\(\int \frac {(a+b \tan (e+f x))^3 (A+B \tan (e+f x)+C \tan ^2(e+f x))}{\sqrt {c+d \tan (e+f x)}} \, dx\) [110]

Optimal result

Integrand size = 47, antiderivative size = 407 \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx=\frac {(i a+b)^3 (A-i B-C) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d} f}-\frac {(i a-b)^3 (A+i B-C) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d} f}+\frac {2 \left (72 a^3 C d^3-6 a^2 b d^2 (32 c C-49 B d)+21 a b^2 d \left (8 c^2 C-10 B c d+15 (A-C) d^2\right )-b^3 \left (48 c^3 C-56 B c^2 d+70 c (A-C) d^2+105 B d^3\right )\right ) \sqrt {c+d \tan (e+f x)}}{105 d^4 f}+\frac {2 b \left (35 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-7 b B d-6 a C d)\right ) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{105 d^3 f}-\frac {2 (6 b c C-7 b B d-6 a C d) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{35 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f} \]

[Out]

(I*a+b)^3*(A-I*B-C)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/f/(c-I*d)^(1/2)-(I*a-b)^3*(A+I*B-C)*arctanh(
(c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/f/(c+I*d)^(1/2)+2/105*(72*a^3*C*d^3-6*a^2*b*d^2*(-49*B*d+32*C*c)+21*a*b^
2*d*(8*c^2*C-10*B*c*d+15*(A-C)*d^2)-b^3*(48*c^3*C-56*B*c^2*d+70*c*(A-C)*d^2+105*B*d^3))*(c+d*tan(f*x+e))^(1/2)
/d^4/f+2/105*b*(35*b*(A*b+B*a-C*b)*d^2+4*(-a*d+b*c)*(-7*B*b*d-6*C*a*d+6*C*b*c))*(c+d*tan(f*x+e))^(1/2)*tan(f*x
+e)/d^3/f-2/35*(-7*B*b*d-6*C*a*d+6*C*b*c)*(c+d*tan(f*x+e))^(1/2)*(a+b*tan(f*x+e))^2/d^2/f+2/7*C*(c+d*tan(f*x+e
))^(1/2)*(a+b*tan(f*x+e))^3/d/f

Rubi [A] (verified)

Time = 1.91 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.149, Rules used = {3728, 3718, 3711, 3620, 3618, 65, 214} \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx=\frac {2 \sqrt {c+d \tan (e+f x)} \left (72 a^3 C d^3-6 a^2 b d^2 (32 c C-49 B d)+21 a b^2 d \left (15 d^2 (A-C)-10 B c d+8 c^2 C\right )-\left (b^3 \left (70 c d^2 (A-C)-56 B c^2 d+105 B d^3+48 c^3 C\right )\right )\right )}{105 d^4 f}-\frac {(-b+i a)^3 (A+i B-C) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}+\frac {(b+i a)^3 (A-i B-C) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}+\frac {2 b \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (35 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-7 b B d+6 b c C)\right )}{105 d^3 f}-\frac {2 (-6 a C d-7 b B d+6 b c C) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{35 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f} \]

[In]

Int[((a + b*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/Sqrt[c + d*Tan[e + f*x]],x]

[Out]

((I*a + b)^3*(A - I*B - C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(Sqrt[c - I*d]*f) - ((I*a - b)^3*(
A + I*B - C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(Sqrt[c + I*d]*f) + (2*(72*a^3*C*d^3 - 6*a^2*b*d
^2*(32*c*C - 49*B*d) + 21*a*b^2*d*(8*c^2*C - 10*B*c*d + 15*(A - C)*d^2) - b^3*(48*c^3*C - 56*B*c^2*d + 70*c*(A
- C)*d^2 + 105*B*d^3))*Sqrt[c + d*Tan[e + f*x]])/(105*d^4*f) + (2*b*(35*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*
d)*(6*b*c*C - 7*b*B*d - 6*a*C*d))*Tan[e + f*x]*Sqrt[c + d*Tan[e + f*x]])/(105*d^3*f) - (2*(6*b*c*C - 7*b*B*d -
6*a*C*d)*(a + b*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]])/(35*d^2*f) + (2*C*(a + b*Tan[e + f*x])^3*Sqrt[c + d
*Tan[e + f*x]])/(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 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 3711

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])
^m*Simp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0]
&&  !LeQ[m, -1]

Rule 3718

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*tan[(e
_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[b*C*Tan[e + f*x]*((c + d*Tan[e + f*x])
^(n + 1)/(d*f*(n + 2))), x] - Dist[1/(d*(n + 2)), Int[(c + d*Tan[e + f*x])^n*Simp[b*c*C - a*A*d*(n + 2) - (A*b
+ a*B - b*C)*d*(n + 2)*Tan[e + f*x] - (a*C*d*(n + 2) - b*(c*C - B*d*(n + 2)))*Tan[e + f*x]^2, x], x], x] /; F
reeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] &&  !LtQ[n, -1]

Rule 3728

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*
tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*(a + b*Tan[e + f*x])^m*((c + d
*Tan[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c +
d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C*(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f
*x] - (C*m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}
, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !Intege
rQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rubi steps \begin{align*} \text {integral}& = \frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}+\frac {2 \int \frac {(a+b \tan (e+f x))^2 \left (\frac {1}{2} (-6 b c C+a (7 A-C) d)+\frac {7}{2} (A b+a B-b C) d \tan (e+f x)-\frac {1}{2} (6 b c C-7 b B d-6 a C d) \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx}{7 d} \\ & = -\frac {2 (6 b c C-7 b B d-6 a C d) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{35 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}+\frac {4 \int \frac {(a+b \tan (e+f x)) \left (\frac {1}{4} (-5 a d (6 b c C-a (7 A-C) d)+(4 b c+a d) (6 b c C-7 b B d-6 a C d))+\frac {35}{4} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2 \tan (e+f x)+\frac {1}{4} \left (35 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-7 b B d-6 a C d)\right ) \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx}{35 d^2} \\ & = \frac {2 b \left (35 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-7 b B d-6 a C d)\right ) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{105 d^3 f}-\frac {2 (6 b c C-7 b B d-6 a C d) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{35 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}-\frac {8 \int \frac {\frac {1}{8} \left (-3 a^3 (35 A-11 C) d^3-42 a b^2 c d (4 c C-5 B d)+3 a^2 b d^2 (64 c C+7 B d)+2 b^3 c \left (24 c^2 C-28 B c d+35 (A-C) d^2\right )\right )-\frac {105}{8} \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) d^3 \tan (e+f x)-\frac {1}{8} \left (72 a^3 C d^3-6 a^2 b d^2 (32 c C-49 B d)+21 a b^2 d \left (8 c^2 C-10 B c d+15 (A-C) d^2\right )-b^3 \left (48 c^3 C-56 B c^2 d+70 c (A-C) d^2+105 B d^3\right )\right ) \tan ^2(e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{105 d^3} \\ & = \frac {2 \left (72 a^3 C d^3-6 a^2 b d^2 (32 c C-49 B d)+21 a b^2 d \left (8 c^2 C-10 B c d+15 (A-C) d^2\right )-b^3 \left (48 c^3 C-56 B c^2 d+70 c (A-C) d^2+105 B d^3\right )\right ) \sqrt {c+d \tan (e+f x)}}{105 d^4 f}+\frac {2 b \left (35 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-7 b B d-6 a C d)\right ) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{105 d^3 f}-\frac {2 (6 b c C-7 b B d-6 a C d) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{35 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}-\frac {8 \int \frac {\frac {105}{8} \left (3 a^2 b B-b^3 B-a^3 (A-C)+3 a b^2 (A-C)\right ) d^3-\frac {105}{8} \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) d^3 \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{105 d^3} \\ & = \frac {2 \left (72 a^3 C d^3-6 a^2 b d^2 (32 c C-49 B d)+21 a b^2 d \left (8 c^2 C-10 B c d+15 (A-C) d^2\right )-b^3 \left (48 c^3 C-56 B c^2 d+70 c (A-C) d^2+105 B d^3\right )\right ) \sqrt {c+d \tan (e+f x)}}{105 d^4 f}+\frac {2 b \left (35 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-7 b B d-6 a C d)\right ) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{105 d^3 f}-\frac {2 (6 b c C-7 b B d-6 a C d) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{35 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}+\frac {1}{2} \left ((a-i b)^3 (A-i B-C)\right ) \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx+\frac {1}{2} \left ((a+i b)^3 (A+i B-C)\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx \\ & = \frac {2 \left (72 a^3 C d^3-6 a^2 b d^2 (32 c C-49 B d)+21 a b^2 d \left (8 c^2 C-10 B c d+15 (A-C) d^2\right )-b^3 \left (48 c^3 C-56 B c^2 d+70 c (A-C) d^2+105 B d^3\right )\right ) \sqrt {c+d \tan (e+f x)}}{105 d^4 f}+\frac {2 b \left (35 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-7 b B d-6 a C d)\right ) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{105 d^3 f}-\frac {2 (6 b c C-7 b B d-6 a C d) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{35 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}-\frac {\left (i (a+i b)^3 (A+i B-C)\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)^3 (i A+B-i C)\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 {2 \left (72 a^3 C d^3-6 a^2 b d^2 (32 c C-49 B d)+21 a b^2 d \left (8 c^2 C-10 B c d+15 (A-C) d^2\right )-b^3 \left (48 c^3 C-56 B c^2 d+70 c (A-C) d^2+105 B d^3\right )\right ) \sqrt {c+d \tan (e+f x)}}{105 d^4 f}+\frac {2 b \left (35 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-7 b B d-6 a C d)\right ) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{105 d^3 f}-\frac {2 (6 b c C-7 b B d-6 a C d) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{35 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}-\frac {\left ((a-i b)^3 (A-i B-C)\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)^3 (A+i B-C)\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 {(a-i b)^3 (i A+B-i C) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d} f}-\frac {(i a-b)^3 (A+i B-C) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d} f}+\frac {2 \left (72 a^3 C d^3-6 a^2 b d^2 (32 c C-49 B d)+21 a b^2 d \left (8 c^2 C-10 B c d+15 (A-C) d^2\right )-b^3 \left (48 c^3 C-56 B c^2 d+70 c (A-C) d^2+105 B d^3\right )\right ) \sqrt {c+d \tan (e+f x)}}{105 d^4 f}+\frac {2 b \left (35 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-7 b B d-6 a C d)\right ) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{105 d^3 f}-\frac {2 (6 b c C-7 b B d-6 a C d) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{35 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1200\) vs. \(2(407)=814\).

