3*b^3*c^4*d*f^2 - 600*a^4*b^2*c*d^4*
f^2 - 480*a^5*b*c^2*d^3*f^2)^2/4 - (16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4
*f^4 + 80*c^8*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) -
4*a^6*c^5*f^2 + 4*b^6*c^5*f^2 - 24*a*b^5*d^5*f^2 - 24*a^5*b*d^5*f^2 - 20*a^6*c*d^4*f^2 + 20*b^6*c*d^4*f^2 - 6
0*a^2*b^4*c^5*f^2 + 60*a^4*b^2*c^5*f^2 + 80*a^3*b^3*d^5*f^2 + 40*a^6*c^3*d^2*f^2 - 40*b^6*c^3*d^2*f^2 + 600*a^
2*b^4*c^3*d^2*f^2 - 800*a^3*b^3*c^2*d^3*f^2 - 600*a^4*b^2*c^3*d^2*f^2 - 120*a*b^5*c^4*d*f^2 - 120*a^5*b*c^4*d*
f^2 + 240*a*b^5*c^2*d^3*f^2 - 300*a^2*b^4*c*d^4*f^2 + 400*a^3*b^3*c^4*d*f^2 + 300*a^4*b^2*c*d^4*f^2 + 240*a^5*
b*c^2*d^3*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4)))^(
1/2)*((c + d*tan(e + f*x))^(1/2)*((((8*a^6*c^5*f^2 - 8*b^6*c^5*f^2 + 48*a*b^5*d^5*f^2 + 48*a^5*b*d^5*f^2 + 40*
a^6*c*d^4*f^2 - 40*b^6*c*d^4*f^2 + 120*a^2*b^4*c^5*f^2 - 120*a^4*b^2*c^5*f^2 - 160*a^3*b^3*d^5*f^2 - 80*a^6*c^
3*d^2*f^2 + 80*b^6*c^3*d^2*f^2 - 1200*a^2*b^4*c^3*d^2*f^2 + 1600*a^3*b^3*c^2*d^3*f^2 + 1200*a^4*b^2*c^3*d^2*f^
2 + 240*a*b^5*c^4*d*f^2 + 240*a^5*b*c^4*d*f^2 - 480*a*b^5*c^2*d^3*f^2 + 600*a^2*b^4*c*d^4*f^2 - 800*a^3*b^3*c^
4*d*f^2 - 600*a^4*b^2*c*d^4*f^2 - 480*a^5*b*c^2*d^3*f^2)^2/4 - (16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 1
60*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^
8*b^4 + 6*a^10*b^2))^(1/2) - 4*a^6*c^5*f^2 + 4*b^6*c^5*f^2 - 24*a*b^5*d^5*f^2 - 24*a^5*b*d^5*f^2 - 20*a^6*c*d^
4*f^2 + 20*b^6*c*d^4*f^2 - 60*a^2*b^4*c^5*f^2 + 60*a^4*b^2*c^5*f^2 + 80*a^3*b^3*d^5*f^2 + 40*a^6*c^3*d^2*f^2 -
40*b^6*c^3*d^2*f^2 + 600*a^2*b^4*c^3*d^2*f^2 - 800*a^3*b^3*c^2*d^3*f^2 - 600*a^4*b^2*c^3*d^2*f^2 - 120*a*b^5*
c^4*d*f^2 - 120*a^5*b*c^4*d*f^2 + 240*a*b^5*c^2*d^3*f^2 - 300*a^2*b^4*c*d^4*f^2 + 400*a^3*b^3*c^4*d*f^2 + 300*
a^4*b^2*c*d^4*f^2 + 240*a^5*b*c^2*d^3*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*
d^4*f^4 + 5*c^8*d^2*f^4)))^(1/2)*(64*c*d^22*f^5 + 640*c^3*d^20*f^5 + 2880*c^5*d^18*f^5 + 7680*c^7*d^16*f^5 + 1
3440*c^9*d^14*f^5 + 16128*c^11*d^12*f^5 + 13440*c^13*d^10*f^5 + 7680*c^15*d^8*f^5 + 2880*c^17*d^6*f^5 + 640*c^
19*d^4*f^5 + 64*c^21*d^2*f^5) - 32*a^3*d^21*f^4 + 96*a*b^2*d^21*f^4 - 96*b^3*c*d^20*f^4 - 160*a^3*c^2*d^19*f^4
- 128*a^3*c^4*d^17*f^4 + 896*a^3*c^6*d^15*f^4 + 3136*a^3*c^8*d^13*f^4 + 4928*a^3*c^10*d^11*f^4 + 4480*a^3*c^1
2*d^9*f^4 + 2432*a^3*c^14*d^7*f^4 + 736*a^3*c^16*d^5*f^4 + 96*a^3*c^18*d^3*f^4 - 736*b^3*c^3*d^18*f^4 - 2432*b
^3*c^5*d^16*f^4 - 4480*b^3*c^7*d^14*f^4 - 4928*b^3*c^9*d^12*f^4 - 3136*b^3*c^11*d^10*f^4 - 896*b^3*c^13*d^8*f^
4 + 128*b^3*c^15*d^6*f^4 + 160*b^3*c^17*d^4*f^4 + 32*b^3*c^19*d^2*f^4 + 288*a^2*b*c*d^20*f^4 + 480*a*b^2*c^2*d
^19*f^4 + 384*a*b^2*c^4*d^17*f^4 - 2688*a*b^2*c^6*d^15*f^4 - 9408*a*b^2*c^8*d^13*f^4 - 14784*a*b^2*c^10*d^11*f
^4 - 13440*a*b^2*c^12*d^9*f^4 - 7296*a*b^2*c^14*d^7*f^4 - 2208*a*b^2*c^16*d^5*f^4 - 288*a*b^2*c^18*d^3*f^4 + 2
208*a^2*b*c^3*d^18*f^4 + 7296*a^2*b*c^5*d^16*f^4 + 13440*a^2*b*c^7*d^14*f^4 + 14784*a^2*b*c^9*d^12*f^4 + 9408*
a^2*b*c^11*d^10*f^4 + 2688*a^2*b*c^13*d^8*f^4 - 384*a^2*b*c^15*d^6*f^4 - 480*a^2*b*c^17*d^4*f^4 - 96*a^2*b*c^1
9*d^2*f^4) + (c + d*tan(e + f*x))^(1/2)*(16*a^6*d^18*f^3 - 16*b^6*d^18*f^3 + 240*a^2*b^4*d^18*f^3 - 240*a^4*b^
2*d^18*f^3 - 320*a^6*c^4*d^14*f^3 - 1024*a^6*c^6*d^12*f^3 - 1440*a^6*c^8*d^10*f^3 - 1024*a^6*c^10*d^8*f^3 - 32
0*a^6*c^12*d^6*f^3 + 16*a^6*c^16*d^2*f^3 + 320*b^6*c^4*d^14*f^3 + 1024*b^6*c^6*d^12*f^3 + 1440*b^6*c^8*d^10*f^
3 + 1024*b^6*c^10*d^8*f^3 + 320*b^6*c^12*d^6*f^3 - 16*b^6*c^16*d^2*f^3 - 4800*a^2*b^4*c^4*d^14*f^3 - 15360*a^2
*b^4*c^6*d^12*f^3 - 21600*a^2*b^4*c^8*d^10*f^3 - 15360*a^2*b^4*c^10*d^8*f^3 - 4800*a^2*b^4*c^12*d^6*f^3 + 240*
a^2*b^4*c^16*d^2*f^3 + 6400*a^3*b^3*c^3*d^15*f^3 + 11520*a^3*b^3*c^5*d^13*f^3 + 6400*a^3*b^3*c^7*d^11*f^3 - 64
00*a^3*b^3*c^9*d^9*f^3 - 11520*a^3*b^3*c^11*d^7*f^3 - 6400*a^3*b^3*c^13*d^5*f^3 - 1280*a^3*b^3*c^15*d^3*f^3 +
4800*a^4*b^2*c^4*d^14*f^3 + 15360*a^4*b^2*c^6*d^12*f^3 + 21600*a^4*b^2*c^8*d^10*f^3 + 15360*a^4*b^2*c^10*d^8*f
^3 + 4800*a^4*b^2*c^12*d^6*f^3 - 240*a^4*b^2*c^16*d^2*f^3 - 384*a*b^5*c*d^17*f^3 - 384*a^5*b*c*d^17*f^3 - 1920
*a*b^5*c^3*d^15*f^3 - 3456*a*b^5*c^5*d^13*f^3 - 1920*a*b^5*c^7*d^11*f^3 + 1920*a*b^5*c^9*d^9*f^3 + 3456*a*b^5*
c^11*d^7*f^3 + 1920*a*b^5*c^13*d^5*f^3 + 384*a*b^5*c^15*d^3*f^3 + 1280*a^3*b^3*c*d^17*f^3 - 1920*a^5*b*c^3*d^1
5*f^3 - 3456*a^5*b*c^5*d^13*f^3 - 1920*a^5*b*c^7*d^11*f^3 + 1920*a^5*b*c^9*d^9*f^3 + 3456*a^5*b*c^11*d^7*f^3 +
1920*a^5*b*c^13*d^5*f^3 + 384*a^5*b*c^15*d^3*f^3))*((((8*a^6*c^5*f^2 - 8*b^6*c^5*f^2 + 48*a*b^5*d^5*f^2 + 48*
a^5*b*d^5*f^2 + 40*a^6*c*d^4*f^2 - 40*b^6*c*d^4*f^2 + 120*a^2*b^4*c^5*f^2 - 120*a^4*b^2*c^5*f^2 - 160*a^3*b^3*
d^5*f^2 - 80*a^6*c^3*d^2*f^2 + 80*b^6*c^3*d^2*f^2 - 1200*a^2*b^4*c^3*d^2*f^2 + 1600*a^3*b^3*c^2*d^3*f^2 + 1200
*a^4*b^2*c^3*d^2*f^2 + 240*a*b^5*c^4*d*f^2 + 240*a^5*b*c^4*d*f^2 - 480*a*b^5*c^2*d^3*f^2 + 600*a^2*b^4*c*d^4*f
^2 - 800*a^3*b^3*c^4*d*f^2 - 600*a^4*b^2*c*d^4*f^2 - 480*a^5*b*c^2*d^3*f^2)^2/4 - (16*c^10*f^4 + 16*d^10*f^4 +
80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 +
20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) - 4*a^6*c^5*f^2 + 4*b^6*c^5*f^2 - 24*a*b^5*d^5*f^2 - 24*a^5*b*d^
5*f^2 - 20*a^6*c*d^4*f^2 + 20*b^6*c*d^4*f^2 - 60*a^2*b^4*c^5*f^2 + 60*a^4*b^2*c^5*f^2 + 80*a^3*b^3*d^5*f^2 + 4