Time = 6.51 (sec) , antiderivative size = 1200, normalized size of antiderivative = 2.95 \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx=\frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}+\frac {2 \left (\frac {(-6 b c C+7 b B d+6 a C d) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}+\frac {2 \left (\frac {b \left (35 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-7 b B d-6 a C d)\right ) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{6 d f}-\frac {2 \left (\frac {i \sqrt {c-i d} \left (\frac {1}{4} b c \left (35 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-7 b B d-6 a C d)\right )+\frac {3}{8} a d \left (35 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-7 b B d-6 a C d)\right )-\frac {3}{8} a d (-5 a d (6 b c C-a (7 A-C) d)+(4 b c+a d) (6 b c C-7 b B d-6 a C d))-b \left (-\frac {105}{8} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+\frac {1}{4} c \left (35 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-7 b B d-6 a C d)\right )\right )+\frac {3}{2} i d \left (\frac {35}{4} a \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2-\frac {1}{4} b \left (35 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-7 b B d-6 a C d)\right )+\frac {1}{4} b (-5 a d (6 b c C-a (7 A-C) d)+(4 b c+a d) (6 b c C-7 b B d-6 a C d))\right )\right ) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(-c+i d) f}-\frac {i \sqrt {c+i d} \left (\frac {1}{4} b c \left (35 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-7 b B d-6 a C d)\right )+\frac {3}{8} a d \left (35 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-7 b B d-6 a C d)\right )-\frac {3}{8} a d (-5 a d (6 b c C-a (7 A-C) d)+(4 b c+a d) (6 b c C-7 b B d-6 a C d))-b \left (-\frac {105}{8} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+\frac {1}{4} c \left (35 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-7 b B d-6 a C d)\right )\right )-\frac {3}{2} i d \left (\frac {35}{4} a \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2-\frac {1}{4} b \left (35 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-7 b B d-6 a C d)\right )+\frac {1}{4} b (-5 a d (6 b c C-a (7 A-C) d)+(4 b c+a d) (6 b c C-7 b B d-6 a C d))\right )\right ) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(-c-i d) f}+\frac {2 \left (-\frac {3}{8} a d \left (35 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-7 b B d-6 a C d)\right )+b \left (-\frac {105}{8} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+\frac {1}{4} c \left (35 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-7 b B d-6 a C d)\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{d f}\right )}{3 d}\right )}{5 d}\right )}{7 d} \]

[In]

Integrate[((a + b*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/Sqrt[c + d*Tan[e + f*x]],x]

[Out]

(2*C*(a + b*Tan[e + f*x])^3*Sqrt[c + d*Tan[e + f*x]])/(7*d*f) + (2*(((-6*b*c*C + 7*b*B*d + 6*a*C*d)*(a + b*Tan
[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]])/(5*d*f) + (2*((b*(35*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C -
7*b*B*d - 6*a*C*d))*Tan[e + f*x]*Sqrt[c + d*Tan[e + f*x]])/(6*d*f) - (2*((I*Sqrt[c - I*d]*((b*c*(35*b*(A*b +
a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 7*b*B*d - 6*a*C*d)))/4 + (3*a*d*(35*b*(A*b + a*B - b*C)*d^2 + 4*(b*c
- a*d)*(6*b*c*C - 7*b*B*d - 6*a*C*d)))/8 - (3*a*d*(-5*a*d*(6*b*c*C - a*(7*A - C)*d) + (4*b*c + a*d)*(6*b*c*C
- 7*b*B*d - 6*a*C*d)))/8 - b*((-105*(a^2*B - b^2*B + 2*a*b*(A - C))*d^3)/8 + (c*(35*b*(A*b + a*B - b*C)*d^2 +
4*(b*c - a*d)*(6*b*c*C - 7*b*B*d - 6*a*C*d)))/4) + ((3*I)/2)*d*((35*a*(a^2*B - b^2*B + 2*a*b*(A - C))*d^2)/4 -
(b*(35*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 7*b*B*d - 6*a*C*d)))/4 + (b*(-5*a*d*(6*b*c*C - a*(7
*A - C)*d) + (4*b*c + a*d)*(6*b*c*C - 7*b*B*d - 6*a*C*d)))/4))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]]
)/((-c + I*d)*f) - (I*Sqrt[c + I*d]*((b*c*(35*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 7*b*B*d - 6*a
*C*d)))/4 + (3*a*d*(35*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 7*b*B*d - 6*a*C*d)))/8 - (3*a*d*(-5*
a*d*(6*b*c*C - a*(7*A - C)*d) + (4*b*c + a*d)*(6*b*c*C - 7*b*B*d - 6*a*C*d)))/8 - b*((-105*(a^2*B - b^2*B + 2*
a*b*(A - C))*d^3)/8 + (c*(35*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 7*b*B*d - 6*a*C*d)))/4) - ((3*
I)/2)*d*((35*a*(a^2*B - b^2*B + 2*a*b*(A - C))*d^2)/4 - (b*(35*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*
C - 7*b*B*d - 6*a*C*d)))/4 + (b*(-5*a*d*(6*b*c*C - a*(7*A - C)*d) + (4*b*c + a*d)*(6*b*c*C - 7*b*B*d - 6*a*C*d
)))/4))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((-c - I*d)*f) + (2*((-3*a*d*(35*b*(A*b + a*B - b*C)*
d^2 + 4*(b*c - a*d)*(6*b*c*C - 7*b*B*d - 6*a*C*d)))/8 + b*((-105*(a^2*B - b^2*B + 2*a*b*(A - C))*d^3)/8 + (c*(
35*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 7*b*B*d - 6*a*C*d)))/4))*Sqrt[c + d*Tan[e + f*x]])/(d*f)
))/(3*d)))/(5*d)))/(7*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(5977\) vs. \(2(371)=742\).

Time = 0.31 (sec) , antiderivative size = 5978, normalized size of antiderivative = 14.69

method	result	size
parts	\(\text {Expression too large to display}\)	\(5978\)
derivativedivides	\(\text {Expression too large to display}\)	\(25426\)
default	\(\text {Expression too large to display}\)	\(25426\)

[In]

int((a+b*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 37247 vs. \(2 (363) = 726\).

Time = 11.35 (sec) , antiderivative size = 37247, normalized size of antiderivative = 91.52 \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx=\text {Too large to display} \]

[In]

integrate((a+b*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F]

\[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx=\int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{3} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )}{\sqrt {c + d \tan {\left (e + f x \right )}}}\, dx \]

[In]

integrate((a+b*tan(f*x+e))**3*(A+B*tan(f*x+e)+C*tan(f*x+e)**2)/(c+d*tan(f*x+e))**(1/2),x)

[Out]

Integral((a + b*tan(e + f*x))**3*(A + B*tan(e + f*x) + C*tan(e + f*x)**2)/sqrt(c + d*tan(e + f*x)), x)

Maxima [F(-1)]

Timed out. \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx=\text {Timed out} \]

[In]

integrate((a+b*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

Timed out

Giac [F(-1)]

Timed out. \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx=\text {Timed out} \]

[In]

integrate((a+b*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(1/2),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 112.14 (sec) , antiderivative size = 28858, normalized size of antiderivative = 70.90 \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx=\text {Too large to display} \]

[In]

int(((a + b*tan(e + f*x))^3*(A + B*tan(e + f*x) + C*tan(e + f*x)^2))/(c + d*tan(e + f*x))^(1/2),x)

[Out]

atan(((((8*(4*A*a^3*d^3*f^2 - 12*A*a*b^2*d^3*f^2 + 4*A*b^3*c*d^2*f^2 - 12*A*a^2*b*c*d^2*f^2))/f^3 - 64*c*d^2*(
c + d*tan(e + f*x))^(1/2)*((((8*A^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*a*b^5*d*f^2 + 48*A^2*a^5*b*d*f^2 + 12
0*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12
+ A^4*b^12 + 6*A^4*a^2*b^10 + 15*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a^8*b^4 + 6*A^4*a^10*b^2))^(1/2) - 4*A
^2*a^6*c*f^2 + 4*A^2*b^6*c*f^2 - 24*A^2*a*b^5*d*f^2 - 24*A^2*a^5*b*d*f^2 - 60*A^2*a^2*b^4*c*f^2 + 60*A^2*a^4*b
^2*c*f^2 + 80*A^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2))*((((8*A^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A
^2*a*b^5*d*f^2 + 48*A^2*a^5*b*d*f^2 + 120*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2
/4 - (16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12 + A^4*b^12 + 6*A^4*a^2*b^10 + 15*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^
4*a^8*b^4 + 6*A^4*a^10*b^2))^(1/2) - 4*A^2*a^6*c*f^2 + 4*A^2*b^6*c*f^2 - 24*A^2*a*b^5*d*f^2 - 24*A^2*a^5*b*d*f
^2 - 60*A^2*a^2*b^4*c*f^2 + 60*A^2*a^4*b^2*c*f^2 + 80*A^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2) - (16
*(c + d*tan(e + f*x))^(1/2)*(A^2*a^6*d^2 - A^2*b^6*d^2 + 15*A^2*a^2*b^4*d^2 - 15*A^2*a^4*b^2*d^2))/f^2)*((((8*
A^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*a*b^5*d*f^2 + 48*A^2*a^5*b*d*f^2 + 120*A^2*a^2*b^4*c*f^2 - 120*A^2*a^
4*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12 + A^4*b^12 + 6*A^4*a^2*b^10 + 1
5*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a^8*b^4 + 6*A^4*a^10*b^2))^(1/2) - 4*A^2*a^6*c*f^2 + 4*A^2*b^6*c*f^2 -
24*A^2*a*b^5*d*f^2 - 24*A^2*a^5*b*d*f^2 - 60*A^2*a^2*b^4*c*f^2 + 60*A^2*a^4*b^2*c*f^2 + 80*A^2*a^3*b^3*d*f^2)
/(16*(c^2*f^4 + d^2*f^4)))^(1/2)*1i - (((8*(4*A*a^3*d^3*f^2 - 12*A*a*b^2*d^3*f^2 + 4*A*b^3*c*d^2*f^2 - 12*A*a^
2*b*c*d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((((8*A^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*a*b^5
*d*f^2 + 48*A^2*a^5*b*d*f^2 + 120*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4 - (16
*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12 + A^4*b^12 + 6*A^4*a^2*b^10 + 15*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a^8*b^
4 + 6*A^4*a^10*b^2))^(1/2) - 4*A^2*a^6*c*f^2 + 4*A^2*b^6*c*f^2 - 24*A^2*a*b^5*d*f^2 - 24*A^2*a^5*b*d*f^2 - 60*
A^2*a^2*b^4*c*f^2 + 60*A^2*a^4*b^2*c*f^2 + 80*A^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2))*((((8*A^2*a^
6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*a*b^5*d*f^2 + 48*A^2*a^5*b*d*f^2 + 120*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4*b^2*
c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12 + A^4*b^12 + 6*A^4*a^2*b^10 + 15*A^4*
a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a^8*b^4 + 6*A^4*a^10*b^2))^(1/2) - 4*A^2*a^6*c*f^2 + 4*A^2*b^6*c*f^2 - 24*A^
2*a*b^5*d*f^2 - 24*A^2*a^5*b*d*f^2 - 60*A^2*a^2*b^4*c*f^2 + 60*A^2*a^4*b^2*c*f^2 + 80*A^2*a^3*b^3*d*f^2)/(16*(
c^2*f^4 + d^2*f^4)))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(A^2*a^6*d^2 - A^2*b^6*d^2 + 15*A^2*a^2*b^4*d^2 -
15*A^2*a^4*b^2*d^2))/f^2)*((((8*A^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*a*b^5*d*f^2 + 48*A^2*a^5*b*d*f^2 + 12
0*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12
+ A^4*b^12 + 6*A^4*a^2*b^10 + 15*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a^8*b^4 + 6*A^4*a^10*b^2))^(1/2) - 4*A
^2*a^6*c*f^2 + 4*A^2*b^6*c*f^2 - 24*A^2*a*b^5*d*f^2 - 24*A^2*a^5*b*d*f^2 - 60*A^2*a^2*b^4*c*f^2 + 60*A^2*a^4*b
^2*c*f^2 + 80*A^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2)*1i)/((16*(6*A^3*a^4*b^5*d^2 - A^3*b^9*d^2 + 8
*A^3*a^6*b^3*d^2 + 3*A^3*a^8*b*d^2))/f^3 + (((8*(4*A*a^3*d^3*f^2 - 12*A*a*b^2*d^3*f^2 + 4*A*b^3*c*d^2*f^2 - 12
*A*a^2*b*c*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((((8*A^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*
a*b^5*d*f^2 + 48*A^2*a^5*b*d*f^2 + 120*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4
- (16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12 + A^4*b^12 + 6*A^4*a^2*b^10 + 15*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a
^8*b^4 + 6*A^4*a^10*b^2))^(1/2) - 4*A^2*a^6*c*f^2 + 4*A^2*b^6*c*f^2 - 24*A^2*a*b^5*d*f^2 - 24*A^2*a^5*b*d*f^2
- 60*A^2*a^2*b^4*c*f^2 + 60*A^2*a^4*b^2*c*f^2 + 80*A^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2))*((((8*A
^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*a*b^5*d*f^2 + 48*A^2*a^5*b*d*f^2 + 120*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4
*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12 + A^4*b^12 + 6*A^4*a^2*b^10 + 15
*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a^8*b^4 + 6*A^4*a^10*b^2))^(1/2) - 4*A^2*a^6*c*f^2 + 4*A^2*b^6*c*f^2 -
24*A^2*a*b^5*d*f^2 - 24*A^2*a^5*b*d*f^2 - 60*A^2*a^2*b^4*c*f^2 + 60*A^2*a^4*b^2*c*f^2 + 80*A^2*a^3*b^3*d*f^2)/
(16*(c^2*f^4 + d^2*f^4)))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(A^2*a^6*d^2 - A^2*b^6*d^2 + 15*A^2*a^2*b^4*d
^2 - 15*A^2*a^4*b^2*d^2))/f^2)*((((8*A^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*a*b^5*d*f^2 + 48*A^2*a^5*b*d*f^2
+ 120*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(A^4
*a^12 + A^4*b^12 + 6*A^4*a^2*b^10 + 15*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a^8*b^4 + 6*A^4*a^10*b^2))^(1/2)
- 4*A^2*a^6*c*f^2 + 4*A^2*b^6*c*f^2 - 24*A^2*a*b^5*d*f^2 - 24*A^2*a^5*b*d*f^2 - 60*A^2*a^2*b^4*c*f^2 + 60*A^2*
a^4*b^2*c*f^2 + 80*A^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2) + (((8*(4*A*a^3*d^3*f^2 - 12*A*a*b^2*d^3
*f^2 + 4*A*b^3*c*d^2*f^2 - 12*A*a^2*b*c*d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((((8*A^2*a^6*c*f^
2 - 8*A^2*b^6*c*f^2 + 48*A^2*a*b^5*d*f^2 + 48*A^2*a^5*b*d*f^2 + 120*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4*b^2*c*f^2
- 160*A^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12 + A^4*b^12 + 6*A^4*a^2*b^10 + 15*A^4*a^4*b^
8 + 20*A^4*a^6*b^6 + 15*A^4*a^8*b^4 + 6*A^4*a^10*b^2))^(1/2) - 4*A^2*a^6*c*f^2 + 4*A^2*b^6*c*f^2 - 24*A^2*a*b^
5*d*f^2 - 24*A^2*a^5*b*d*f^2 - 60*A^2*a^2*b^4*c*f^2 + 60*A^2*a^4*b^2*c*f^2 + 80*A^2*a^3*b^3*d*f^2)/(16*(c^2*f^
4 + d^2*f^4)))^(1/2))*((((8*A^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*a*b^5*d*f^2 + 48*A^2*a^5*b*d*f^2 + 120*A^
2*a^2*b^4*c*f^2 - 120*A^2*a^4*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12 + A
^4*b^12 + 6*A^4*a^2*b^10 + 15*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a^8*b^4 + 6*A^4*a^10*b^2))^(1/2) - 4*A^2*a
^6*c*f^2 + 4*A^2*b^6*c*f^2 - 24*A^2*a*b^5*d*f^2 - 24*A^2*a^5*b*d*f^2 - 60*A^2*a^2*b^4*c*f^2 + 60*A^2*a^4*b^2*c
*f^2 + 80*A^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(A^2*a^6*d^2 - A
^2*b^6*d^2 + 15*A^2*a^2*b^4*d^2 - 15*A^2*a^4*b^2*d^2))/f^2)*((((8*A^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*a*b
^5*d*f^2 + 48*A^2*a^5*b*d*f^2 + 120*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4 - (
16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12 + A^4*b^12 + 6*A^4*a^2*b^10 + 15*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a^8*
b^4 + 6*A^4*a^10*b^2))^(1/2) - 4*A^2*a^6*c*f^2 + 4*A^2*b^6*c*f^2 - 24*A^2*a*b^5*d*f^2 - 24*A^2*a^5*b*d*f^2 - 6
0*A^2*a^2*b^4*c*f^2 + 60*A^2*a^4*b^2*c*f^2 + 80*A^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2)))*((((8*A^2
*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*a*b^5*d*f^2 + 48*A^2*a^5*b*d*f^2 + 120*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4*b
^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12 + A^4*b^12 + 6*A^4*a^2*b^10 + 15*A
^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a^8*b^4 + 6*A^4*a^10*b^2))^(1/2) - 4*A^2*a^6*c*f^2 + 4*A^2*b^6*c*f^2 - 24
*A^2*a*b^5*d*f^2 - 24*A^2*a^5*b*d*f^2 - 60*A^2*a^2*b^4*c*f^2 + 60*A^2*a^4*b^2*c*f^2 + 80*A^2*a^3*b^3*d*f^2)/(1
6*(c^2*f^4 + d^2*f^4)))^(1/2)*2i + atan(((((8*(4*A*a^3*d^3*f^2 - 12*A*a*b^2*d^3*f^2 + 4*A*b^3*c*d^2*f^2 - 12*A