0*a^6*c^3*d^2*f^2 - 40*b^6*c^3*d^2*f^2 + 600*a^2*b^4*c^3*d^2*f^2 - 800*a^3*b^3*c^2*d^3*f^2 - 600*a^4*b^2*c^3*d
^2*f^2 - 120*a*b^5*c^4*d*f^2 - 120*a^5*b*c^4*d*f^2 + 240*a*b^5*c^2*d^3*f^2 - 300*a^2*b^4*c*d^4*f^2 + 400*a^3*b
^3*c^4*d*f^2 + 300*a^4*b^2*c*d^4*f^2 + 240*a^5*b*c^2*d^3*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^
4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4)))^(1/2) - (((((8*a^6*c^5*f^2 - 8*b^6*c^5*f^2 + 48*a*b^5*d^5*f^2 +
48*a^5*b*d^5*f^2 + 40*a^6*c*d^4*f^2 - 40*b^6*c*d^4*f^2 + 120*a^2*b^4*c^5*f^2 - 120*a^4*b^2*c^5*f^2 - 160*a^3*b
^3*d^5*f^2 - 80*a^6*c^3*d^2*f^2 + 80*b^6*c^3*d^2*f^2 - 1200*a^2*b^4*c^3*d^2*f^2 + 1600*a^3*b^3*c^2*d^3*f^2 + 1
200*a^4*b^2*c^3*d^2*f^2 + 240*a*b^5*c^4*d*f^2 + 240*a^5*b*c^4*d*f^2 - 480*a*b^5*c^2*d^3*f^2 + 600*a^2*b^4*c*d^
4*f^2 - 800*a^3*b^3*c^4*d*f^2 - 600*a^4*b^2*c*d^4*f^2 - 480*a^5*b*c^2*d^3*f^2)^2/4 - (16*c^10*f^4 + 16*d^10*f^
4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^
8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) - 4*a^6*c^5*f^2 + 4*b^6*c^5*f^2 - 24*a*b^5*d^5*f^2 - 24*a^5*b
*d^5*f^2 - 20*a^6*c*d^4*f^2 + 20*b^6*c*d^4*f^2 - 60*a^2*b^4*c^5*f^2 + 60*a^4*b^2*c^5*f^2 + 80*a^3*b^3*d^5*f^2
+ 40*a^6*c^3*d^2*f^2 - 40*b^6*c^3*d^2*f^2 + 600*a^2*b^4*c^3*d^2*f^2 - 800*a^3*b^3*c^2*d^3*f^2 - 600*a^4*b^2*c^
3*d^2*f^2 - 120*a*b^5*c^4*d*f^2 - 120*a^5*b*c^4*d*f^2 + 240*a*b^5*c^2*d^3*f^2 - 300*a^2*b^4*c*d^4*f^2 + 400*a^
3*b^3*c^4*d*f^2 + 300*a^4*b^2*c*d^4*f^2 + 240*a^5*b*c^2*d^3*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10
*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4)))^(1/2)*(32*a^3*d^21*f^4 + (c + d*tan(e + f*x))^(1/2)*((((8*a^6
*c^5*f^2 - 8*b^6*c^5*f^2 + 48*a*b^5*d^5*f^2 + 48*a^5*b*d^5*f^2 + 40*a^6*c*d^4*f^2 - 40*b^6*c*d^4*f^2 + 120*a^2
*b^4*c^5*f^2 - 120*a^4*b^2*c^5*f^2 - 160*a^3*b^3*d^5*f^2 - 80*a^6*c^3*d^2*f^2 + 80*b^6*c^3*d^2*f^2 - 1200*a^2*
b^4*c^3*d^2*f^2 + 1600*a^3*b^3*c^2*d^3*f^2 + 1200*a^4*b^2*c^3*d^2*f^2 + 240*a*b^5*c^4*d*f^2 + 240*a^5*b*c^4*d*
f^2 - 480*a*b^5*c^2*d^3*f^2 + 600*a^2*b^4*c*d^4*f^2 - 800*a^3*b^3*c^4*d*f^2 - 600*a^4*b^2*c*d^4*f^2 - 480*a^5*
b*c^2*d^3*f^2)^2/4 - (16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*
d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) - 4*a^6*c^5*f^2
+ 4*b^6*c^5*f^2 - 24*a*b^5*d^5*f^2 - 24*a^5*b*d^5*f^2 - 20*a^6*c*d^4*f^2 + 20*b^6*c*d^4*f^2 - 60*a^2*b^4*c^5*
f^2 + 60*a^4*b^2*c^5*f^2 + 80*a^3*b^3*d^5*f^2 + 40*a^6*c^3*d^2*f^2 - 40*b^6*c^3*d^2*f^2 + 600*a^2*b^4*c^3*d^2*
f^2 - 800*a^3*b^3*c^2*d^3*f^2 - 600*a^4*b^2*c^3*d^2*f^2 - 120*a*b^5*c^4*d*f^2 - 120*a^5*b*c^4*d*f^2 + 240*a*b^
5*c^2*d^3*f^2 - 300*a^2*b^4*c*d^4*f^2 + 400*a^3*b^3*c^4*d*f^2 + 300*a^4*b^2*c*d^4*f^2 + 240*a^5*b*c^2*d^3*f^2)
/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4)))^(1/2)*(64*c*d^2
2*f^5 + 640*c^3*d^20*f^5 + 2880*c^5*d^18*f^5 + 7680*c^7*d^16*f^5 + 13440*c^9*d^14*f^5 + 16128*c^11*d^12*f^5 +
13440*c^13*d^10*f^5 + 7680*c^15*d^8*f^5 + 2880*c^17*d^6*f^5 + 640*c^19*d^4*f^5 + 64*c^21*d^2*f^5) - 96*a*b^2*d
^21*f^4 + 96*b^3*c*d^20*f^4 + 160*a^3*c^2*d^19*f^4 + 128*a^3*c^4*d^17*f^4 - 896*a^3*c^6*d^15*f^4 - 3136*a^3*c^
8*d^13*f^4 - 4928*a^3*c^10*d^11*f^4 - 4480*a^3*c^12*d^9*f^4 - 2432*a^3*c^14*d^7*f^4 - 736*a^3*c^16*d^5*f^4 - 9
6*a^3*c^18*d^3*f^4 + 736*b^3*c^3*d^18*f^4 + 2432*b^3*c^5*d^16*f^4 + 4480*b^3*c^7*d^14*f^4 + 4928*b^3*c^9*d^12*
f^4 + 3136*b^3*c^11*d^10*f^4 + 896*b^3*c^13*d^8*f^4 - 128*b^3*c^15*d^6*f^4 - 160*b^3*c^17*d^4*f^4 - 32*b^3*c^1
9*d^2*f^4 - 288*a^2*b*c*d^20*f^4 - 480*a*b^2*c^2*d^19*f^4 - 384*a*b^2*c^4*d^17*f^4 + 2688*a*b^2*c^6*d^15*f^4 +
9408*a*b^2*c^8*d^13*f^4 + 14784*a*b^2*c^10*d^11*f^4 + 13440*a*b^2*c^12*d^9*f^4 + 7296*a*b^2*c^14*d^7*f^4 + 22
08*a*b^2*c^16*d^5*f^4 + 288*a*b^2*c^18*d^3*f^4 - 2208*a^2*b*c^3*d^18*f^4 - 7296*a^2*b*c^5*d^16*f^4 - 13440*a^2
*b*c^7*d^14*f^4 - 14784*a^2*b*c^9*d^12*f^4 - 9408*a^2*b*c^11*d^10*f^4 - 2688*a^2*b*c^13*d^8*f^4 + 384*a^2*b*c^
15*d^6*f^4 + 480*a^2*b*c^17*d^4*f^4 + 96*a^2*b*c^19*d^2*f^4) + (c + d*tan(e + f*x))^(1/2)*(16*a^6*d^18*f^3 - 1
6*b^6*d^18*f^3 + 240*a^2*b^4*d^18*f^3 - 240*a^4*b^2*d^18*f^3 - 320*a^6*c^4*d^14*f^3 - 1024*a^6*c^6*d^12*f^3 -
1440*a^6*c^8*d^10*f^3 - 1024*a^6*c^10*d^8*f^3 - 320*a^6*c^12*d^6*f^3 + 16*a^6*c^16*d^2*f^3 + 320*b^6*c^4*d^14*
f^3 + 1024*b^6*c^6*d^12*f^3 + 1440*b^6*c^8*d^10*f^3 + 1024*b^6*c^10*d^8*f^3 + 320*b^6*c^12*d^6*f^3 - 16*b^6*c^
16*d^2*f^3 - 4800*a^2*b^4*c^4*d^14*f^3 - 15360*a^2*b^4*c^6*d^12*f^3 - 21600*a^2*b^4*c^8*d^10*f^3 - 15360*a^2*b
^4*c^10*d^8*f^3 - 4800*a^2*b^4*c^12*d^6*f^3 + 240*a^2*b^4*c^16*d^2*f^3 + 6400*a^3*b^3*c^3*d^15*f^3 + 11520*a^3
*b^3*c^5*d^13*f^3 + 6400*a^3*b^3*c^7*d^11*f^3 - 6400*a^3*b^3*c^9*d^9*f^3 - 11520*a^3*b^3*c^11*d^7*f^3 - 6400*a
^3*b^3*c^13*d^5*f^3 - 1280*a^3*b^3*c^15*d^3*f^3 + 4800*a^4*b^2*c^4*d^14*f^3 + 15360*a^4*b^2*c^6*d^12*f^3 + 216
00*a^4*b^2*c^8*d^10*f^3 + 15360*a^4*b^2*c^10*d^8*f^3 + 4800*a^4*b^2*c^12*d^6*f^3 - 240*a^4*b^2*c^16*d^2*f^3 -
384*a*b^5*c*d^17*f^3 - 384*a^5*b*c*d^17*f^3 - 1920*a*b^5*c^3*d^15*f^3 - 3456*a*b^5*c^5*d^13*f^3 - 1920*a*b^5*c
^7*d^11*f^3 + 1920*a*b^5*c^9*d^9*f^3 + 3456*a*b^5*c^11*d^7*f^3 + 1920*a*b^5*c^13*d^5*f^3 + 384*a*b^5*c^15*d^3*
f^3 + 1280*a^3*b^3*c*d^17*f^3 - 1920*a^5*b*c^3*d^15*f^3 - 3456*a^5*b*c^5*d^13*f^3 - 1920*a^5*b*c^7*d^11*f^3 +