*a^2*b*c*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(((8*A^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*a
*b^5*d*f^2 + 48*A^2*a^5*b*d*f^2 + 120*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4 -
(16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12 + A^4*b^12 + 6*A^4*a^2*b^10 + 15*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a^
8*b^4 + 6*A^4*a^10*b^2))^(1/2) + 4*A^2*a^6*c*f^2 - 4*A^2*b^6*c*f^2 + 24*A^2*a*b^5*d*f^2 + 24*A^2*a^5*b*d*f^2 +
60*A^2*a^2*b^4*c*f^2 - 60*A^2*a^4*b^2*c*f^2 - 80*A^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2))*(-(((8*A
^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*a*b^5*d*f^2 + 48*A^2*a^5*b*d*f^2 + 120*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4
*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12 + A^4*b^12 + 6*A^4*a^2*b^10 + 15
*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a^8*b^4 + 6*A^4*a^10*b^2))^(1/2) + 4*A^2*a^6*c*f^2 - 4*A^2*b^6*c*f^2 +
24*A^2*a*b^5*d*f^2 + 24*A^2*a^5*b*d*f^2 + 60*A^2*a^2*b^4*c*f^2 - 60*A^2*a^4*b^2*c*f^2 - 80*A^2*a^3*b^3*d*f^2)/
(16*(c^2*f^4 + d^2*f^4)))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(A^2*a^6*d^2 - A^2*b^6*d^2 + 15*A^2*a^2*b^4*d
^2 - 15*A^2*a^4*b^2*d^2))/f^2)*(-(((8*A^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*a*b^5*d*f^2 + 48*A^2*a^5*b*d*f^
2 + 120*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(A^
4*a^12 + A^4*b^12 + 6*A^4*a^2*b^10 + 15*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a^8*b^4 + 6*A^4*a^10*b^2))^(1/2)
+ 4*A^2*a^6*c*f^2 - 4*A^2*b^6*c*f^2 + 24*A^2*a*b^5*d*f^2 + 24*A^2*a^5*b*d*f^2 + 60*A^2*a^2*b^4*c*f^2 - 60*A^2
*a^4*b^2*c*f^2 - 80*A^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2)*1i - (((8*(4*A*a^3*d^3*f^2 - 12*A*a*b^2
*d^3*f^2 + 4*A*b^3*c*d^2*f^2 - 12*A*a^2*b*c*d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(((8*A^2*a^6
*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*a*b^5*d*f^2 + 48*A^2*a^5*b*d*f^2 + 120*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4*b^2*c
*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12 + A^4*b^12 + 6*A^4*a^2*b^10 + 15*A^4*a
^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a^8*b^4 + 6*A^4*a^10*b^2))^(1/2) + 4*A^2*a^6*c*f^2 - 4*A^2*b^6*c*f^2 + 24*A^2
*a*b^5*d*f^2 + 24*A^2*a^5*b*d*f^2 + 60*A^2*a^2*b^4*c*f^2 - 60*A^2*a^4*b^2*c*f^2 - 80*A^2*a^3*b^3*d*f^2)/(16*(c
^2*f^4 + d^2*f^4)))^(1/2))*(-(((8*A^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*a*b^5*d*f^2 + 48*A^2*a^5*b*d*f^2 +
120*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^
12 + A^4*b^12 + 6*A^4*a^2*b^10 + 15*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a^8*b^4 + 6*A^4*a^10*b^2))^(1/2) + 4
*A^2*a^6*c*f^2 - 4*A^2*b^6*c*f^2 + 24*A^2*a*b^5*d*f^2 + 24*A^2*a^5*b*d*f^2 + 60*A^2*a^2*b^4*c*f^2 - 60*A^2*a^4
*b^2*c*f^2 - 80*A^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(A^2*a^6*d
^2 - A^2*b^6*d^2 + 15*A^2*a^2*b^4*d^2 - 15*A^2*a^4*b^2*d^2))/f^2)*(-(((8*A^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*
A^2*a*b^5*d*f^2 + 48*A^2*a^5*b*d*f^2 + 120*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^
2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12 + A^4*b^12 + 6*A^4*a^2*b^10 + 15*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A
^4*a^8*b^4 + 6*A^4*a^10*b^2))^(1/2) + 4*A^2*a^6*c*f^2 - 4*A^2*b^6*c*f^2 + 24*A^2*a*b^5*d*f^2 + 24*A^2*a^5*b*d*
f^2 + 60*A^2*a^2*b^4*c*f^2 - 60*A^2*a^4*b^2*c*f^2 - 80*A^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2)*1i)/
((16*(6*A^3*a^4*b^5*d^2 - A^3*b^9*d^2 + 8*A^3*a^6*b^3*d^2 + 3*A^3*a^8*b*d^2))/f^3 + (((8*(4*A*a^3*d^3*f^2 - 12
*A*a*b^2*d^3*f^2 + 4*A*b^3*c*d^2*f^2 - 12*A*a^2*b*c*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(((8
*A^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*a*b^5*d*f^2 + 48*A^2*a^5*b*d*f^2 + 120*A^2*a^2*b^4*c*f^2 - 120*A^2*a
^4*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12 + A^4*b^12 + 6*A^4*a^2*b^10 +
15*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a^8*b^4 + 6*A^4*a^10*b^2))^(1/2) + 4*A^2*a^6*c*f^2 - 4*A^2*b^6*c*f^2
+ 24*A^2*a*b^5*d*f^2 + 24*A^2*a^5*b*d*f^2 + 60*A^2*a^2*b^4*c*f^2 - 60*A^2*a^4*b^2*c*f^2 - 80*A^2*a^3*b^3*d*f^2
)/(16*(c^2*f^4 + d^2*f^4)))^(1/2))*(-(((8*A^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*a*b^5*d*f^2 + 48*A^2*a^5*b*
d*f^2 + 120*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)
*(A^4*a^12 + A^4*b^12 + 6*A^4*a^2*b^10 + 15*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a^8*b^4 + 6*A^4*a^10*b^2))^(
1/2) + 4*A^2*a^6*c*f^2 - 4*A^2*b^6*c*f^2 + 24*A^2*a*b^5*d*f^2 + 24*A^2*a^5*b*d*f^2 + 60*A^2*a^2*b^4*c*f^2 - 60
*A^2*a^4*b^2*c*f^2 - 80*A^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(A
^2*a^6*d^2 - A^2*b^6*d^2 + 15*A^2*a^2*b^4*d^2 - 15*A^2*a^4*b^2*d^2))/f^2)*(-(((8*A^2*a^6*c*f^2 - 8*A^2*b^6*c*f
^2 + 48*A^2*a*b^5*d*f^2 + 48*A^2*a^5*b*d*f^2 + 120*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4*b^2*c*f^2 - 160*A^2*a^3*b^3
*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12 + A^4*b^12 + 6*A^4*a^2*b^10 + 15*A^4*a^4*b^8 + 20*A^4*a^6*b^
6 + 15*A^4*a^8*b^4 + 6*A^4*a^10*b^2))^(1/2) + 4*A^2*a^6*c*f^2 - 4*A^2*b^6*c*f^2 + 24*A^2*a*b^5*d*f^2 + 24*A^2*
a^5*b*d*f^2 + 60*A^2*a^2*b^4*c*f^2 - 60*A^2*a^4*b^2*c*f^2 - 80*A^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1
/2) + (((8*(4*A*a^3*d^3*f^2 - 12*A*a*b^2*d^3*f^2 + 4*A*b^3*c*d^2*f^2 - 12*A*a^2*b*c*d^2*f^2))/f^3 + 64*c*d^2*(
c + d*tan(e + f*x))^(1/2)*(-(((8*A^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*a*b^5*d*f^2 + 48*A^2*a^5*b*d*f^2 + 1
20*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^1
2 + A^4*b^12 + 6*A^4*a^2*b^10 + 15*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a^8*b^4 + 6*A^4*a^10*b^2))^(1/2) + 4*
A^2*a^6*c*f^2 - 4*A^2*b^6*c*f^2 + 24*A^2*a*b^5*d*f^2 + 24*A^2*a^5*b*d*f^2 + 60*A^2*a^2*b^4*c*f^2 - 60*A^2*a^4*
b^2*c*f^2 - 80*A^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2))*(-(((8*A^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48
*A^2*a*b^5*d*f^2 + 48*A^2*a^5*b*d*f^2 + 120*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)
^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12 + A^4*b^12 + 6*A^4*a^2*b^10 + 15*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*
A^4*a^8*b^4 + 6*A^4*a^10*b^2))^(1/2) + 4*A^2*a^6*c*f^2 - 4*A^2*b^6*c*f^2 + 24*A^2*a*b^5*d*f^2 + 24*A^2*a^5*b*d
*f^2 + 60*A^2*a^2*b^4*c*f^2 - 60*A^2*a^4*b^2*c*f^2 - 80*A^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2) + (
16*(c + d*tan(e + f*x))^(1/2)*(A^2*a^6*d^2 - A^2*b^6*d^2 + 15*A^2*a^2*b^4*d^2 - 15*A^2*a^4*b^2*d^2))/f^2)*(-((
(8*A^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*a*b^5*d*f^2 + 48*A^2*a^5*b*d*f^2 + 120*A^2*a^2*b^4*c*f^2 - 120*A^2
*a^4*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12 + A^4*b^12 + 6*A^4*a^2*b^10
+ 15*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a^8*b^4 + 6*A^4*a^10*b^2))^(1/2) + 4*A^2*a^6*c*f^2 - 4*A^2*b^6*c*f^
2 + 24*A^2*a*b^5*d*f^2 + 24*A^2*a^5*b*d*f^2 + 60*A^2*a^2*b^4*c*f^2 - 60*A^2*a^4*b^2*c*f^2 - 80*A^2*a^3*b^3*d*f
^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2)))*(-(((8*A^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*a*b^5*d*f^2 + 48*A^2*a^5
*b*d*f^2 + 120*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f
^4)*(A^4*a^12 + A^4*b^12 + 6*A^4*a^2*b^10 + 15*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a^8*b^4 + 6*A^4*a^10*b^2)
)^(1/2) + 4*A^2*a^6*c*f^2 - 4*A^2*b^6*c*f^2 + 24*A^2*a*b^5*d*f^2 + 24*A^2*a^5*b*d*f^2 + 60*A^2*a^2*b^4*c*f^2 -
60*A^2*a^4*b^2*c*f^2 - 80*A^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2)*2i - (c + d*tan(e + f*x))^(1/2)*
(2*c*((8*B*b^3*c - 6*B*a*b^2*d)/(d^3*f) - (4*B*b^3*c)/(d^3*f)) - (12*B*b^3*c^2 + 6*B*a^2*b*d^2 - 18*B*a*b^2*c*
d)/(d^3*f) + (2*B*b^3*(d^5*f + c^2*d^3*f))/(d^6*f^2)) + atan(((((8*(4*B*b^3*d^3*f^2 - 12*B*a^2*b*d^3*f^2 - 4*B
*a^3*c*d^2*f^2 + 12*B*a*b^2*c*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(((8*B^2*a^6*c*f^2 - 8*B^2
*b^6*c*f^2 + 48*B^2*a*b^5*d*f^2 + 48*B^2*a^5*b*d*f^2 + 120*B^2*a^2*b^4*c*f^2 - 120*B^2*a^4*b^2*c*f^2 - 160*B^2
*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(B^4*a^12 + B^4*b^12 + 6*B^4*a^2*b^10 + 15*B^4*a^4*b^8 + 20*B^
4*a^6*b^6 + 15*B^4*a^8*b^4 + 6*B^4*a^10*b^2))^(1/2) - 4*B^2*a^6*c*f^2 + 4*B^2*b^6*c*f^2 - 24*B^2*a*b^5*d*f^2 -
24*B^2*a^5*b*d*f^2 - 60*B^2*a^2*b^4*c*f^2 + 60*B^2*a^4*b^2*c*f^2 + 80*B^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f
^4)))^(1/2))*(-(((8*B^2*a^6*c*f^2 - 8*B^2*b^6*c*f^2 + 48*B^2*a*b^5*d*f^2 + 48*B^2*a^5*b*d*f^2 + 120*B^2*a^2*b^
4*c*f^2 - 120*B^2*a^4*b^2*c*f^2 - 160*B^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(B^4*a^12 + B^4*b^12
+ 6*B^4*a^2*b^10 + 15*B^4*a^4*b^8 + 20*B^4*a^6*b^6 + 15*B^4*a^8*b^4 + 6*B^4*a^10*b^2))^(1/2) - 4*B^2*a^6*c*f^2
+ 4*B^2*b^6*c*f^2 - 24*B^2*a*b^5*d*f^2 - 24*B^2*a^5*b*d*f^2 - 60*B^2*a^2*b^4*c*f^2 + 60*B^2*a^4*b^2*c*f^2 + 8
0*B^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(B^2*a^6*d^2 - B^2*b^6*d
^2 + 15*B^2*a^2*b^4*d^2 - 15*B^2*a^4*b^2*d^2))/f^2)*(-(((8*B^2*a^6*c*f^2 - 8*B^2*b^6*c*f^2 + 48*B^2*a*b^5*d*f^
2 + 48*B^2*a^5*b*d*f^2 + 120*B^2*a^2*b^4*c*f^2 - 120*B^2*a^4*b^2*c*f^2 - 160*B^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*
f^4 + 16*d^2*f^4)*(B^4*a^12 + B^4*b^12 + 6*B^4*a^2*b^10 + 15*B^4*a^4*b^8 + 20*B^4*a^6*b^6 + 15*B^4*a^8*b^4 + 6
*B^4*a^10*b^2))^(1/2) - 4*B^2*a^6*c*f^2 + 4*B^2*b^6*c*f^2 - 24*B^2*a*b^5*d*f^2 - 24*B^2*a^5*b*d*f^2 - 60*B^2*a
^2*b^4*c*f^2 + 60*B^2*a^4*b^2*c*f^2 + 80*B^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2)*1i - (((8*(4*B*b^3
*d^3*f^2 - 12*B*a^2*b*d^3*f^2 - 4*B*a^3*c*d^2*f^2 + 12*B*a*b^2*c*d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))
^(1/2)*(-(((8*B^2*a^6*c*f^2 - 8*B^2*b^6*c*f^2 + 48*B^2*a*b^5*d*f^2 + 48*B^2*a^5*b*d*f^2 + 120*B^2*a^2*b^4*c*f^
2 - 120*B^2*a^4*b^2*c*f^2 - 160*B^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(B^4*a^12 + B^4*b^12 + 6*B^
4*a^2*b^10 + 15*B^4*a^4*b^8 + 20*B^4*a^6*b^6 + 15*B^4*a^8*b^4 + 6*B^4*a^10*b^2))^(1/2) - 4*B^2*a^6*c*f^2 + 4*B
^2*b^6*c*f^2 - 24*B^2*a*b^5*d*f^2 - 24*B^2*a^5*b*d*f^2 - 60*B^2*a^2*b^4*c*f^2 + 60*B^2*a^4*b^2*c*f^2 + 80*B^2*
a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2))*(-(((8*B^2*a^6*c*f^2 - 8*B^2*b^6*c*f^2 + 48*B^2*a*b^5*d*f^2 +
48*B^2*a^5*b*d*f^2 + 120*B^2*a^2*b^4*c*f^2 - 120*B^2*a^4*b^2*c*f^2 - 160*B^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4
+ 16*d^2*f^4)*(B^4*a^12 + B^4*b^12 + 6*B^4*a^2*b^10 + 15*B^4*a^4*b^8 + 20*B^4*a^6*b^6 + 15*B^4*a^8*b^4 + 6*B^4
*a^10*b^2))^(1/2) - 4*B^2*a^6*c*f^2 + 4*B^2*b^6*c*f^2 - 24*B^2*a*b^5*d*f^2 - 24*B^2*a^5*b*d*f^2 - 60*B^2*a^2*b
^4*c*f^2 + 60*B^2*a^4*b^2*c*f^2 + 80*B^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2) - (16*(c + d*tan(e + f
*x))^(1/2)*(B^2*a^6*d^2 - B^2*b^6*d^2 + 15*B^2*a^2*b^4*d^2 - 15*B^2*a^4*b^2*d^2))/f^2)*(-(((8*B^2*a^6*c*f^2 -
8*B^2*b^6*c*f^2 + 48*B^2*a*b^5*d*f^2 + 48*B^2*a^5*b*d*f^2 + 120*B^2*a^2*b^4*c*f^2 - 120*B^2*a^4*b^2*c*f^2 - 16
0*B^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(B^4*a^12 + B^4*b^12 + 6*B^4*a^2*b^10 + 15*B^4*a^4*b^8 +
20*B^4*a^6*b^6 + 15*B^4*a^8*b^4 + 6*B^4*a^10*b^2))^(1/2) - 4*B^2*a^6*c*f^2 + 4*B^2*b^6*c*f^2 - 24*B^2*a*b^5*d*
f^2 - 24*B^2*a^5*b*d*f^2 - 60*B^2*a^2*b^4*c*f^2 + 60*B^2*a^4*b^2*c*f^2 + 80*B^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 +
d^2*f^4)))^(1/2)*1i)/((((8*(4*B*b^3*d^3*f^2 - 12*B*a^2*b*d^3*f^2 - 4*B*a^3*c*d^2*f^2 + 12*B*a*b^2*c*d^2*f^2))/
f^3 - 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(((8*B^2*a^6*c*f^2 - 8*B^2*b^6*c*f^2 + 48*B^2*a*b^5*d*f^2 + 48*B^2
*a^5*b*d*f^2 + 120*B^2*a^2*b^4*c*f^2 - 120*B^2*a^4*b^2*c*f^2 - 160*B^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d
^2*f^4)*(B^4*a^12 + B^4*b^12 + 6*B^4*a^2*b^10 + 15*B^4*a^4*b^8 + 20*B^4*a^6*b^6 + 15*B^4*a^8*b^4 + 6*B^4*a^10*
b^2))^(1/2) - 4*B^2*a^6*c*f^2 + 4*B^2*b^6*c*f^2 - 24*B^2*a*b^5*d*f^2 - 24*B^2*a^5*b*d*f^2 - 60*B^2*a^2*b^4*c*f
^2 + 60*B^2*a^4*b^2*c*f^2 + 80*B^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2))*(-(((8*B^2*a^6*c*f^2 - 8*B^
2*b^6*c*f^2 + 48*B^2*a*b^5*d*f^2 + 48*B^2*a^5*b*d*f^2 + 120*B^2*a^2*b^4*c*f^2 - 120*B^2*a^4*b^2*c*f^2 - 160*B^
2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(B^4*a^12 + B^4*b^12 + 6*B^4*a^2*b^10 + 15*B^4*a^4*b^8 + 20*B
^4*a^6*b^6 + 15*B^4*a^8*b^4 + 6*B^4*a^10*b^2))^(1/2) - 4*B^2*a^6*c*f^2 + 4*B^2*b^6*c*f^2 - 24*B^2*a*b^5*d*f^2
- 24*B^2*a^5*b*d*f^2 - 60*B^2*a^2*b^4*c*f^2 + 60*B^2*a^4*b^2*c*f^2 + 80*B^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*
f^4)))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(B^2*a^6*d^2 - B^2*b^6*d^2 + 15*B^2*a^2*b^4*d^2 - 15*B^2*a^4*b^2
*d^2))/f^2)*(-(((8*B^2*a^6*c*f^2 - 8*B^2*b^6*c*f^2 + 48*B^2*a*b^5*d*f^2 + 48*B^2*a^5*b*d*f^2 + 120*B^2*a^2*b^4
*c*f^2 - 120*B^2*a^4*b^2*c*f^2 - 160*B^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(B^4*a^12 + B^4*b^12 +
6*B^4*a^2*b^10 + 15*B^4*a^4*b^8 + 20*B^4*a^6*b^6 + 15*B^4*a^8*b^4 + 6*B^4*a^10*b^2))^(1/2) - 4*B^2*a^6*c*f^2
+ 4*B^2*b^6*c*f^2 - 24*B^2*a*b^5*d*f^2 - 24*B^2*a^5*b*d*f^2 - 60*B^2*a^2*b^4*c*f^2 + 60*B^2*a^4*b^2*c*f^2 + 80
*B^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2) - (16*(8*B^3*a^3*b^6*d^2 - B^3*a^9*d^2 + 6*B^3*a^5*b^4*d^2
+ 3*B^3*a*b^8*d^2))/f^3 + (((8*(4*B*b^3*d^3*f^2 - 12*B*a^2*b*d^3*f^2 - 4*B*a^3*c*d^2*f^2 + 12*B*a*b^2*c*d^2*f
^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(((8*B^2*a^6*c*f^2 - 8*B^2*b^6*c*f^2 + 48*B^2*a*b^5*d*f^2 + 4
8*B^2*a^5*b*d*f^2 + 120*B^2*a^2*b^4*c*f^2 - 120*B^2*a^4*b^2*c*f^2 - 160*B^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 +
16*d^2*f^4)*(B^4*a^12 + B^4*b^12 + 6*B^4*a^2*b^10 + 15*B^4*a^4*b^8 + 20*B^4*a^6*b^6 + 15*B^4*a^8*b^4 + 6*B^4*
a^10*b^2))^(1/2) - 4*B^2*a^6*c*f^2 + 4*B^2*b^6*c*f^2 - 24*B^2*a*b^5*d*f^2 - 24*B^2*a^5*b*d*f^2 - 60*B^2*a^2*b^
4*c*f^2 + 60*B^2*a^4*b^2*c*f^2 + 80*B^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2))*(-(((8*B^2*a^6*c*f^2 -
8*B^2*b^6*c*f^2 + 48*B^2*a*b^5*d*f^2 + 48*B^2*a^5*b*d*f^2 + 120*B^2*a^2*b^4*c*f^2 - 120*B^2*a^4*b^2*c*f^2 - 1
60*B^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(B^4*a^12 + B^4*b^12 + 6*B^4*a^2*b^10 + 15*B^4*a^4*b^8 +
20*B^4*a^6*b^6 + 15*B^4*a^8*b^4 + 6*B^4*a^10*b^2))^(1/2) - 4*B^2*a^6*c*f^2 + 4*B^2*b^6*c*f^2 - 24*B^2*a*b^5*d
*f^2 - 24*B^2*a^5*b*d*f^2 - 60*B^2*a^2*b^4*c*f^2 + 60*B^2*a^4*b^2*c*f^2 + 80*B^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 +
d^2*f^4)))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(B^2*a^6*d^2 - B^2*b^6*d^2 + 15*B^2*a^2*b^4*d^2 - 15*B^2*a^
4*b^2*d^2))/f^2)*(-(((8*B^2*a^6*c*f^2 - 8*B^2*b^6*c*f^2 + 48*B^2*a*b^5*d*f^2 + 48*B^2*a^5*b*d*f^2 + 120*B^2*a^
2*b^4*c*f^2 - 120*B^2*a^4*b^2*c*f^2 - 160*B^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(B^4*a^12 + B^4*b
^12 + 6*B^4*a^2*b^10 + 15*B^4*a^4*b^8 + 20*B^4*a^6*b^6 + 15*B^4*a^8*b^4 + 6*B^4*a^10*b^2))^(1/2) - 4*B^2*a^6*c
*f^2 + 4*B^2*b^6*c*f^2 - 24*B^2*a*b^5*d*f^2 - 24*B^2*a^5*b*d*f^2 - 60*B^2*a^2*b^4*c*f^2 + 60*B^2*a^4*b^2*c*f^2
+ 80*B^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2)))*(-(((8*B^2*a^6*c*f^2 - 8*B^2*b^6*c*f^2 + 48*B^2*a*b
^5*d*f^2 + 48*B^2*a^5*b*d*f^2 + 120*B^2*a^2*b^4*c*f^2 - 120*B^2*a^4*b^2*c*f^2 - 160*B^2*a^3*b^3*d*f^2)^2/4 - (
16*c^2*f^4 + 16*d^2*f^4)*(B^4*a^12 + B^4*b^12 + 6*B^4*a^2*b^10 + 15*B^4*a^4*b^8 + 20*B^4*a^6*b^6 + 15*B^4*a^8*
b^4 + 6*B^4*a^10*b^2))^(1/2) - 4*B^2*a^6*c*f^2 + 4*B^2*b^6*c*f^2 - 24*B^2*a*b^5*d*f^2 - 24*B^2*a^5*b*d*f^2 - 6
0*B^2*a^2*b^4*c*f^2 + 60*B^2*a^4*b^2*c*f^2 + 80*B^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2)*2i + atan((
(((8*(4*B*b^3*d^3*f^2 - 12*B*a^2*b*d^3*f^2 - 4*B*a^3*c*d^2*f^2 + 12*B*a*b^2*c*d^2*f^2))/f^3 - 64*c*d^2*(c + d*
tan(e + f*x))^(1/2)*((((8*B^2*a^6*c*f^2 - 8*B^2*b^6*c*f^2 + 48*B^2*a*b^5*d*f^2 + 48*B^2*a^5*b*d*f^2 + 120*B^2*