1920*a^5*b*c^9*d^9*f^3 + 3456*a^5*b*c^11*d^7*f^3 + 1920*a^5*b*c^13*d^5*f^3 + 384*a^5*b*c^15*d^3*f^3))*((((8*a^
6*c^5*f^2 - 8*b^6*c^5*f^2 + 48*a*b^5*d^5*f^2 + 48*a^5*b*d^5*f^2 + 40*a^6*c*d^4*f^2 - 40*b^6*c*d^4*f^2 + 120*a^
2*b^4*c^5*f^2 - 120*a^4*b^2*c^5*f^2 - 160*a^3*b^3*d^5*f^2 - 80*a^6*c^3*d^2*f^2 + 80*b^6*c^3*d^2*f^2 - 1200*a^2
*b^4*c^3*d^2*f^2 + 1600*a^3*b^3*c^2*d^3*f^2 + 1200*a^4*b^2*c^3*d^2*f^2 + 240*a*b^5*c^4*d*f^2 + 240*a^5*b*c^4*d
*f^2 - 480*a*b^5*c^2*d^3*f^2 + 600*a^2*b^4*c*d^4*f^2 - 800*a^3*b^3*c^4*d*f^2 - 600*a^4*b^2*c*d^4*f^2 - 480*a^5
*b*c^2*d^3*f^2)^2/4 - (16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8
*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) - 4*a^6*c^5*f^
2 + 4*b^6*c^5*f^2 - 24*a*b^5*d^5*f^2 - 24*a^5*b*d^5*f^2 - 20*a^6*c*d^4*f^2 + 20*b^6*c*d^4*f^2 - 60*a^2*b^4*c^5
*f^2 + 60*a^4*b^2*c^5*f^2 + 80*a^3*b^3*d^5*f^2 + 40*a^6*c^3*d^2*f^2 - 40*b^6*c^3*d^2*f^2 + 600*a^2*b^4*c^3*d^2
*f^2 - 800*a^3*b^3*c^2*d^3*f^2 - 600*a^4*b^2*c^3*d^2*f^2 - 120*a*b^5*c^4*d*f^2 - 120*a^5*b*c^4*d*f^2 + 240*a*b
^5*c^2*d^3*f^2 - 300*a^2*b^4*c*d^4*f^2 + 400*a^3*b^3*c^4*d*f^2 + 300*a^4*b^2*c*d^4*f^2 + 240*a^5*b*c^2*d^3*f^2
)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4)))^(1/2) + 16*b^9
*d^16*f^2 - 48*a^8*b*d^16*f^2 - 32*a^9*c*d^15*f^2 - 96*a^4*b^5*d^16*f^2 - 128*a^6*b^3*d^16*f^2 - 192*a^9*c^3*d
^13*f^2 - 480*a^9*c^5*d^11*f^2 - 640*a^9*c^7*d^9*f^2 - 480*a^9*c^9*d^7*f^2 - 192*a^9*c^11*d^5*f^2 - 32*a^9*c^1
3*d^3*f^2 + 80*b^9*c^2*d^14*f^2 + 144*b^9*c^4*d^12*f^2 + 80*b^9*c^6*d^10*f^2 - 80*b^9*c^8*d^8*f^2 - 144*b^9*c^
10*d^6*f^2 - 80*b^9*c^12*d^4*f^2 - 16*b^9*c^14*d^2*f^2 + 1536*a^3*b^6*c^3*d^13*f^2 + 3840*a^3*b^6*c^5*d^11*f^2
+ 5120*a^3*b^6*c^7*d^9*f^2 + 3840*a^3*b^6*c^9*d^7*f^2 + 1536*a^3*b^6*c^11*d^5*f^2 + 256*a^3*b^6*c^13*d^3*f^2
- 480*a^4*b^5*c^2*d^14*f^2 - 864*a^4*b^5*c^4*d^12*f^2 - 480*a^4*b^5*c^6*d^10*f^2 + 480*a^4*b^5*c^8*d^8*f^2 + 8
64*a^4*b^5*c^10*d^6*f^2 + 480*a^4*b^5*c^12*d^4*f^2 + 96*a^4*b^5*c^14*d^2*f^2 + 1152*a^5*b^4*c^3*d^13*f^2 + 288
0*a^5*b^4*c^5*d^11*f^2 + 3840*a^5*b^4*c^7*d^9*f^2 + 2880*a^5*b^4*c^9*d^7*f^2 + 1152*a^5*b^4*c^11*d^5*f^2 + 192
*a^5*b^4*c^13*d^3*f^2 - 640*a^6*b^3*c^2*d^14*f^2 - 1152*a^6*b^3*c^4*d^12*f^2 - 640*a^6*b^3*c^6*d^10*f^2 + 640*
a^6*b^3*c^8*d^8*f^2 + 1152*a^6*b^3*c^10*d^6*f^2 + 640*a^6*b^3*c^12*d^4*f^2 + 128*a^6*b^3*c^14*d^2*f^2 + 96*a*b
^8*c*d^15*f^2 + 576*a*b^8*c^3*d^13*f^2 + 1440*a*b^8*c^5*d^11*f^2 + 1920*a*b^8*c^7*d^9*f^2 + 1440*a*b^8*c^9*d^7
*f^2 + 576*a*b^8*c^11*d^5*f^2 + 96*a*b^8*c^13*d^3*f^2 + 256*a^3*b^6*c*d^15*f^2 + 192*a^5*b^4*c*d^15*f^2 - 240*
a^8*b*c^2*d^14*f^2 - 432*a^8*b*c^4*d^12*f^2 - 240*a^8*b*c^6*d^10*f^2 + 240*a^8*b*c^8*d^8*f^2 + 432*a^8*b*c^10*
d^6*f^2 + 240*a^8*b*c^12*d^4*f^2 + 48*a^8*b*c^14*d^2*f^2))*((((8*a^6*c^5*f^2 - 8*b^6*c^5*f^2 + 48*a*b^5*d^5*f^
2 + 48*a^5*b*d^5*f^2 + 40*a^6*c*d^4*f^2 - 40*b^6*c*d^4*f^2 + 120*a^2*b^4*c^5*f^2 - 120*a^4*b^2*c^5*f^2 - 160*a
^3*b^3*d^5*f^2 - 80*a^6*c^3*d^2*f^2 + 80*b^6*c^3*d^2*f^2 - 1200*a^2*b^4*c^3*d^2*f^2 + 1600*a^3*b^3*c^2*d^3*f^2
+ 1200*a^4*b^2*c^3*d^2*f^2 + 240*a*b^5*c^4*d*f^2 + 240*a^5*b*c^4*d*f^2 - 480*a*b^5*c^2*d^3*f^2 + 600*a^2*b^4*
c*d^4*f^2 - 800*a^3*b^3*c^4*d*f^2 - 600*a^4*b^2*c*d^4*f^2 - 480*a^5*b*c^2*d^3*f^2)^2/4 - (16*c^10*f^4 + 16*d^1
0*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^
4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) - 4*a^6*c^5*f^2 + 4*b^6*c^5*f^2 - 24*a*b^5*d^5*f^2 - 24*a
^5*b*d^5*f^2 - 20*a^6*c*d^4*f^2 + 20*b^6*c*d^4*f^2 - 60*a^2*b^4*c^5*f^2 + 60*a^4*b^2*c^5*f^2 + 80*a^3*b^3*d^5*
f^2 + 40*a^6*c^3*d^2*f^2 - 40*b^6*c^3*d^2*f^2 + 600*a^2*b^4*c^3*d^2*f^2 - 800*a^3*b^3*c^2*d^3*f^2 - 600*a^4*b^
2*c^3*d^2*f^2 - 120*a*b^5*c^4*d*f^2 - 120*a^5*b*c^4*d*f^2 + 240*a*b^5*c^2*d^3*f^2 - 300*a^2*b^4*c*d^4*f^2 + 40
0*a^3*b^3*c^4*d*f^2 + 300*a^4*b^2*c*d^4*f^2 + 240*a^5*b*c^2*d^3*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4
+ 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4)))^(1/2)*2i - atan((((-(((8*a^6*c^5*f^2 - 8*b^6*c^5*f^2 + 48
*a*b^5*d^5*f^2 + 48*a^5*b*d^5*f^2 + 40*a^6*c*d^4*f^2 - 40*b^6*c*d^4*f^2 + 120*a^2*b^4*c^5*f^2 - 120*a^4*b^2*c^
5*f^2 - 160*a^3*b^3*d^5*f^2 - 80*a^6*c^3*d^2*f^2 + 80*b^6*c^3*d^2*f^2 - 1200*a^2*b^4*c^3*d^2*f^2 + 1600*a^3*b^
3*c^2*d^3*f^2 + 1200*a^4*b^2*c^3*d^2*f^2 + 240*a*b^5*c^4*d*f^2 + 240*a^5*b*c^4*d*f^2 - 480*a*b^5*c^2*d^3*f^2 +
600*a^2*b^4*c*d^4*f^2 - 800*a^3*b^3*c^4*d*f^2 - 600*a^4*b^2*c*d^4*f^2 - 480*a^5*b*c^2*d^3*f^2)^2/4 - (16*c^10
*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4)*(a^12 + b^12 + 6*a^2
*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) + 4*a^6*c^5*f^2 - 4*b^6*c^5*f^2 + 24*a*b^5*d
^5*f^2 + 24*a^5*b*d^5*f^2 + 20*a^6*c*d^4*f^2 - 20*b^6*c*d^4*f^2 + 60*a^2*b^4*c^5*f^2 - 60*a^4*b^2*c^5*f^2 - 80
*a^3*b^3*d^5*f^2 - 40*a^6*c^3*d^2*f^2 + 40*b^6*c^3*d^2*f^2 - 600*a^2*b^4*c^3*d^2*f^2 + 800*a^3*b^3*c^2*d^3*f^2
+ 600*a^4*b^2*c^3*d^2*f^2 + 120*a*b^5*c^4*d*f^2 + 120*a^5*b*c^4*d*f^2 - 240*a*b^5*c^2*d^3*f^2 + 300*a^2*b^4*c
*d^4*f^2 - 400*a^3*b^3*c^4*d*f^2 - 300*a^4*b^2*c*d^4*f^2 - 240*a^5*b*c^2*d^3*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5
*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4)))^(1/2)*((c + d*tan(e + f*x))^(1/2)*(-(((8*a^6
*c^5*f^2 - 8*b^6*c^5*f^2 + 48*a*b^5*d^5*f^2 + 48*a^5*b*d^5*f^2 + 40*a^6*c*d^4*f^2 - 40*b^6*c*d^4*f^2 + 120*a^2