a^2*b^4*c*f^2 - 120*B^2*a^4*b^2*c*f^2 - 160*B^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(B^4*a^12 + B^4
*b^12 + 6*B^4*a^2*b^10 + 15*B^4*a^4*b^8 + 20*B^4*a^6*b^6 + 15*B^4*a^8*b^4 + 6*B^4*a^10*b^2))^(1/2) + 4*B^2*a^6
*c*f^2 - 4*B^2*b^6*c*f^2 + 24*B^2*a*b^5*d*f^2 + 24*B^2*a^5*b*d*f^2 + 60*B^2*a^2*b^4*c*f^2 - 60*B^2*a^4*b^2*c*f
^2 - 80*B^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2))*((((8*B^2*a^6*c*f^2 - 8*B^2*b^6*c*f^2 + 48*B^2*a*b
^5*d*f^2 + 48*B^2*a^5*b*d*f^2 + 120*B^2*a^2*b^4*c*f^2 - 120*B^2*a^4*b^2*c*f^2 - 160*B^2*a^3*b^3*d*f^2)^2/4 - (
16*c^2*f^4 + 16*d^2*f^4)*(B^4*a^12 + B^4*b^12 + 6*B^4*a^2*b^10 + 15*B^4*a^4*b^8 + 20*B^4*a^6*b^6 + 15*B^4*a^8*
b^4 + 6*B^4*a^10*b^2))^(1/2) + 4*B^2*a^6*c*f^2 - 4*B^2*b^6*c*f^2 + 24*B^2*a*b^5*d*f^2 + 24*B^2*a^5*b*d*f^2 + 6
0*B^2*a^2*b^4*c*f^2 - 60*B^2*a^4*b^2*c*f^2 - 80*B^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2) + (16*(c +
d*tan(e + f*x))^(1/2)*(B^2*a^6*d^2 - B^2*b^6*d^2 + 15*B^2*a^2*b^4*d^2 - 15*B^2*a^4*b^2*d^2))/f^2)*((((8*B^2*a^
6*c*f^2 - 8*B^2*b^6*c*f^2 + 48*B^2*a*b^5*d*f^2 + 48*B^2*a^5*b*d*f^2 + 120*B^2*a^2*b^4*c*f^2 - 120*B^2*a^4*b^2*
c*f^2 - 160*B^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(B^4*a^12 + B^4*b^12 + 6*B^4*a^2*b^10 + 15*B^4*
a^4*b^8 + 20*B^4*a^6*b^6 + 15*B^4*a^8*b^4 + 6*B^4*a^10*b^2))^(1/2) + 4*B^2*a^6*c*f^2 - 4*B^2*b^6*c*f^2 + 24*B^
2*a*b^5*d*f^2 + 24*B^2*a^5*b*d*f^2 + 60*B^2*a^2*b^4*c*f^2 - 60*B^2*a^4*b^2*c*f^2 - 80*B^2*a^3*b^3*d*f^2)/(16*(
c^2*f^4 + d^2*f^4)))^(1/2)*1i - (((8*(4*B*b^3*d^3*f^2 - 12*B*a^2*b*d^3*f^2 - 4*B*a^3*c*d^2*f^2 + 12*B*a*b^2*c*
d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((((8*B^2*a^6*c*f^2 - 8*B^2*b^6*c*f^2 + 48*B^2*a*b^5*d*f^2
+ 48*B^2*a^5*b*d*f^2 + 120*B^2*a^2*b^4*c*f^2 - 120*B^2*a^4*b^2*c*f^2 - 160*B^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f
^4 + 16*d^2*f^4)*(B^4*a^12 + B^4*b^12 + 6*B^4*a^2*b^10 + 15*B^4*a^4*b^8 + 20*B^4*a^6*b^6 + 15*B^4*a^8*b^4 + 6*
B^4*a^10*b^2))^(1/2) + 4*B^2*a^6*c*f^2 - 4*B^2*b^6*c*f^2 + 24*B^2*a*b^5*d*f^2 + 24*B^2*a^5*b*d*f^2 + 60*B^2*a^
2*b^4*c*f^2 - 60*B^2*a^4*b^2*c*f^2 - 80*B^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2))*((((8*B^2*a^6*c*f^
2 - 8*B^2*b^6*c*f^2 + 48*B^2*a*b^5*d*f^2 + 48*B^2*a^5*b*d*f^2 + 120*B^2*a^2*b^4*c*f^2 - 120*B^2*a^4*b^2*c*f^2
- 160*B^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(B^4*a^12 + B^4*b^12 + 6*B^4*a^2*b^10 + 15*B^4*a^4*b^
8 + 20*B^4*a^6*b^6 + 15*B^4*a^8*b^4 + 6*B^4*a^10*b^2))^(1/2) + 4*B^2*a^6*c*f^2 - 4*B^2*b^6*c*f^2 + 24*B^2*a*b^
5*d*f^2 + 24*B^2*a^5*b*d*f^2 + 60*B^2*a^2*b^4*c*f^2 - 60*B^2*a^4*b^2*c*f^2 - 80*B^2*a^3*b^3*d*f^2)/(16*(c^2*f^
4 + d^2*f^4)))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(B^2*a^6*d^2 - B^2*b^6*d^2 + 15*B^2*a^2*b^4*d^2 - 15*B^2
*a^4*b^2*d^2))/f^2)*((((8*B^2*a^6*c*f^2 - 8*B^2*b^6*c*f^2 + 48*B^2*a*b^5*d*f^2 + 48*B^2*a^5*b*d*f^2 + 120*B^2*
a^2*b^4*c*f^2 - 120*B^2*a^4*b^2*c*f^2 - 160*B^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(B^4*a^12 + B^4
*b^12 + 6*B^4*a^2*b^10 + 15*B^4*a^4*b^8 + 20*B^4*a^6*b^6 + 15*B^4*a^8*b^4 + 6*B^4*a^10*b^2))^(1/2) + 4*B^2*a^6
*c*f^2 - 4*B^2*b^6*c*f^2 + 24*B^2*a*b^5*d*f^2 + 24*B^2*a^5*b*d*f^2 + 60*B^2*a^2*b^4*c*f^2 - 60*B^2*a^4*b^2*c*f
^2 - 80*B^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2)*1i)/((((8*(4*B*b^3*d^3*f^2 - 12*B*a^2*b*d^3*f^2 - 4
*B*a^3*c*d^2*f^2 + 12*B*a*b^2*c*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((((8*B^2*a^6*c*f^2 - 8*B^
2*b^6*c*f^2 + 48*B^2*a*b^5*d*f^2 + 48*B^2*a^5*b*d*f^2 + 120*B^2*a^2*b^4*c*f^2 - 120*B^2*a^4*b^2*c*f^2 - 160*B^
2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(B^4*a^12 + B^4*b^12 + 6*B^4*a^2*b^10 + 15*B^4*a^4*b^8 + 20*B
^4*a^6*b^6 + 15*B^4*a^8*b^4 + 6*B^4*a^10*b^2))^(1/2) + 4*B^2*a^6*c*f^2 - 4*B^2*b^6*c*f^2 + 24*B^2*a*b^5*d*f^2
+ 24*B^2*a^5*b*d*f^2 + 60*B^2*a^2*b^4*c*f^2 - 60*B^2*a^4*b^2*c*f^2 - 80*B^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*
f^4)))^(1/2))*((((8*B^2*a^6*c*f^2 - 8*B^2*b^6*c*f^2 + 48*B^2*a*b^5*d*f^2 + 48*B^2*a^5*b*d*f^2 + 120*B^2*a^2*b^
4*c*f^2 - 120*B^2*a^4*b^2*c*f^2 - 160*B^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(B^4*a^12 + B^4*b^12
+ 6*B^4*a^2*b^10 + 15*B^4*a^4*b^8 + 20*B^4*a^6*b^6 + 15*B^4*a^8*b^4 + 6*B^4*a^10*b^2))^(1/2) + 4*B^2*a^6*c*f^2
- 4*B^2*b^6*c*f^2 + 24*B^2*a*b^5*d*f^2 + 24*B^2*a^5*b*d*f^2 + 60*B^2*a^2*b^4*c*f^2 - 60*B^2*a^4*b^2*c*f^2 - 8
0*B^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(B^2*a^6*d^2 - B^2*b^6*d
^2 + 15*B^2*a^2*b^4*d^2 - 15*B^2*a^4*b^2*d^2))/f^2)*((((8*B^2*a^6*c*f^2 - 8*B^2*b^6*c*f^2 + 48*B^2*a*b^5*d*f^2
+ 48*B^2*a^5*b*d*f^2 + 120*B^2*a^2*b^4*c*f^2 - 120*B^2*a^4*b^2*c*f^2 - 160*B^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f
^4 + 16*d^2*f^4)*(B^4*a^12 + B^4*b^12 + 6*B^4*a^2*b^10 + 15*B^4*a^4*b^8 + 20*B^4*a^6*b^6 + 15*B^4*a^8*b^4 + 6*
B^4*a^10*b^2))^(1/2) + 4*B^2*a^6*c*f^2 - 4*B^2*b^6*c*f^2 + 24*B^2*a*b^5*d*f^2 + 24*B^2*a^5*b*d*f^2 + 60*B^2*a^
2*b^4*c*f^2 - 60*B^2*a^4*b^2*c*f^2 - 80*B^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2) - (16*(8*B^3*a^3*b^
6*d^2 - B^3*a^9*d^2 + 6*B^3*a^5*b^4*d^2 + 3*B^3*a*b^8*d^2))/f^3 + (((8*(4*B*b^3*d^3*f^2 - 12*B*a^2*b*d^3*f^2 -
4*B*a^3*c*d^2*f^2 + 12*B*a*b^2*c*d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((((8*B^2*a^6*c*f^2 - 8*
B^2*b^6*c*f^2 + 48*B^2*a*b^5*d*f^2 + 48*B^2*a^5*b*d*f^2 + 120*B^2*a^2*b^4*c*f^2 - 120*B^2*a^4*b^2*c*f^2 - 160*
B^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(B^4*a^12 + B^4*b^12 + 6*B^4*a^2*b^10 + 15*B^4*a^4*b^8 + 20
*B^4*a^6*b^6 + 15*B^4*a^8*b^4 + 6*B^4*a^10*b^2))^(1/2) + 4*B^2*a^6*c*f^2 - 4*B^2*b^6*c*f^2 + 24*B^2*a*b^5*d*f^
2 + 24*B^2*a^5*b*d*f^2 + 60*B^2*a^2*b^4*c*f^2 - 60*B^2*a^4*b^2*c*f^2 - 80*B^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^
2*f^4)))^(1/2))*((((8*B^2*a^6*c*f^2 - 8*B^2*b^6*c*f^2 + 48*B^2*a*b^5*d*f^2 + 48*B^2*a^5*b*d*f^2 + 120*B^2*a^2*
b^4*c*f^2 - 120*B^2*a^4*b^2*c*f^2 - 160*B^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(B^4*a^12 + B^4*b^1
2 + 6*B^4*a^2*b^10 + 15*B^4*a^4*b^8 + 20*B^4*a^6*b^6 + 15*B^4*a^8*b^4 + 6*B^4*a^10*b^2))^(1/2) + 4*B^2*a^6*c*f
^2 - 4*B^2*b^6*c*f^2 + 24*B^2*a*b^5*d*f^2 + 24*B^2*a^5*b*d*f^2 + 60*B^2*a^2*b^4*c*f^2 - 60*B^2*a^4*b^2*c*f^2 -
80*B^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(B^2*a^6*d^2 - B^2*b^6
*d^2 + 15*B^2*a^2*b^4*d^2 - 15*B^2*a^4*b^2*d^2))/f^2)*((((8*B^2*a^6*c*f^2 - 8*B^2*b^6*c*f^2 + 48*B^2*a*b^5*d*f
^2 + 48*B^2*a^5*b*d*f^2 + 120*B^2*a^2*b^4*c*f^2 - 120*B^2*a^4*b^2*c*f^2 - 160*B^2*a^3*b^3*d*f^2)^2/4 - (16*c^2
*f^4 + 16*d^2*f^4)*(B^4*a^12 + B^4*b^12 + 6*B^4*a^2*b^10 + 15*B^4*a^4*b^8 + 20*B^4*a^6*b^6 + 15*B^4*a^8*b^4 +
6*B^4*a^10*b^2))^(1/2) + 4*B^2*a^6*c*f^2 - 4*B^2*b^6*c*f^2 + 24*B^2*a*b^5*d*f^2 + 24*B^2*a^5*b*d*f^2 + 60*B^2*
a^2*b^4*c*f^2 - 60*B^2*a^4*b^2*c*f^2 - 80*B^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2)))*((((8*B^2*a^6*c
*f^2 - 8*B^2*b^6*c*f^2 + 48*B^2*a*b^5*d*f^2 + 48*B^2*a^5*b*d*f^2 + 120*B^2*a^2*b^4*c*f^2 - 120*B^2*a^4*b^2*c*f
^2 - 160*B^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(B^4*a^12 + B^4*b^12 + 6*B^4*a^2*b^10 + 15*B^4*a^4
*b^8 + 20*B^4*a^6*b^6 + 15*B^4*a^8*b^4 + 6*B^4*a^10*b^2))^(1/2) + 4*B^2*a^6*c*f^2 - 4*B^2*b^6*c*f^2 + 24*B^2*a
*b^5*d*f^2 + 24*B^2*a^5*b*d*f^2 + 60*B^2*a^2*b^4*c*f^2 - 60*B^2*a^4*b^2*c*f^2 - 80*B^2*a^3*b^3*d*f^2)/(16*(c^2
*f^4 + d^2*f^4)))^(1/2)*2i - (c + d*tan(e + f*x))^(3/2)*((2*c*((10*C*b^3*c - 6*C*a*b^2*d)/(d^4*f) - (4*C*b^3*c
)/(d^4*f)))/3 - (20*C*b^3*c^2 + 6*C*a^2*b*d^2 - 24*C*a*b^2*c*d)/(3*d^4*f) + (2*C*b^3*(d^6*f + c^2*d^4*f))/(3*d
^8*f^2)) - atan(((((8*(4*C*a^3*d^3*f^2 - 12*C*a*b^2*d^3*f^2 + 4*C*b^3*c*d^2*f^2 - 12*C*a^2*b*c*d^2*f^2))/f^3 -
64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((((8*C^2*a^6*c*f^2 - 8*C^2*b^6*c*f^2 + 48*C^2*a*b^5*d*f^2 + 48*C^2*a^5*b
*d*f^2 + 120*C^2*a^2*b^4*c*f^2 - 120*C^2*a^4*b^2*c*f^2 - 160*C^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4
)*(C^4*a^12 + C^4*b^12 + 6*C^4*a^2*b^10 + 15*C^4*a^4*b^8 + 20*C^4*a^6*b^6 + 15*C^4*a^8*b^4 + 6*C^4*a^10*b^2))^
(1/2) - 4*C^2*a^6*c*f^2 + 4*C^2*b^6*c*f^2 - 24*C^2*a*b^5*d*f^2 - 24*C^2*a^5*b*d*f^2 - 60*C^2*a^2*b^4*c*f^2 + 6
0*C^2*a^4*b^2*c*f^2 + 80*C^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2))*((((8*C^2*a^6*c*f^2 - 8*C^2*b^6*c
*f^2 + 48*C^2*a*b^5*d*f^2 + 48*C^2*a^5*b*d*f^2 + 120*C^2*a^2*b^4*c*f^2 - 120*C^2*a^4*b^2*c*f^2 - 160*C^2*a^3*b
^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(C^4*a^12 + C^4*b^12 + 6*C^4*a^2*b^10 + 15*C^4*a^4*b^8 + 20*C^4*a^6*
b^6 + 15*C^4*a^8*b^4 + 6*C^4*a^10*b^2))^(1/2) - 4*C^2*a^6*c*f^2 + 4*C^2*b^6*c*f^2 - 24*C^2*a*b^5*d*f^2 - 24*C^
2*a^5*b*d*f^2 - 60*C^2*a^2*b^4*c*f^2 + 60*C^2*a^4*b^2*c*f^2 + 80*C^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^
(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(C^2*a^6*d^2 - C^2*b^6*d^2 + 15*C^2*a^2*b^4*d^2 - 15*C^2*a^4*b^2*d^2))/
f^2)*((((8*C^2*a^6*c*f^2 - 8*C^2*b^6*c*f^2 + 48*C^2*a*b^5*d*f^2 + 48*C^2*a^5*b*d*f^2 + 120*C^2*a^2*b^4*c*f^2 -
120*C^2*a^4*b^2*c*f^2 - 160*C^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(C^4*a^12 + C^4*b^12 + 6*C^4*a
^2*b^10 + 15*C^4*a^4*b^8 + 20*C^4*a^6*b^6 + 15*C^4*a^8*b^4 + 6*C^4*a^10*b^2))^(1/2) - 4*C^2*a^6*c*f^2 + 4*C^2*
b^6*c*f^2 - 24*C^2*a*b^5*d*f^2 - 24*C^2*a^5*b*d*f^2 - 60*C^2*a^2*b^4*c*f^2 + 60*C^2*a^4*b^2*c*f^2 + 80*C^2*a^3
*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2)*1i - (((8*(4*C*a^3*d^3*f^2 - 12*C*a*b^2*d^3*f^2 + 4*C*b^3*c*d^2*f^
2 - 12*C*a^2*b*c*d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((((8*C^2*a^6*c*f^2 - 8*C^2*b^6*c*f^2 + 4
8*C^2*a*b^5*d*f^2 + 48*C^2*a^5*b*d*f^2 + 120*C^2*a^2*b^4*c*f^2 - 120*C^2*a^4*b^2*c*f^2 - 160*C^2*a^3*b^3*d*f^2
)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(C^4*a^12 + C^4*b^12 + 6*C^4*a^2*b^10 + 15*C^4*a^4*b^8 + 20*C^4*a^6*b^6 + 15
*C^4*a^8*b^4 + 6*C^4*a^10*b^2))^(1/2) - 4*C^2*a^6*c*f^2 + 4*C^2*b^6*c*f^2 - 24*C^2*a*b^5*d*f^2 - 24*C^2*a^5*b*
d*f^2 - 60*C^2*a^2*b^4*c*f^2 + 60*C^2*a^4*b^2*c*f^2 + 80*C^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2))*(
(((8*C^2*a^6*c*f^2 - 8*C^2*b^6*c*f^2 + 48*C^2*a*b^5*d*f^2 + 48*C^2*a^5*b*d*f^2 + 120*C^2*a^2*b^4*c*f^2 - 120*C
^2*a^4*b^2*c*f^2 - 160*C^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(C^4*a^12 + C^4*b^12 + 6*C^4*a^2*b^1
0 + 15*C^4*a^4*b^8 + 20*C^4*a^6*b^6 + 15*C^4*a^8*b^4 + 6*C^4*a^10*b^2))^(1/2) - 4*C^2*a^6*c*f^2 + 4*C^2*b^6*c*
f^2 - 24*C^2*a*b^5*d*f^2 - 24*C^2*a^5*b*d*f^2 - 60*C^2*a^2*b^4*c*f^2 + 60*C^2*a^4*b^2*c*f^2 + 80*C^2*a^3*b^3*d
*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(C^2*a^6*d^2 - C^2*b^6*d^2 + 15*C^2*a^2
*b^4*d^2 - 15*C^2*a^4*b^2*d^2))/f^2)*((((8*C^2*a^6*c*f^2 - 8*C^2*b^6*c*f^2 + 48*C^2*a*b^5*d*f^2 + 48*C^2*a^5*b
*d*f^2 + 120*C^2*a^2*b^4*c*f^2 - 120*C^2*a^4*b^2*c*f^2 - 160*C^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4
)*(C^4*a^12 + C^4*b^12 + 6*C^4*a^2*b^10 + 15*C^4*a^4*b^8 + 20*C^4*a^6*b^6 + 15*C^4*a^8*b^4 + 6*C^4*a^10*b^2))^
(1/2) - 4*C^2*a^6*c*f^2 + 4*C^2*b^6*c*f^2 - 24*C^2*a*b^5*d*f^2 - 24*C^2*a^5*b*d*f^2 - 60*C^2*a^2*b^4*c*f^2 + 6
0*C^2*a^4*b^2*c*f^2 + 80*C^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2)*1i)/((((8*(4*C*a^3*d^3*f^2 - 12*C*
a*b^2*d^3*f^2 + 4*C*b^3*c*d^2*f^2 - 12*C*a^2*b*c*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((((8*C^2
*a^6*c*f^2 - 8*C^2*b^6*c*f^2 + 48*C^2*a*b^5*d*f^2 + 48*C^2*a^5*b*d*f^2 + 120*C^2*a^2*b^4*c*f^2 - 120*C^2*a^4*b
^2*c*f^2 - 160*C^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(C^4*a^12 + C^4*b^12 + 6*C^4*a^2*b^10 + 15*C
^4*a^4*b^8 + 20*C^4*a^6*b^6 + 15*C^4*a^8*b^4 + 6*C^4*a^10*b^2))^(1/2) - 4*C^2*a^6*c*f^2 + 4*C^2*b^6*c*f^2 - 24
*C^2*a*b^5*d*f^2 - 24*C^2*a^5*b*d*f^2 - 60*C^2*a^2*b^4*c*f^2 + 60*C^2*a^4*b^2*c*f^2 + 80*C^2*a^3*b^3*d*f^2)/(1
6*(c^2*f^4 + d^2*f^4)))^(1/2))*((((8*C^2*a^6*c*f^2 - 8*C^2*b^6*c*f^2 + 48*C^2*a*b^5*d*f^2 + 48*C^2*a^5*b*d*f^2
+ 120*C^2*a^2*b^4*c*f^2 - 120*C^2*a^4*b^2*c*f^2 - 160*C^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(C^4
*a^12 + C^4*b^12 + 6*C^4*a^2*b^10 + 15*C^4*a^4*b^8 + 20*C^4*a^6*b^6 + 15*C^4*a^8*b^4 + 6*C^4*a^10*b^2))^(1/2)
- 4*C^2*a^6*c*f^2 + 4*C^2*b^6*c*f^2 - 24*C^2*a*b^5*d*f^2 - 24*C^2*a^5*b*d*f^2 - 60*C^2*a^2*b^4*c*f^2 + 60*C^2*
a^4*b^2*c*f^2 + 80*C^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(C^2*a^
6*d^2 - C^2*b^6*d^2 + 15*C^2*a^2*b^4*d^2 - 15*C^2*a^4*b^2*d^2))/f^2)*((((8*C^2*a^6*c*f^2 - 8*C^2*b^6*c*f^2 + 4
8*C^2*a*b^5*d*f^2 + 48*C^2*a^5*b*d*f^2 + 120*C^2*a^2*b^4*c*f^2 - 120*C^2*a^4*b^2*c*f^2 - 160*C^2*a^3*b^3*d*f^2
)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(C^4*a^12 + C^4*b^12 + 6*C^4*a^2*b^10 + 15*C^4*a^4*b^8 + 20*C^4*a^6*b^6 + 15
*C^4*a^8*b^4 + 6*C^4*a^10*b^2))^(1/2) - 4*C^2*a^6*c*f^2 + 4*C^2*b^6*c*f^2 - 24*C^2*a*b^5*d*f^2 - 24*C^2*a^5*b*
d*f^2 - 60*C^2*a^2*b^4*c*f^2 + 60*C^2*a^4*b^2*c*f^2 + 80*C^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2) +
(((8*(4*C*a^3*d^3*f^2 - 12*C*a*b^2*d^3*f^2 + 4*C*b^3*c*d^2*f^2 - 12*C*a^2*b*c*d^2*f^2))/f^3 + 64*c*d^2*(c + d*
tan(e + f*x))^(1/2)*((((8*C^2*a^6*c*f^2 - 8*C^2*b^6*c*f^2 + 48*C^2*a*b^5*d*f^2 + 48*C^2*a^5*b*d*f^2 + 120*C^2*
a^2*b^4*c*f^2 - 120*C^2*a^4*b^2*c*f^2 - 160*C^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(C^4*a^12 + C^4
*b^12 + 6*C^4*a^2*b^10 + 15*C^4*a^4*b^8 + 20*C^4*a^6*b^6 + 15*C^4*a^8*b^4 + 6*C^4*a^10*b^2))^(1/2) - 4*C^2*a^6
*c*f^2 + 4*C^2*b^6*c*f^2 - 24*C^2*a*b^5*d*f^2 - 24*C^2*a^5*b*d*f^2 - 60*C^2*a^2*b^4*c*f^2 + 60*C^2*a^4*b^2*c*f
^2 + 80*C^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2))*((((8*C^2*a^6*c*f^2 - 8*C^2*b^6*c*f^2 + 48*C^2*a*b
^5*d*f^2 + 48*C^2*a^5*b*d*f^2 + 120*C^2*a^2*b^4*c*f^2 - 120*C^2*a^4*b^2*c*f^2 - 160*C^2*a^3*b^3*d*f^2)^2/4 - (
16*c^2*f^4 + 16*d^2*f^4)*(C^4*a^12 + C^4*b^12 + 6*C^4*a^2*b^10 + 15*C^4*a^4*b^8 + 20*C^4*a^6*b^6 + 15*C^4*a^8*
b^4 + 6*C^4*a^10*b^2))^(1/2) - 4*C^2*a^6*c*f^2 + 4*C^2*b^6*c*f^2 - 24*C^2*a*b^5*d*f^2 - 24*C^2*a^5*b*d*f^2 - 6
0*C^2*a^2*b^4*c*f^2 + 60*C^2*a^4*b^2*c*f^2 + 80*C^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2) + (16*(c +
d*tan(e + f*x))^(1/2)*(C^2*a^6*d^2 - C^2*b^6*d^2 + 15*C^2*a^2*b^4*d^2 - 15*C^2*a^4*b^2*d^2))/f^2)*((((8*C^2*a^
6*c*f^2 - 8*C^2*b^6*c*f^2 + 48*C^2*a*b^5*d*f^2 + 48*C^2*a^5*b*d*f^2 + 120*C^2*a^2*b^4*c*f^2 - 120*C^2*a^4*b^2*
c*f^2 - 160*C^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(C^4*a^12 + C^4*b^12 + 6*C^4*a^2*b^10 + 15*C^4*
a^4*b^8 + 20*C^4*a^6*b^6 + 15*C^4*a^8*b^4 + 6*C^4*a^10*b^2))^(1/2) - 4*C^2*a^6*c*f^2 + 4*C^2*b^6*c*f^2 - 24*C^
2*a*b^5*d*f^2 - 24*C^2*a^5*b*d*f^2 - 60*C^2*a^2*b^4*c*f^2 + 60*C^2*a^4*b^2*c*f^2 + 80*C^2*a^3*b^3*d*f^2)/(16*(
c^2*f^4 + d^2*f^4)))^(1/2) + (16*(6*C^3*a^4*b^5*d^2 - C^3*b^9*d^2 + 8*C^3*a^6*b^3*d^2 + 3*C^3*a^8*b*d^2))/f^3)
)*((((8*C^2*a^6*c*f^2 - 8*C^2*b^6*c*f^2 + 48*C^2*a*b^5*d*f^2 + 48*C^2*a^5*b*d*f^2 + 120*C^2*a^2*b^4*c*f^2 - 12
0*C^2*a^4*b^2*c*f^2 - 160*C^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(C^4*a^12 + C^4*b^12 + 6*C^4*a^2*
b^10 + 15*C^4*a^4*b^8 + 20*C^4*a^6*b^6 + 15*C^4*a^8*b^4 + 6*C^4*a^10*b^2))^(1/2) - 4*C^2*a^6*c*f^2 + 4*C^2*b^6
*c*f^2 - 24*C^2*a*b^5*d*f^2 - 24*C^2*a^5*b*d*f^2 - 60*C^2*a^2*b^4*c*f^2 + 60*C^2*a^4*b^2*c*f^2 + 80*C^2*a^3*b^
3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2)*2i - atan(((((8*(4*C*a^3*d^3*f^2 - 12*C*a*b^2*d^3*f^2 + 4*C*b^3*c*d^2
*f^2 - 12*C*a^2*b*c*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(((8*C^2*a^6*c*f^2 - 8*C^2*b^6*c*f^2
+ 48*C^2*a*b^5*d*f^2 + 48*C^2*a^5*b*d*f^2 + 120*C^2*a^2*b^4*c*f^2 - 120*C^2*a^4*b^2*c*f^2 - 160*C^2*a^3*b^3*d
*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(C^4*a^12 + C^4*b^12 + 6*C^4*a^2*b^10 + 15*C^4*a^4*b^8 + 20*C^4*a^6*b^6
+ 15*C^4*a^8*b^4 + 6*C^4*a^10*b^2))^(1/2) + 4*C^2*a^6*c*f^2 - 4*C^2*b^6*c*f^2 + 24*C^2*a*b^5*d*f^2 + 24*C^2*a^