*b^4*c^5*f^2 - 120*a^4*b^2*c^5*f^2 - 160*a^3*b^3*d^5*f^2 - 80*a^6*c^3*d^2*f^2 + 80*b^6*c^3*d^2*f^2 - 1200*a^2*
b^4*c^3*d^2*f^2 + 1600*a^3*b^3*c^2*d^3*f^2 + 1200*a^4*b^2*c^3*d^2*f^2 + 240*a*b^5*c^4*d*f^2 + 240*a^5*b*c^4*d*
f^2 - 480*a*b^5*c^2*d^3*f^2 + 600*a^2*b^4*c*d^4*f^2 - 800*a^3*b^3*c^4*d*f^2 - 600*a^4*b^2*c*d^4*f^2 - 480*a^5*
b*c^2*d^3*f^2)^2/4 - (16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*
d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) + 4*a^6*c^5*f^2
- 4*b^6*c^5*f^2 + 24*a*b^5*d^5*f^2 + 24*a^5*b*d^5*f^2 + 20*a^6*c*d^4*f^2 - 20*b^6*c*d^4*f^2 + 60*a^2*b^4*c^5*
f^2 - 60*a^4*b^2*c^5*f^2 - 80*a^3*b^3*d^5*f^2 - 40*a^6*c^3*d^2*f^2 + 40*b^6*c^3*d^2*f^2 - 600*a^2*b^4*c^3*d^2*
f^2 + 800*a^3*b^3*c^2*d^3*f^2 + 600*a^4*b^2*c^3*d^2*f^2 + 120*a*b^5*c^4*d*f^2 + 120*a^5*b*c^4*d*f^2 - 240*a*b^
5*c^2*d^3*f^2 + 300*a^2*b^4*c*d^4*f^2 - 400*a^3*b^3*c^4*d*f^2 - 300*a^4*b^2*c*d^4*f^2 - 240*a^5*b*c^2*d^3*f^2)
/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4)))^(1/2)*(64*c*d^2
2*f^5 + 640*c^3*d^20*f^5 + 2880*c^5*d^18*f^5 + 7680*c^7*d^16*f^5 + 13440*c^9*d^14*f^5 + 16128*c^11*d^12*f^5 +
13440*c^13*d^10*f^5 + 7680*c^15*d^8*f^5 + 2880*c^17*d^6*f^5 + 640*c^19*d^4*f^5 + 64*c^21*d^2*f^5) - 32*a^3*d^2
1*f^4 + 96*a*b^2*d^21*f^4 - 96*b^3*c*d^20*f^4 - 160*a^3*c^2*d^19*f^4 - 128*a^3*c^4*d^17*f^4 + 896*a^3*c^6*d^15
*f^4 + 3136*a^3*c^8*d^13*f^4 + 4928*a^3*c^10*d^11*f^4 + 4480*a^3*c^12*d^9*f^4 + 2432*a^3*c^14*d^7*f^4 + 736*a^
3*c^16*d^5*f^4 + 96*a^3*c^18*d^3*f^4 - 736*b^3*c^3*d^18*f^4 - 2432*b^3*c^5*d^16*f^4 - 4480*b^3*c^7*d^14*f^4 -
4928*b^3*c^9*d^12*f^4 - 3136*b^3*c^11*d^10*f^4 - 896*b^3*c^13*d^8*f^4 + 128*b^3*c^15*d^6*f^4 + 160*b^3*c^17*d^
4*f^4 + 32*b^3*c^19*d^2*f^4 + 288*a^2*b*c*d^20*f^4 + 480*a*b^2*c^2*d^19*f^4 + 384*a*b^2*c^4*d^17*f^4 - 2688*a*
b^2*c^6*d^15*f^4 - 9408*a*b^2*c^8*d^13*f^4 - 14784*a*b^2*c^10*d^11*f^4 - 13440*a*b^2*c^12*d^9*f^4 - 7296*a*b^2
*c^14*d^7*f^4 - 2208*a*b^2*c^16*d^5*f^4 - 288*a*b^2*c^18*d^3*f^4 + 2208*a^2*b*c^3*d^18*f^4 + 7296*a^2*b*c^5*d^
16*f^4 + 13440*a^2*b*c^7*d^14*f^4 + 14784*a^2*b*c^9*d^12*f^4 + 9408*a^2*b*c^11*d^10*f^4 + 2688*a^2*b*c^13*d^8*
f^4 - 384*a^2*b*c^15*d^6*f^4 - 480*a^2*b*c^17*d^4*f^4 - 96*a^2*b*c^19*d^2*f^4) + (c + d*tan(e + f*x))^(1/2)*(1
6*a^6*d^18*f^3 - 16*b^6*d^18*f^3 + 240*a^2*b^4*d^18*f^3 - 240*a^4*b^2*d^18*f^3 - 320*a^6*c^4*d^14*f^3 - 1024*a
^6*c^6*d^12*f^3 - 1440*a^6*c^8*d^10*f^3 - 1024*a^6*c^10*d^8*f^3 - 320*a^6*c^12*d^6*f^3 + 16*a^6*c^16*d^2*f^3 +
320*b^6*c^4*d^14*f^3 + 1024*b^6*c^6*d^12*f^3 + 1440*b^6*c^8*d^10*f^3 + 1024*b^6*c^10*d^8*f^3 + 320*b^6*c^12*d
^6*f^3 - 16*b^6*c^16*d^2*f^3 - 4800*a^2*b^4*c^4*d^14*f^3 - 15360*a^2*b^4*c^6*d^12*f^3 - 21600*a^2*b^4*c^8*d^10
*f^3 - 15360*a^2*b^4*c^10*d^8*f^3 - 4800*a^2*b^4*c^12*d^6*f^3 + 240*a^2*b^4*c^16*d^2*f^3 + 6400*a^3*b^3*c^3*d^
15*f^3 + 11520*a^3*b^3*c^5*d^13*f^3 + 6400*a^3*b^3*c^7*d^11*f^3 - 6400*a^3*b^3*c^9*d^9*f^3 - 11520*a^3*b^3*c^1
1*d^7*f^3 - 6400*a^3*b^3*c^13*d^5*f^3 - 1280*a^3*b^3*c^15*d^3*f^3 + 4800*a^4*b^2*c^4*d^14*f^3 + 15360*a^4*b^2*
c^6*d^12*f^3 + 21600*a^4*b^2*c^8*d^10*f^3 + 15360*a^4*b^2*c^10*d^8*f^3 + 4800*a^4*b^2*c^12*d^6*f^3 - 240*a^4*b
^2*c^16*d^2*f^3 - 384*a*b^5*c*d^17*f^3 - 384*a^5*b*c*d^17*f^3 - 1920*a*b^5*c^3*d^15*f^3 - 3456*a*b^5*c^5*d^13*
f^3 - 1920*a*b^5*c^7*d^11*f^3 + 1920*a*b^5*c^9*d^9*f^3 + 3456*a*b^5*c^11*d^7*f^3 + 1920*a*b^5*c^13*d^5*f^3 + 3
84*a*b^5*c^15*d^3*f^3 + 1280*a^3*b^3*c*d^17*f^3 - 1920*a^5*b*c^3*d^15*f^3 - 3456*a^5*b*c^5*d^13*f^3 - 1920*a^5
*b*c^7*d^11*f^3 + 1920*a^5*b*c^9*d^9*f^3 + 3456*a^5*b*c^11*d^7*f^3 + 1920*a^5*b*c^13*d^5*f^3 + 384*a^5*b*c^15*
d^3*f^3))*(-(((8*a^6*c^5*f^2 - 8*b^6*c^5*f^2 + 48*a*b^5*d^5*f^2 + 48*a^5*b*d^5*f^2 + 40*a^6*c*d^4*f^2 - 40*b^6
*c*d^4*f^2 + 120*a^2*b^4*c^5*f^2 - 120*a^4*b^2*c^5*f^2 - 160*a^3*b^3*d^5*f^2 - 80*a^6*c^3*d^2*f^2 + 80*b^6*c^3
*d^2*f^2 - 1200*a^2*b^4*c^3*d^2*f^2 + 1600*a^3*b^3*c^2*d^3*f^2 + 1200*a^4*b^2*c^3*d^2*f^2 + 240*a*b^5*c^4*d*f^
2 + 240*a^5*b*c^4*d*f^2 - 480*a*b^5*c^2*d^3*f^2 + 600*a^2*b^4*c*d^4*f^2 - 800*a^3*b^3*c^4*d*f^2 - 600*a^4*b^2*
c*d^4*f^2 - 480*a^5*b*c^2*d^3*f^2)^2/4 - (16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c
^6*d^4*f^4 + 80*c^8*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(
1/2) + 4*a^6*c^5*f^2 - 4*b^6*c^5*f^2 + 24*a*b^5*d^5*f^2 + 24*a^5*b*d^5*f^2 + 20*a^6*c*d^4*f^2 - 20*b^6*c*d^4*f
^2 + 60*a^2*b^4*c^5*f^2 - 60*a^4*b^2*c^5*f^2 - 80*a^3*b^3*d^5*f^2 - 40*a^6*c^3*d^2*f^2 + 40*b^6*c^3*d^2*f^2 -
600*a^2*b^4*c^3*d^2*f^2 + 800*a^3*b^3*c^2*d^3*f^2 + 600*a^4*b^2*c^3*d^2*f^2 + 120*a*b^5*c^4*d*f^2 + 120*a^5*b*
c^4*d*f^2 - 240*a*b^5*c^2*d^3*f^2 + 300*a^2*b^4*c*d^4*f^2 - 400*a^3*b^3*c^4*d*f^2 - 300*a^4*b^2*c*d^4*f^2 - 24
0*a^5*b*c^2*d^3*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^
4)))^(1/2)*1i + ((-(((8*a^6*c^5*f^2 - 8*b^6*c^5*f^2 + 48*a*b^5*d^5*f^2 + 48*a^5*b*d^5*f^2 + 40*a^6*c*d^4*f^2 -
40*b^6*c*d^4*f^2 + 120*a^2*b^4*c^5*f^2 - 120*a^4*b^2*c^5*f^2 - 160*a^3*b^3*d^5*f^2 - 80*a^6*c^3*d^2*f^2 + 80*
b^6*c^3*d^2*f^2 - 1200*a^2*b^4*c^3*d^2*f^2 + 1600*a^3*b^3*c^2*d^3*f^2 + 1200*a^4*b^2*c^3*d^2*f^2 + 240*a*b^5*c
^4*d*f^2 + 240*a^5*b*c^4*d*f^2 - 480*a*b^5*c^2*d^3*f^2 + 600*a^2*b^4*c*d^4*f^2 - 800*a^3*b^3*c^4*d*f^2 - 600*a
^4*b^2*c*d^4*f^2 - 480*a^5*b*c^2*d^3*f^2)^2/4 - (16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4
+ 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*
b^2))^(1/2) + 4*a^6*c^5*f^2 - 4*b^6*c^5*f^2 + 24*a*b^5*d^5*f^2 + 24*a^5*b*d^5*f^2 + 20*a^6*c*d^4*f^2 - 20*b^6*
c*d^4*f^2 + 60*a^2*b^4*c^5*f^2 - 60*a^4*b^2*c^5*f^2 - 80*a^3*b^3*d^5*f^2 - 40*a^6*c^3*d^2*f^2 + 40*b^6*c^3*d^2
*f^2 - 600*a^2*b^4*c^3*d^2*f^2 + 800*a^3*b^3*c^2*d^3*f^2 + 600*a^4*b^2*c^3*d^2*f^2 + 120*a*b^5*c^4*d*f^2 + 120
*a^5*b*c^4*d*f^2 - 240*a*b^5*c^2*d^3*f^2 + 300*a^2*b^4*c*d^4*f^2 - 400*a^3*b^3*c^4*d*f^2 - 300*a^4*b^2*c*d^4*f
^2 - 240*a^5*b*c^2*d^3*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8
*d^2*f^4)))^(1/2)*(32*a^3*d^21*f^4 + (c + d*tan(e + f*x))^(1/2)*(-(((8*a^6*c^5*f^2 - 8*b^6*c^5*f^2 + 48*a*b^5*
d^5*f^2 + 48*a^5*b*d^5*f^2 + 40*a^6*c*d^4*f^2 - 40*b^6*c*d^4*f^2 + 120*a^2*b^4*c^5*f^2 - 120*a^4*b^2*c^5*f^2 -
160*a^3*b^3*d^5*f^2 - 80*a^6*c^3*d^2*f^2 + 80*b^6*c^3*d^2*f^2 - 1200*a^2*b^4*c^3*d^2*f^2 + 1600*a^3*b^3*c^2*d
^3*f^2 + 1200*a^4*b^2*c^3*d^2*f^2 + 240*a*b^5*c^4*d*f^2 + 240*a^5*b*c^4*d*f^2 - 480*a*b^5*c^2*d^3*f^2 + 600*a^
2*b^4*c*d^4*f^2 - 800*a^3*b^3*c^4*d*f^2 - 600*a^4*b^2*c*d^4*f^2 - 480*a^5*b*c^2*d^3*f^2)^2/4 - (16*c^10*f^4 +
16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 +
15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) + 4*a^6*c^5*f^2 - 4*b^6*c^5*f^2 + 24*a*b^5*d^5*f^2
+ 24*a^5*b*d^5*f^2 + 20*a^6*c*d^4*f^2 - 20*b^6*c*d^4*f^2 + 60*a^2*b^4*c^5*f^2 - 60*a^4*b^2*c^5*f^2 - 80*a^3*b^
3*d^5*f^2 - 40*a^6*c^3*d^2*f^2 + 40*b^6*c^3*d^2*f^2 - 600*a^2*b^4*c^3*d^2*f^2 + 800*a^3*b^3*c^2*d^3*f^2 + 600*
a^4*b^2*c^3*d^2*f^2 + 120*a*b^5*c^4*d*f^2 + 120*a^5*b*c^4*d*f^2 - 240*a*b^5*c^2*d^3*f^2 + 300*a^2*b^4*c*d^4*f^
2 - 400*a^3*b^3*c^4*d*f^2 - 300*a^4*b^2*c*d^4*f^2 - 240*a^5*b*c^2*d^3*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^
8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4)))^(1/2)*(64*c*d^22*f^5 + 640*c^3*d^20*f^5 + 2880*c^5*
d^18*f^5 + 7680*c^7*d^16*f^5 + 13440*c^9*d^14*f^5 + 16128*c^11*d^12*f^5 + 13440*c^13*d^10*f^5 + 7680*c^15*d^8*
f^5 + 2880*c^17*d^6*f^5 + 640*c^19*d^4*f^5 + 64*c^21*d^2*f^5) - 96*a*b^2*d^21*f^4 + 96*b^3*c*d^20*f^4 + 160*a^
3*c^2*d^19*f^4 + 128*a^3*c^4*d^17*f^4 - 896*a^3*c^6*d^15*f^4 - 3136*a^3*c^8*d^13*f^4 - 4928*a^3*c^10*d^11*f^4
- 4480*a^3*c^12*d^9*f^4 - 2432*a^3*c^14*d^7*f^4 - 736*a^3*c^16*d^5*f^4 - 96*a^3*c^18*d^3*f^4 + 736*b^3*c^3*d^1
8*f^4 + 2432*b^3*c^5*d^16*f^4 + 4480*b^3*c^7*d^14*f^4 + 4928*b^3*c^9*d^12*f^4 + 3136*b^3*c^11*d^10*f^4 + 896*b
^3*c^13*d^8*f^4 - 128*b^3*c^15*d^6*f^4 - 160*b^3*c^17*d^4*f^4 - 32*b^3*c^19*d^2*f^4 - 288*a^2*b*c*d^20*f^4 - 4
80*a*b^2*c^2*d^19*f^4 - 384*a*b^2*c^4*d^17*f^4 + 2688*a*b^2*c^6*d^15*f^4 + 9408*a*b^2*c^8*d^13*f^4 + 14784*a*b
^2*c^10*d^11*f^4 + 13440*a*b^2*c^12*d^9*f^4 + 7296*a*b^2*c^14*d^7*f^4 + 2208*a*b^2*c^16*d^5*f^4 + 288*a*b^2*c^
18*d^3*f^4 - 2208*a^2*b*c^3*d^18*f^4 - 7296*a^2*b*c^5*d^16*f^4 - 13440*a^2*b*c^7*d^14*f^4 - 14784*a^2*b*c^9*d^
12*f^4 - 9408*a^2*b*c^11*d^10*f^4 - 2688*a^2*b*c^13*d^8*f^4 + 384*a^2*b*c^15*d^6*f^4 + 480*a^2*b*c^17*d^4*f^4
+ 96*a^2*b*c^19*d^2*f^4) + (c + d*tan(e + f*x))^(1/2)*(16*a^6*d^18*f^3 - 16*b^6*d^18*f^3 + 240*a^2*b^4*d^18*f^
3 - 240*a^4*b^2*d^18*f^3 - 320*a^6*c^4*d^14*f^3 - 1024*a^6*c^6*d^12*f^3 - 1440*a^6*c^8*d^10*f^3 - 1024*a^6*c^1
0*d^8*f^3 - 320*a^6*c^12*d^6*f^3 + 16*a^6*c^16*d^2*f^3 + 320*b^6*c^4*d^14*f^3 + 1024*b^6*c^6*d^12*f^3 + 1440*b
^6*c^8*d^10*f^3 + 1024*b^6*c^10*d^8*f^3 + 320*b^6*c^12*d^6*f^3 - 16*b^6*c^16*d^2*f^3 - 4800*a^2*b^4*c^4*d^14*f
^3 - 15360*a^2*b^4*c^6*d^12*f^3 - 21600*a^2*b^4*c^8*d^10*f^3 - 15360*a^2*b^4*c^10*d^8*f^3 - 4800*a^2*b^4*c^12*
d^6*f^3 + 240*a^2*b^4*c^16*d^2*f^3 + 6400*a^3*b^3*c^3*d^15*f^3 + 11520*a^3*b^3*c^5*d^13*f^3 + 6400*a^3*b^3*c^7
*d^11*f^3 - 6400*a^3*b^3*c^9*d^9*f^3 - 11520*a^3*b^3*c^11*d^7*f^3 - 6400*a^3*b^3*c^13*d^5*f^3 - 1280*a^3*b^3*c
^15*d^3*f^3 + 4800*a^4*b^2*c^4*d^14*f^3 + 15360*a^4*b^2*c^6*d^12*f^3 + 21600*a^4*b^2*c^8*d^10*f^3 + 15360*a^4*
b^2*c^10*d^8*f^3 + 4800*a^4*b^2*c^12*d^6*f^3 - 240*a^4*b^2*c^16*d^2*f^3 - 384*a*b^5*c*d^17*f^3 - 384*a^5*b*c*d
^17*f^3 - 1920*a*b^5*c^3*d^15*f^3 - 3456*a*b^5*c^5*d^13*f^3 - 1920*a*b^5*c^7*d^11*f^3 + 1920*a*b^5*c^9*d^9*f^3
+ 3456*a*b^5*c^11*d^7*f^3 + 1920*a*b^5*c^13*d^5*f^3 + 384*a*b^5*c^15*d^3*f^3 + 1280*a^3*b^3*c*d^17*f^3 - 1920
*a^5*b*c^3*d^15*f^3 - 3456*a^5*b*c^5*d^13*f^3 - 1920*a^5*b*c^7*d^11*f^3 + 1920*a^5*b*c^9*d^9*f^3 + 3456*a^5*b*
c^11*d^7*f^3 + 1920*a^5*b*c^13*d^5*f^3 + 384*a^5*b*c^15*d^3*f^3))*(-(((8*a^6*c^5*f^2 - 8*b^6*c^5*f^2 + 48*a*b^
5*d^5*f^2 + 48*a^5*b*d^5*f^2 + 40*a^6*c*d^4*f^2 - 40*b^6*c*d^4*f^2 + 120*a^2*b^4*c^5*f^2 - 120*a^4*b^2*c^5*f^2
- 160*a^3*b^3*d^5*f^2 - 80*a^6*c^3*d^2*f^2 + 80*b^6*c^3*d^2*f^2 - 1200*a^2*b^4*c^3*d^2*f^2 + 1600*a^3*b^3*c^2
*d^3*f^2 + 1200*a^4*b^2*c^3*d^2*f^2 + 240*a*b^5*c^4*d*f^2 + 240*a^5*b*c^4*d*f^2 - 480*a*b^5*c^2*d^3*f^2 + 600*
a^2*b^4*c*d^4*f^2 - 800*a^3*b^3*c^4*d*f^2 - 600*a^4*b^2*c*d^4*f^2 - 480*a^5*b*c^2*d^3*f^2)^2/4 - (16*c^10*f^4
+ 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10
+ 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) + 4*a^6*c^5*f^2 - 4*b^6*c^5*f^2 + 24*a*b^5*d^5*f^
2 + 24*a^5*b*d^5*f^2 + 20*a^6*c*d^4*f^2 - 20*b^6*c*d^4*f^2 + 60*a^2*b^4*c^5*f^2 - 60*a^4*b^2*c^5*f^2 - 80*a^3*