5*b*d*f^2 + 60*C^2*a^2*b^4*c*f^2 - 60*C^2*a^4*b^2*c*f^2 - 80*C^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2
))*(-(((8*C^2*a^6*c*f^2 - 8*C^2*b^6*c*f^2 + 48*C^2*a*b^5*d*f^2 + 48*C^2*a^5*b*d*f^2 + 120*C^2*a^2*b^4*c*f^2 -
120*C^2*a^4*b^2*c*f^2 - 160*C^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(C^4*a^12 + C^4*b^12 + 6*C^4*a^
2*b^10 + 15*C^4*a^4*b^8 + 20*C^4*a^6*b^6 + 15*C^4*a^8*b^4 + 6*C^4*a^10*b^2))^(1/2) + 4*C^2*a^6*c*f^2 - 4*C^2*b
^6*c*f^2 + 24*C^2*a*b^5*d*f^2 + 24*C^2*a^5*b*d*f^2 + 60*C^2*a^2*b^4*c*f^2 - 60*C^2*a^4*b^2*c*f^2 - 80*C^2*a^3*
b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(C^2*a^6*d^2 - C^2*b^6*d^2 + 15*C^
2*a^2*b^4*d^2 - 15*C^2*a^4*b^2*d^2))/f^2)*(-(((8*C^2*a^6*c*f^2 - 8*C^2*b^6*c*f^2 + 48*C^2*a*b^5*d*f^2 + 48*C^2
*a^5*b*d*f^2 + 120*C^2*a^2*b^4*c*f^2 - 120*C^2*a^4*b^2*c*f^2 - 160*C^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d
^2*f^4)*(C^4*a^12 + C^4*b^12 + 6*C^4*a^2*b^10 + 15*C^4*a^4*b^8 + 20*C^4*a^6*b^6 + 15*C^4*a^8*b^4 + 6*C^4*a^10*
b^2))^(1/2) + 4*C^2*a^6*c*f^2 - 4*C^2*b^6*c*f^2 + 24*C^2*a*b^5*d*f^2 + 24*C^2*a^5*b*d*f^2 + 60*C^2*a^2*b^4*c*f
^2 - 60*C^2*a^4*b^2*c*f^2 - 80*C^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2)*1i - (((8*(4*C*a^3*d^3*f^2 -
12*C*a*b^2*d^3*f^2 + 4*C*b^3*c*d^2*f^2 - 12*C*a^2*b*c*d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(
((8*C^2*a^6*c*f^2 - 8*C^2*b^6*c*f^2 + 48*C^2*a*b^5*d*f^2 + 48*C^2*a^5*b*d*f^2 + 120*C^2*a^2*b^4*c*f^2 - 120*C^
2*a^4*b^2*c*f^2 - 160*C^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(C^4*a^12 + C^4*b^12 + 6*C^4*a^2*b^10
+ 15*C^4*a^4*b^8 + 20*C^4*a^6*b^6 + 15*C^4*a^8*b^4 + 6*C^4*a^10*b^2))^(1/2) + 4*C^2*a^6*c*f^2 - 4*C^2*b^6*c*f
^2 + 24*C^2*a*b^5*d*f^2 + 24*C^2*a^5*b*d*f^2 + 60*C^2*a^2*b^4*c*f^2 - 60*C^2*a^4*b^2*c*f^2 - 80*C^2*a^3*b^3*d*
f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2))*(-(((8*C^2*a^6*c*f^2 - 8*C^2*b^6*c*f^2 + 48*C^2*a*b^5*d*f^2 + 48*C^2*a^5
*b*d*f^2 + 120*C^2*a^2*b^4*c*f^2 - 120*C^2*a^4*b^2*c*f^2 - 160*C^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f
^4)*(C^4*a^12 + C^4*b^12 + 6*C^4*a^2*b^10 + 15*C^4*a^4*b^8 + 20*C^4*a^6*b^6 + 15*C^4*a^8*b^4 + 6*C^4*a^10*b^2)
)^(1/2) + 4*C^2*a^6*c*f^2 - 4*C^2*b^6*c*f^2 + 24*C^2*a*b^5*d*f^2 + 24*C^2*a^5*b*d*f^2 + 60*C^2*a^2*b^4*c*f^2 -
60*C^2*a^4*b^2*c*f^2 - 80*C^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)
*(C^2*a^6*d^2 - C^2*b^6*d^2 + 15*C^2*a^2*b^4*d^2 - 15*C^2*a^4*b^2*d^2))/f^2)*(-(((8*C^2*a^6*c*f^2 - 8*C^2*b^6*
c*f^2 + 48*C^2*a*b^5*d*f^2 + 48*C^2*a^5*b*d*f^2 + 120*C^2*a^2*b^4*c*f^2 - 120*C^2*a^4*b^2*c*f^2 - 160*C^2*a^3*
b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(C^4*a^12 + C^4*b^12 + 6*C^4*a^2*b^10 + 15*C^4*a^4*b^8 + 20*C^4*a^6
*b^6 + 15*C^4*a^8*b^4 + 6*C^4*a^10*b^2))^(1/2) + 4*C^2*a^6*c*f^2 - 4*C^2*b^6*c*f^2 + 24*C^2*a*b^5*d*f^2 + 24*C
^2*a^5*b*d*f^2 + 60*C^2*a^2*b^4*c*f^2 - 60*C^2*a^4*b^2*c*f^2 - 80*C^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))
^(1/2)*1i)/((((8*(4*C*a^3*d^3*f^2 - 12*C*a*b^2*d^3*f^2 + 4*C*b^3*c*d^2*f^2 - 12*C*a^2*b*c*d^2*f^2))/f^3 - 64*c
*d^2*(c + d*tan(e + f*x))^(1/2)*(-(((8*C^2*a^6*c*f^2 - 8*C^2*b^6*c*f^2 + 48*C^2*a*b^5*d*f^2 + 48*C^2*a^5*b*d*f
^2 + 120*C^2*a^2*b^4*c*f^2 - 120*C^2*a^4*b^2*c*f^2 - 160*C^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(C
^4*a^12 + C^4*b^12 + 6*C^4*a^2*b^10 + 15*C^4*a^4*b^8 + 20*C^4*a^6*b^6 + 15*C^4*a^8*b^4 + 6*C^4*a^10*b^2))^(1/2
) + 4*C^2*a^6*c*f^2 - 4*C^2*b^6*c*f^2 + 24*C^2*a*b^5*d*f^2 + 24*C^2*a^5*b*d*f^2 + 60*C^2*a^2*b^4*c*f^2 - 60*C^
2*a^4*b^2*c*f^2 - 80*C^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2))*(-(((8*C^2*a^6*c*f^2 - 8*C^2*b^6*c*f^
2 + 48*C^2*a*b^5*d*f^2 + 48*C^2*a^5*b*d*f^2 + 120*C^2*a^2*b^4*c*f^2 - 120*C^2*a^4*b^2*c*f^2 - 160*C^2*a^3*b^3*
d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(C^4*a^12 + C^4*b^12 + 6*C^4*a^2*b^10 + 15*C^4*a^4*b^8 + 20*C^4*a^6*b^6
+ 15*C^4*a^8*b^4 + 6*C^4*a^10*b^2))^(1/2) + 4*C^2*a^6*c*f^2 - 4*C^2*b^6*c*f^2 + 24*C^2*a*b^5*d*f^2 + 24*C^2*a
^5*b*d*f^2 + 60*C^2*a^2*b^4*c*f^2 - 60*C^2*a^4*b^2*c*f^2 - 80*C^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/
2) - (16*(c + d*tan(e + f*x))^(1/2)*(C^2*a^6*d^2 - C^2*b^6*d^2 + 15*C^2*a^2*b^4*d^2 - 15*C^2*a^4*b^2*d^2))/f^2
)*(-(((8*C^2*a^6*c*f^2 - 8*C^2*b^6*c*f^2 + 48*C^2*a*b^5*d*f^2 + 48*C^2*a^5*b*d*f^2 + 120*C^2*a^2*b^4*c*f^2 - 1
20*C^2*a^4*b^2*c*f^2 - 160*C^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(C^4*a^12 + C^4*b^12 + 6*C^4*a^2
*b^10 + 15*C^4*a^4*b^8 + 20*C^4*a^6*b^6 + 15*C^4*a^8*b^4 + 6*C^4*a^10*b^2))^(1/2) + 4*C^2*a^6*c*f^2 - 4*C^2*b^
6*c*f^2 + 24*C^2*a*b^5*d*f^2 + 24*C^2*a^5*b*d*f^2 + 60*C^2*a^2*b^4*c*f^2 - 60*C^2*a^4*b^2*c*f^2 - 80*C^2*a^3*b
^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2) + (((8*(4*C*a^3*d^3*f^2 - 12*C*a*b^2*d^3*f^2 + 4*C*b^3*c*d^2*f^2 - 1
2*C*a^2*b*c*d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(((8*C^2*a^6*c*f^2 - 8*C^2*b^6*c*f^2 + 48*C^
2*a*b^5*d*f^2 + 48*C^2*a^5*b*d*f^2 + 120*C^2*a^2*b^4*c*f^2 - 120*C^2*a^4*b^2*c*f^2 - 160*C^2*a^3*b^3*d*f^2)^2/
4 - (16*c^2*f^4 + 16*d^2*f^4)*(C^4*a^12 + C^4*b^12 + 6*C^4*a^2*b^10 + 15*C^4*a^4*b^8 + 20*C^4*a^6*b^6 + 15*C^4
*a^8*b^4 + 6*C^4*a^10*b^2))^(1/2) + 4*C^2*a^6*c*f^2 - 4*C^2*b^6*c*f^2 + 24*C^2*a*b^5*d*f^2 + 24*C^2*a^5*b*d*f^
2 + 60*C^2*a^2*b^4*c*f^2 - 60*C^2*a^4*b^2*c*f^2 - 80*C^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2))*(-(((
8*C^2*a^6*c*f^2 - 8*C^2*b^6*c*f^2 + 48*C^2*a*b^5*d*f^2 + 48*C^2*a^5*b*d*f^2 + 120*C^2*a^2*b^4*c*f^2 - 120*C^2*
a^4*b^2*c*f^2 - 160*C^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(C^4*a^12 + C^4*b^12 + 6*C^4*a^2*b^10 +
15*C^4*a^4*b^8 + 20*C^4*a^6*b^6 + 15*C^4*a^8*b^4 + 6*C^4*a^10*b^2))^(1/2) + 4*C^2*a^6*c*f^2 - 4*C^2*b^6*c*f^2
+ 24*C^2*a*b^5*d*f^2 + 24*C^2*a^5*b*d*f^2 + 60*C^2*a^2*b^4*c*f^2 - 60*C^2*a^4*b^2*c*f^2 - 80*C^2*a^3*b^3*d*f^
2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(C^2*a^6*d^2 - C^2*b^6*d^2 + 15*C^2*a^2*b^
4*d^2 - 15*C^2*a^4*b^2*d^2))/f^2)*(-(((8*C^2*a^6*c*f^2 - 8*C^2*b^6*c*f^2 + 48*C^2*a*b^5*d*f^2 + 48*C^2*a^5*b*d
*f^2 + 120*C^2*a^2*b^4*c*f^2 - 120*C^2*a^4*b^2*c*f^2 - 160*C^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*
(C^4*a^12 + C^4*b^12 + 6*C^4*a^2*b^10 + 15*C^4*a^4*b^8 + 20*C^4*a^6*b^6 + 15*C^4*a^8*b^4 + 6*C^4*a^10*b^2))^(1
/2) + 4*C^2*a^6*c*f^2 - 4*C^2*b^6*c*f^2 + 24*C^2*a*b^5*d*f^2 + 24*C^2*a^5*b*d*f^2 + 60*C^2*a^2*b^4*c*f^2 - 60*
C^2*a^4*b^2*c*f^2 - 80*C^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2) + (16*(6*C^3*a^4*b^5*d^2 - C^3*b^9*d
^2 + 8*C^3*a^6*b^3*d^2 + 3*C^3*a^8*b*d^2))/f^3))*(-(((8*C^2*a^6*c*f^2 - 8*C^2*b^6*c*f^2 + 48*C^2*a*b^5*d*f^2 +
48*C^2*a^5*b*d*f^2 + 120*C^2*a^2*b^4*c*f^2 - 120*C^2*a^4*b^2*c*f^2 - 160*C^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4
+ 16*d^2*f^4)*(C^4*a^12 + C^4*b^12 + 6*C^4*a^2*b^10 + 15*C^4*a^4*b^8 + 20*C^4*a^6*b^6 + 15*C^4*a^8*b^4 + 6*C^
4*a^10*b^2))^(1/2) + 4*C^2*a^6*c*f^2 - 4*C^2*b^6*c*f^2 + 24*C^2*a*b^5*d*f^2 + 24*C^2*a^5*b*d*f^2 + 60*C^2*a^2*
b^4*c*f^2 - 60*C^2*a^4*b^2*c*f^2 - 80*C^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2)*2i - ((6*A*b^3*c - 6*
A*a*b^2*d)/(d^2*f) - (4*A*b^3*c)/(d^2*f))*(c + d*tan(e + f*x))^(1/2) - ((8*B*b^3*c - 6*B*a*b^2*d)/(3*d^3*f) -
(4*B*b^3*c)/(3*d^3*f))*(c + d*tan(e + f*x))^(3/2) - ((10*C*b^3*c - 6*C*a*b^2*d)/(5*d^4*f) - (4*C*b^3*c)/(5*d^4
*f))*(c + d*tan(e + f*x))^(5/2) + (c + d*tan(e + f*x))^(1/2)*((2*C*a^3*d^3 - 20*C*b^3*c^3 + 36*C*a*b^2*c^2*d -
18*C*a^2*b*c*d^2)/(d^4*f) - 2*c*(2*c*((10*C*b^3*c - 6*C*a*b^2*d)/(d^4*f) - (4*C*b^3*c)/(d^4*f)) - (20*C*b^3*c
^2 + 6*C*a^2*b*d^2 - 24*C*a*b^2*c*d)/(d^4*f) + (2*C*b^3*(d^6*f + c^2*d^4*f))/(d^8*f^2)) + ((d^6*f + c^2*d^4*f)
*((10*C*b^3*c - 6*C*a*b^2*d)/(d^4*f) - (4*C*b^3*c)/(d^4*f)))/(d^4*f)) + (2*A*b^3*(c + d*tan(e + f*x))^(3/2))/(
3*d^2*f) + (2*B*b^3*(c + d*tan(e + f*x))^(5/2))/(5*d^3*f) + (2*C*b^3*(c + d*tan(e + f*x))^(7/2))/(7*d^4*f)