b^3*d^5*f^2 - 40*a^6*c^3*d^2*f^2 + 40*b^6*c^3*d^2*f^2 - 600*a^2*b^4*c^3*d^2*f^2 + 800*a^3*b^3*c^2*d^3*f^2 + 60
0*a^4*b^2*c^3*d^2*f^2 + 120*a*b^5*c^4*d*f^2 + 120*a^5*b*c^4*d*f^2 - 240*a*b^5*c^2*d^3*f^2 + 300*a^2*b^4*c*d^4*
f^2 - 400*a^3*b^3*c^4*d*f^2 - 300*a^4*b^2*c*d^4*f^2 - 240*a^5*b*c^2*d^3*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*
d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4)))^(1/2)*1i)/(((-(((8*a^6*c^5*f^2 - 8*b^6*c^5*f^2 +
48*a*b^5*d^5*f^2 + 48*a^5*b*d^5*f^2 + 40*a^6*c*d^4*f^2 - 40*b^6*c*d^4*f^2 + 120*a^2*b^4*c^5*f^2 - 120*a^4*b^2*
c^5*f^2 - 160*a^3*b^3*d^5*f^2 - 80*a^6*c^3*d^2*f^2 + 80*b^6*c^3*d^2*f^2 - 1200*a^2*b^4*c^3*d^2*f^2 + 1600*a^3*
b^3*c^2*d^3*f^2 + 1200*a^4*b^2*c^3*d^2*f^2 + 240*a*b^5*c^4*d*f^2 + 240*a^5*b*c^4*d*f^2 - 480*a*b^5*c^2*d^3*f^2
+ 600*a^2*b^4*c*d^4*f^2 - 800*a^3*b^3*c^4*d*f^2 - 600*a^4*b^2*c*d^4*f^2 - 480*a^5*b*c^2*d^3*f^2)^2/4 - (16*c^
10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4)*(a^12 + b^12 + 6*a
^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) + 4*a^6*c^5*f^2 - 4*b^6*c^5*f^2 + 24*a*b^5
*d^5*f^2 + 24*a^5*b*d^5*f^2 + 20*a^6*c*d^4*f^2 - 20*b^6*c*d^4*f^2 + 60*a^2*b^4*c^5*f^2 - 60*a^4*b^2*c^5*f^2 -
80*a^3*b^3*d^5*f^2 - 40*a^6*c^3*d^2*f^2 + 40*b^6*c^3*d^2*f^2 - 600*a^2*b^4*c^3*d^2*f^2 + 800*a^3*b^3*c^2*d^3*f
^2 + 600*a^4*b^2*c^3*d^2*f^2 + 120*a*b^5*c^4*d*f^2 + 120*a^5*b*c^4*d*f^2 - 240*a*b^5*c^2*d^3*f^2 + 300*a^2*b^4
*c*d^4*f^2 - 400*a^3*b^3*c^4*d*f^2 - 300*a^4*b^2*c*d^4*f^2 - 240*a^5*b*c^2*d^3*f^2)/(16*(c^10*f^4 + d^10*f^4 +
5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4)))^(1/2)*((c + d*tan(e + f*x))^(1/2)*(-(((8*a
^6*c^5*f^2 - 8*b^6*c^5*f^2 + 48*a*b^5*d^5*f^2 + 48*a^5*b*d^5*f^2 + 40*a^6*c*d^4*f^2 - 40*b^6*c*d^4*f^2 + 120*a
^2*b^4*c^5*f^2 - 120*a^4*b^2*c^5*f^2 - 160*a^3*b^3*d^5*f^2 - 80*a^6*c^3*d^2*f^2 + 80*b^6*c^3*d^2*f^2 - 1200*a^
2*b^4*c^3*d^2*f^2 + 1600*a^3*b^3*c^2*d^3*f^2 + 1200*a^4*b^2*c^3*d^2*f^2 + 240*a*b^5*c^4*d*f^2 + 240*a^5*b*c^4*
d*f^2 - 480*a*b^5*c^2*d^3*f^2 + 600*a^2*b^4*c*d^4*f^2 - 800*a^3*b^3*c^4*d*f^2 - 600*a^4*b^2*c*d^4*f^2 - 480*a^
5*b*c^2*d^3*f^2)^2/4 - (16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^
8*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) + 4*a^6*c^5*f
^2 - 4*b^6*c^5*f^2 + 24*a*b^5*d^5*f^2 + 24*a^5*b*d^5*f^2 + 20*a^6*c*d^4*f^2 - 20*b^6*c*d^4*f^2 + 60*a^2*b^4*c^
5*f^2 - 60*a^4*b^2*c^5*f^2 - 80*a^3*b^3*d^5*f^2 - 40*a^6*c^3*d^2*f^2 + 40*b^6*c^3*d^2*f^2 - 600*a^2*b^4*c^3*d^
2*f^2 + 800*a^3*b^3*c^2*d^3*f^2 + 600*a^4*b^2*c^3*d^2*f^2 + 120*a*b^5*c^4*d*f^2 + 120*a^5*b*c^4*d*f^2 - 240*a*
b^5*c^2*d^3*f^2 + 300*a^2*b^4*c*d^4*f^2 - 400*a^3*b^3*c^4*d*f^2 - 300*a^4*b^2*c*d^4*f^2 - 240*a^5*b*c^2*d^3*f^
2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4)))^(1/2)*(64*c*d
^22*f^5 + 640*c^3*d^20*f^5 + 2880*c^5*d^18*f^5 + 7680*c^7*d^16*f^5 + 13440*c^9*d^14*f^5 + 16128*c^11*d^12*f^5
+ 13440*c^13*d^10*f^5 + 7680*c^15*d^8*f^5 + 2880*c^17*d^6*f^5 + 640*c^19*d^4*f^5 + 64*c^21*d^2*f^5) - 32*a^3*d
^21*f^4 + 96*a*b^2*d^21*f^4 - 96*b^3*c*d^20*f^4 - 160*a^3*c^2*d^19*f^4 - 128*a^3*c^4*d^17*f^4 + 896*a^3*c^6*d^
15*f^4 + 3136*a^3*c^8*d^13*f^4 + 4928*a^3*c^10*d^11*f^4 + 4480*a^3*c^12*d^9*f^4 + 2432*a^3*c^14*d^7*f^4 + 736*
a^3*c^16*d^5*f^4 + 96*a^3*c^18*d^3*f^4 - 736*b^3*c^3*d^18*f^4 - 2432*b^3*c^5*d^16*f^4 - 4480*b^3*c^7*d^14*f^4
- 4928*b^3*c^9*d^12*f^4 - 3136*b^3*c^11*d^10*f^4 - 896*b^3*c^13*d^8*f^4 + 128*b^3*c^15*d^6*f^4 + 160*b^3*c^17*
d^4*f^4 + 32*b^3*c^19*d^2*f^4 + 288*a^2*b*c*d^20*f^4 + 480*a*b^2*c^2*d^19*f^4 + 384*a*b^2*c^4*d^17*f^4 - 2688*
a*b^2*c^6*d^15*f^4 - 9408*a*b^2*c^8*d^13*f^4 - 14784*a*b^2*c^10*d^11*f^4 - 13440*a*b^2*c^12*d^9*f^4 - 7296*a*b
^2*c^14*d^7*f^4 - 2208*a*b^2*c^16*d^5*f^4 - 288*a*b^2*c^18*d^3*f^4 + 2208*a^2*b*c^3*d^18*f^4 + 7296*a^2*b*c^5*
d^16*f^4 + 13440*a^2*b*c^7*d^14*f^4 + 14784*a^2*b*c^9*d^12*f^4 + 9408*a^2*b*c^11*d^10*f^4 + 2688*a^2*b*c^13*d^
8*f^4 - 384*a^2*b*c^15*d^6*f^4 - 480*a^2*b*c^17*d^4*f^4 - 96*a^2*b*c^19*d^2*f^4) + (c + d*tan(e + f*x))^(1/2)*
(16*a^6*d^18*f^3 - 16*b^6*d^18*f^3 + 240*a^2*b^4*d^18*f^3 - 240*a^4*b^2*d^18*f^3 - 320*a^6*c^4*d^14*f^3 - 1024
*a^6*c^6*d^12*f^3 - 1440*a^6*c^8*d^10*f^3 - 1024*a^6*c^10*d^8*f^3 - 320*a^6*c^12*d^6*f^3 + 16*a^6*c^16*d^2*f^3
+ 320*b^6*c^4*d^14*f^3 + 1024*b^6*c^6*d^12*f^3 + 1440*b^6*c^8*d^10*f^3 + 1024*b^6*c^10*d^8*f^3 + 320*b^6*c^12
*d^6*f^3 - 16*b^6*c^16*d^2*f^3 - 4800*a^2*b^4*c^4*d^14*f^3 - 15360*a^2*b^4*c^6*d^12*f^3 - 21600*a^2*b^4*c^8*d^
10*f^3 - 15360*a^2*b^4*c^10*d^8*f^3 - 4800*a^2*b^4*c^12*d^6*f^3 + 240*a^2*b^4*c^16*d^2*f^3 + 6400*a^3*b^3*c^3*
d^15*f^3 + 11520*a^3*b^3*c^5*d^13*f^3 + 6400*a^3*b^3*c^7*d^11*f^3 - 6400*a^3*b^3*c^9*d^9*f^3 - 11520*a^3*b^3*c
^11*d^7*f^3 - 6400*a^3*b^3*c^13*d^5*f^3 - 1280*a^3*b^3*c^15*d^3*f^3 + 4800*a^4*b^2*c^4*d^14*f^3 + 15360*a^4*b^
2*c^6*d^12*f^3 + 21600*a^4*b^2*c^8*d^10*f^3 + 15360*a^4*b^2*c^10*d^8*f^3 + 4800*a^4*b^2*c^12*d^6*f^3 - 240*a^4
*b^2*c^16*d^2*f^3 - 384*a*b^5*c*d^17*f^3 - 384*a^5*b*c*d^17*f^3 - 1920*a*b^5*c^3*d^15*f^3 - 3456*a*b^5*c^5*d^1
3*f^3 - 1920*a*b^5*c^7*d^11*f^3 + 1920*a*b^5*c^9*d^9*f^3 + 3456*a*b^5*c^11*d^7*f^3 + 1920*a*b^5*c^13*d^5*f^3 +
384*a*b^5*c^15*d^3*f^3 + 1280*a^3*b^3*c*d^17*f^3 - 1920*a^5*b*c^3*d^15*f^3 - 3456*a^5*b*c^5*d^13*f^3 - 1920*a
^5*b*c^7*d^11*f^3 + 1920*a^5*b*c^9*d^9*f^3 + 3456*a^5*b*c^11*d^7*f^3 + 1920*a^5*b*c^13*d^5*f^3 + 384*a^5*b*c^1
5*d^3*f^3))*(-(((8*a^6*c^5*f^2 - 8*b^6*c^5*f^2 + 48*a*b^5*d^5*f^2 + 48*a^5*b*d^5*f^2 + 40*a^6*c*d^4*f^2 - 40*b
^6*c*d^4*f^2 + 120*a^2*b^4*c^5*f^2 - 120*a^4*b^2*c^5*f^2 - 160*a^3*b^3*d^5*f^2 - 80*a^6*c^3*d^2*f^2 + 80*b^6*c
^3*d^2*f^2 - 1200*a^2*b^4*c^3*d^2*f^2 + 1600*a^3*b^3*c^2*d^3*f^2 + 1200*a^4*b^2*c^3*d^2*f^2 + 240*a*b^5*c^4*d*
f^2 + 240*a^5*b*c^4*d*f^2 - 480*a*b^5*c^2*d^3*f^2 + 600*a^2*b^4*c*d^4*f^2 - 800*a^3*b^3*c^4*d*f^2 - 600*a^4*b^
2*c*d^4*f^2 - 480*a^5*b*c^2*d^3*f^2)^2/4 - (16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160
*c^6*d^4*f^4 + 80*c^8*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))
^(1/2) + 4*a^6*c^5*f^2 - 4*b^6*c^5*f^2 + 24*a*b^5*d^5*f^2 + 24*a^5*b*d^5*f^2 + 20*a^6*c*d^4*f^2 - 20*b^6*c*d^4
*f^2 + 60*a^2*b^4*c^5*f^2 - 60*a^4*b^2*c^5*f^2 - 80*a^3*b^3*d^5*f^2 - 40*a^6*c^3*d^2*f^2 + 40*b^6*c^3*d^2*f^2
- 600*a^2*b^4*c^3*d^2*f^2 + 800*a^3*b^3*c^2*d^3*f^2 + 600*a^4*b^2*c^3*d^2*f^2 + 120*a*b^5*c^4*d*f^2 + 120*a^5*
b*c^4*d*f^2 - 240*a*b^5*c^2*d^3*f^2 + 300*a^2*b^4*c*d^4*f^2 - 400*a^3*b^3*c^4*d*f^2 - 300*a^4*b^2*c*d^4*f^2 -
240*a^5*b*c^2*d^3*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*
f^4)))^(1/2) - ((-(((8*a^6*c^5*f^2 - 8*b^6*c^5*f^2 + 48*a*b^5*d^5*f^2 + 48*a^5*b*d^5*f^2 + 40*a^6*c*d^4*f^2 -
40*b^6*c*d^4*f^2 + 120*a^2*b^4*c^5*f^2 - 120*a^4*b^2*c^5*f^2 - 160*a^3*b^3*d^5*f^2 - 80*a^6*c^3*d^2*f^2 + 80*b
^6*c^3*d^2*f^2 - 1200*a^2*b^4*c^3*d^2*f^2 + 1600*a^3*b^3*c^2*d^3*f^2 + 1200*a^4*b^2*c^3*d^2*f^2 + 240*a*b^5*c^
4*d*f^2 + 240*a^5*b*c^4*d*f^2 - 480*a*b^5*c^2*d^3*f^2 + 600*a^2*b^4*c*d^4*f^2 - 800*a^3*b^3*c^4*d*f^2 - 600*a^
4*b^2*c*d^4*f^2 - 480*a^5*b*c^2*d^3*f^2)^2/4 - (16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 +
160*c^6*d^4*f^4 + 80*c^8*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b
^2))^(1/2) + 4*a^6*c^5*f^2 - 4*b^6*c^5*f^2 + 24*a*b^5*d^5*f^2 + 24*a^5*b*d^5*f^2 + 20*a^6*c*d^4*f^2 - 20*b^6*c
*d^4*f^2 + 60*a^2*b^4*c^5*f^2 - 60*a^4*b^2*c^5*f^2 - 80*a^3*b^3*d^5*f^2 - 40*a^6*c^3*d^2*f^2 + 40*b^6*c^3*d^2*
f^2 - 600*a^2*b^4*c^3*d^2*f^2 + 800*a^3*b^3*c^2*d^3*f^2 + 600*a^4*b^2*c^3*d^2*f^2 + 120*a*b^5*c^4*d*f^2 + 120*
a^5*b*c^4*d*f^2 - 240*a*b^5*c^2*d^3*f^2 + 300*a^2*b^4*c*d^4*f^2 - 400*a^3*b^3*c^4*d*f^2 - 300*a^4*b^2*c*d^4*f^
2 - 240*a^5*b*c^2*d^3*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*
d^2*f^4)))^(1/2)*(32*a^3*d^21*f^4 + (c + d*tan(e + f*x))^(1/2)*(-(((8*a^6*c^5*f^2 - 8*b^6*c^5*f^2 + 48*a*b^5*d
^5*f^2 + 48*a^5*b*d^5*f^2 + 40*a^6*c*d^4*f^2 - 40*b^6*c*d^4*f^2 + 120*a^2*b^4*c^5*f^2 - 120*a^4*b^2*c^5*f^2 -
160*a^3*b^3*d^5*f^2 - 80*a^6*c^3*d^2*f^2 + 80*b^6*c^3*d^2*f^2 - 1200*a^2*b^4*c^3*d^2*f^2 + 1600*a^3*b^3*c^2*d^
3*f^2 + 1200*a^4*b^2*c^3*d^2*f^2 + 240*a*b^5*c^4*d*f^2 + 240*a^5*b*c^4*d*f^2 - 480*a*b^5*c^2*d^3*f^2 + 600*a^2
*b^4*c*d^4*f^2 - 800*a^3*b^3*c^4*d*f^2 - 600*a^4*b^2*c*d^4*f^2 - 480*a^5*b*c^2*d^3*f^2)^2/4 - (16*c^10*f^4 + 1
6*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 +
15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) + 4*a^6*c^5*f^2 - 4*b^6*c^5*f^2 + 24*a*b^5*d^5*f^2 +
24*a^5*b*d^5*f^2 + 20*a^6*c*d^4*f^2 - 20*b^6*c*d^4*f^2 + 60*a^2*b^4*c^5*f^2 - 60*a^4*b^2*c^5*f^2 - 80*a^3*b^3
*d^5*f^2 - 40*a^6*c^3*d^2*f^2 + 40*b^6*c^3*d^2*f^2 - 600*a^2*b^4*c^3*d^2*f^2 + 800*a^3*b^3*c^2*d^3*f^2 + 600*a
^4*b^2*c^3*d^2*f^2 + 120*a*b^5*c^4*d*f^2 + 120*a^5*b*c^4*d*f^2 - 240*a*b^5*c^2*d^3*f^2 + 300*a^2*b^4*c*d^4*f^2
- 400*a^3*b^3*c^4*d*f^2 - 300*a^4*b^2*c*d^4*f^2 - 240*a^5*b*c^2*d^3*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8
*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4)))^(1/2)*(64*c*d^22*f^5 + 640*c^3*d^20*f^5 + 2880*c^5*d
^18*f^5 + 7680*c^7*d^16*f^5 + 13440*c^9*d^14*f^5 + 16128*c^11*d^12*f^5 + 13440*c^13*d^10*f^5 + 7680*c^15*d^8*f
^5 + 2880*c^17*d^6*f^5 + 640*c^19*d^4*f^5 + 64*c^21*d^2*f^5) - 96*a*b^2*d^21*f^4 + 96*b^3*c*d^20*f^4 + 160*a^3
*c^2*d^19*f^4 + 128*a^3*c^4*d^17*f^4 - 896*a^3*c^6*d^15*f^4 - 3136*a^3*c^8*d^13*f^4 - 4928*a^3*c^10*d^11*f^4 -
4480*a^3*c^12*d^9*f^4 - 2432*a^3*c^14*d^7*f^4 - 736*a^3*c^16*d^5*f^4 - 96*a^3*c^18*d^3*f^4 + 736*b^3*c^3*d^18
*f^4 + 2432*b^3*c^5*d^16*f^4 + 4480*b^3*c^7*d^14*f^4 + 4928*b^3*c^9*d^12*f^4 + 3136*b^3*c^11*d^10*f^4 + 896*b^
3*c^13*d^8*f^4 - 128*b^3*c^15*d^6*f^4 - 160*b^3*c^17*d^4*f^4 - 32*b^3*c^19*d^2*f^4 - 288*a^2*b*c*d^20*f^4 - 48
0*a*b^2*c^2*d^19*f^4 - 384*a*b^2*c^4*d^17*f^4 + 2688*a*b^2*c^6*d^15*f^4 + 9408*a*b^2*c^8*d^13*f^4 + 14784*a*b^
2*c^10*d^11*f^4 + 13440*a*b^2*c^12*d^9*f^4 + 7296*a*b^2*c^14*d^7*f^4 + 2208*a*b^2*c^16*d^5*f^4 + 288*a*b^2*c^1
8*d^3*f^4 - 2208*a^2*b*c^3*d^18*f^4 - 7296*a^2*b*c^5*d^16*f^4 - 13440*a^2*b*c^7*d^14*f^4 - 14784*a^2*b*c^9*d^1
2*f^4 - 9408*a^2*b*c^11*d^10*f^4 - 2688*a^2*b*c^13*d^8*f^4 + 384*a^2*b*c^15*d^6*f^4 + 480*a^2*b*c^17*d^4*f^4 +
96*a^2*b*c^19*d^2*f^4) + (c + d*tan(e + f*x))^(1/2)*(16*a^6*d^18*f^3 - 16*b^6*d^18*f^3 + 240*a^2*b^4*d^18*f^3
- 240*a^4*b^2*d^18*f^3 - 320*a^6*c^4*d^14*f^3 - 1024*a^6*c^6*d^12*f^3 - 1440*a^6*c^8*d^10*f^3 - 1024*a^6*c^10
*d^8*f^3 - 320*a^6*c^12*d^6*f^3 + 16*a^6*c^16*d^2*f^3 + 320*b^6*c^4*d^14*f^3 + 1024*b^6*c^6*d^12*f^3 + 1440*b^
6*c^8*d^10*f^3 + 1024*b^6*c^10*d^8*f^3 + 320*b^6*c^12*d^6*f^3 - 16*b^6*c^16*d^2*f^3 - 4800*a^2*b^4*c^4*d^14*f^
3 - 15360*a^2*b^4*c^6*d^12*f^3 - 21600*a^2*b^4*c^8*d^10*f^3 - 15360*a^2*b^4*c^10*d^8*f^3 - 4800*a^2*b^4*c^12*d
^6*f^3 + 240*a^2*b^4*c^16*d^2*f^3 + 6400*a^3*b^3*c^3*d^15*f^3 + 11520*a^3*b^3*c^5*d^13*f^3 + 6400*a^3*b^3*c^7*
d^11*f^3 - 6400*a^3*b^3*c^9*d^9*f^3 - 11520*a^3*b^3*c^11*d^7*f^3 - 6400*a^3*b^3*c^13*d^5*f^3 - 1280*a^3*b^3*c^
15*d^3*f^3 + 4800*a^4*b^2*c^4*d^14*f^3 + 15360*a^4*b^2*c^6*d^12*f^3 + 21600*a^4*b^2*c^8*d^10*f^3 + 15360*a^4*b
^2*c^10*d^8*f^3 + 4800*a^4*b^2*c^12*d^6*f^3 - 240*a^4*b^2*c^16*d^2*f^3 - 384*a*b^5*c*d^17*f^3 - 384*a^5*b*c*d^
17*f^3 - 1920*a*b^5*c^3*d^15*f^3 - 3456*a*b^5*c^5*d^13*f^3 - 1920*a*b^5*c^7*d^11*f^3 + 1920*a*b^5*c^9*d^9*f^3
+ 3456*a*b^5*c^11*d^7*f^3 + 1920*a*b^5*c^13*d^5*f^3 + 384*a*b^5*c^15*d^3*f^3 + 1280*a^3*b^3*c*d^17*f^3 - 1920*
a^5*b*c^3*d^15*f^3 - 3456*a^5*b*c^5*d^13*f^3 - 1920*a^5*b*c^7*d^11*f^3 + 1920*a^5*b*c^9*d^9*f^3 + 3456*a^5*b*c
^11*d^7*f^3 + 1920*a^5*b*c^13*d^5*f^3 + 384*a^5*b*c^15*d^3*f^3))*(-(((8*a^6*c^5*f^2 - 8*b^6*c^5*f^2 + 48*a*b^5
*d^5*f^2 + 48*a^5*b*d^5*f^2 + 40*a^6*c*d^4*f^2 - 40*b^6*c*d^4*f^2 + 120*a^2*b^4*c^5*f^2 - 120*a^4*b^2*c^5*f^2
- 160*a^3*b^3*d^5*f^2 - 80*a^6*c^3*d^2*f^2 + 80*b^6*c^3*d^2*f^2 - 1200*a^2*b^4*c^3*d^2*f^2 + 1600*a^3*b^3*c^2*
d^3*f^2 + 1200*a^4*b^2*c^3*d^2*f^2 + 240*a*b^5*c^4*d*f^2 + 240*a^5*b*c^4*d*f^2 - 480*a*b^5*c^2*d^3*f^2 + 600*a
^2*b^4*c*d^4*f^2 - 800*a^3*b^3*c^4*d*f^2 - 600*a^4*b^2*c*d^4*f^2 - 480*a^5*b*c^2*d^3*f^2)^2/4 - (16*c^10*f^4 +
16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10
+ 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) + 4*a^6*c^5*f^2 - 4*b^6*c^5*f^2 + 24*a*b^5*d^5*f^2
+ 24*a^5*b*d^5*f^2 + 20*a^6*c*d^4*f^2 - 20*b^6*c*d^4*f^2 + 60*a^2*b^4*c^5*f^2 - 60*a^4*b^2*c^5*f^2 - 80*a^3*b
^3*d^5*f^2 - 40*a^6*c^3*d^2*f^2 + 40*b^6*c^3*d^2*f^2 - 600*a^2*b^4*c^3*d^2*f^2 + 800*a^3*b^3*c^2*d^3*f^2 + 600
*a^4*b^2*c^3*d^2*f^2 + 120*a*b^5*c^4*d*f^2 + 120*a^5*b*c^4*d*f^2 - 240*a*b^5*c^2*d^3*f^2 + 300*a^2*b^4*c*d^4*f
^2 - 400*a^3*b^3*c^4*d*f^2 - 300*a^4*b^2*c*d^4*f^2 - 240*a^5*b*c^2*d^3*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d
^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4)))^(1/2) + 16*b^9*d^16*f^2 - 48*a^8*b*d^16*f^2 - 32*a
^9*c*d^15*f^2 - 96*a^4*b^5*d^16*f^2 - 128*a^6*b^3*d^16*f^2 - 192*a^9*c^3*d^13*f^2 - 480*a^9*c^5*d^11*f^2 - 640
*a^9*c^7*d^9*f^2 - 480*a^9*c^9*d^7*f^2 - 192*a^9*c^11*d^5*f^2 - 32*a^9*c^13*d^3*f^2 + 80*b^9*c^2*d^14*f^2 + 14
4*b^9*c^4*d^12*f^2 + 80*b^9*c^6*d^10*f^2 - 80*b^9*c^8*d^8*f^2 - 144*b^9*c^10*d^6*f^2 - 80*b^9*c^12*d^4*f^2 - 1
6*b^9*c^14*d^2*f^2 + 1536*a^3*b^6*c^3*d^13*f^2 + 3840*a^3*b^6*c^5*d^11*f^2 + 5120*a^3*b^6*c^7*d^9*f^2 + 3840*a
^3*b^6*c^9*d^7*f^2 + 1536*a^3*b^6*c^11*d^5*f^2 + 256*a^3*b^6*c^13*d^3*f^2 - 480*a^4*b^5*c^2*d^14*f^2 - 864*a^4
*b^5*c^4*d^12*f^2 - 480*a^4*b^5*c^6*d^10*f^2 + 480*a^4*b^5*c^8*d^8*f^2 + 864*a^4*b^5*c^10*d^6*f^2 + 480*a^4*b^
5*c^12*d^4*f^2 + 96*a^4*b^5*c^14*d^2*f^2 + 1152*a^5*b^4*c^3*d^13*f^2 + 2880*a^5*b^4*c^5*d^11*f^2 + 3840*a^5*b^
4*c^7*d^9*f^2 + 2880*a^5*b^4*c^9*d^7*f^2 + 1152*a^5*b^4*c^11*d^5*f^2 + 192*a^5*b^4*c^13*d^3*f^2 - 640*a^6*b^3*
c^2*d^14*f^2 - 1152*a^6*b^3*c^4*d^12*f^2 - 640*a^6*b^3*c^6*d^10*f^2 + 640*a^6*b^3*c^8*d^8*f^2 + 1152*a^6*b^3*c
^10*d^6*f^2 + 640*a^6*b^3*c^12*d^4*f^2 + 128*a^6*b^3*c^14*d^2*f^2 + 96*a*b^8*c*d^15*f^2 + 576*a*b^8*c^3*d^13*f
^2 + 1440*a*b^8*c^5*d^11*f^2 + 1920*a*b^8*c^7*d^9*f^2 + 1440*a*b^8*c^9*d^7*f^2 + 576*a*b^8*c^11*d^5*f^2 + 96*a
*b^8*c^13*d^3*f^2 + 256*a^3*b^6*c*d^15*f^2 + 192*a^5*b^4*c*d^15*f^2 - 240*a^8*b*c^2*d^14*f^2 - 432*a^8*b*c^4*d
^12*f^2 - 240*a^8*b*c^6*d^10*f^2 + 240*a^8*b*c^8*d^8*f^2 + 432*a^8*b*c^10*d^6*f^2 + 240*a^8*b*c^12*d^4*f^2 + 4
8*a^8*b*c^14*d^2*f^2))*(-(((8*a^6*c^5*f^2 - 8*b^6*c^5*f^2 + 48*a*b^5*d^5*f^2 + 48*a^5*b*d^5*f^2 + 40*a^6*c*d^4
*f^2 - 40*b^6*c*d^4*f^2 + 120*a^2*b^4*c^5*f^2 - 120*a^4*b^2*c^5*f^2 - 160*a^3*b^3*d^5*f^2 - 80*a^6*c^3*d^2*f^2
+ 80*b^6*c^3*d^2*f^2 - 1200*a^2*b^4*c^3*d^2*f^2 + 1600*a^3*b^3*c^2*d^3*f^2 + 1200*a^4*b^2*c^3*d^2*f^2 + 240*a
*b^5*c^4*d*f^2 + 240*a^5*b*c^4*d*f^2 - 480*a*b^5*c^2*d^3*f^2 + 600*a^2*b^4*c*d^4*f^2 - 800*a^3*b^3*c^4*d*f^2 -
600*a^4*b^2*c*d^4*f^2 - 480*a^5*b*c^2*d^3*f^2)^2/4 - (16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^
6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6
*a^10*b^2))^(1/2) + 4*a^6*c^5*f^2 - 4*b^6*c^5*f^2 + 24*a*b^5*d^5*f^2 + 24*a^5*b*d^5*f^2 + 20*a^6*c*d^4*f^2 - 2
0*b^6*c*d^4*f^2 + 60*a^2*b^4*c^5*f^2 - 60*a^4*b^2*c^5*f^2 - 80*a^3*b^3*d^5*f^2 - 40*a^6*c^3*d^2*f^2 + 40*b^6*c
^3*d^2*f^2 - 600*a^2*b^4*c^3*d^2*f^2 + 800*a^3*b^3*c^2*d^3*f^2 + 600*a^4*b^2*c^3*d^2*f^2 + 120*a*b^5*c^4*d*f^2
+ 120*a^5*b*c^4*d*f^2 - 240*a*b^5*c^2*d^3*f^2 + 300*a^2*b^4*c*d^4*f^2 - 400*a^3*b^3*c^4*d*f^2 - 300*a^4*b^2*c
*d^4*f^2 - 240*a^5*b*c^2*d^3*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 +
5*c^8*d^2*f^4)))^(1/2)*2i - ((2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(3*(c^2 + d^2)) + (2*(c
+ d*tan(e + f*x))*(b^3*c^4 + 3*a^2*b*d^4 + 2*a^3*c*d^3 + 3*b^3*c^2*d^2 - 3*a^2*b*c^2*d^2 - 6*a*b^2*c*d^3))/(c^
2 + d^2)^2)/(d^2*f*(c + d*tan(e + f*x))^(3/2))
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{3}}{\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*tan(f*x+e))**3/(c+d*tan(f*x+e))**(5/2),x)
[Out]
Integral((a + b*tan(e + f*x))**3/(c + d*tan(e + f*x))**(5/2), x)
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