\(\int \cot ^4(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx\) [324]

Optimal result

Integrand size = 33, antiderivative size = 279 \[ \int \cot ^4(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\frac {\left (8 a^2 A b-A b^3+16 a^3 B+2 a b^2 B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{8 a^{5/2} d}-\frac {\sqrt {a-i b} (i A+B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {\sqrt {a+i b} (i A-B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {\left (8 a^2 A+A b^2-2 a b B\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 a^2 d}-\frac {(A b+6 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 a d}-\frac {A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d} \]

[Out]

1/8*(8*A*a^2*b-A*b^3+16*B*a^3+2*B*a*b^2)*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/a^(5/2)/d-(I*A+B)*arctanh((a+
b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))*(a-I*b)^(1/2)/d+(I*A-B)*arctanh((a+b*tan(d*x+c))^(1/2)/(a+I*b)^(1/2))*(a+I*
b)^(1/2)/d+1/8*(8*A*a^2+A*b^2-2*B*a*b)*cot(d*x+c)*(a+b*tan(d*x+c))^(1/2)/a^2/d-1/12*(A*b+6*B*a)*cot(d*x+c)^2*(
a+b*tan(d*x+c))^(1/2)/a/d-1/3*A*cot(d*x+c)^3*(a+b*tan(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 1.36 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {3689, 3730, 3734, 3620, 3618, 65, 214, 3715} \[ \int \cot ^4(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\frac {\left (8 a^2 A-2 a b B+A b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 a^2 d}+\frac {\left (16 a^3 B+8 a^2 A b+2 a b^2 B-A b^3\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{8 a^{5/2} d}-\frac {\sqrt {a-i b} (B+i A) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {\sqrt {a+i b} (-B+i A) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {(6 a B+A b) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 a d}-\frac {A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d} \]

[In]

Int[Cot[c + d*x]^4*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]

[Out]

((8*a^2*A*b - A*b^3 + 16*a^3*B + 2*a*b^2*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(8*a^(5/2)*d) - (Sqrt[a
- I*b]*(I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d + (Sqrt[a + I*b]*(I*A - B)*ArcTanh[Sqrt[a
+ b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + ((8*a^2*A + A*b^2 - 2*a*b*B)*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(8*
a^2*d) - ((A*b + 6*a*B)*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(12*a*d) - (A*Cot[c + d*x]^3*Sqrt[a + b*Tan[c
+ d*x]])/(3*d)

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 3689

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

Rule 3715

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*
tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[A/f, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x]
/; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]

Rule 3730

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Ta
n[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*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] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3734

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

Rubi steps \begin{align*} \text {integral}& = -\frac {A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}-\frac {1}{3} \int \frac {\cot ^3(c+d x) \left (\frac {1}{2} (-A b-6 a B)+3 (a A-b B) \tan (c+d x)+\frac {5}{2} A b \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx \\ & = -\frac {(A b+6 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 a d}-\frac {A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {\int \frac {\cot ^2(c+d x) \left (-\frac {3}{4} \left (8 a^2 A+A b^2-2 a b B\right )-6 a (A b+a B) \tan (c+d x)-\frac {3}{4} b (A b+6 a B) \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{6 a} \\ & = \frac {\left (8 a^2 A+A b^2-2 a b B\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 a^2 d}-\frac {(A b+6 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 a d}-\frac {A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}-\frac {\int \frac {\cot (c+d x) \left (\frac {3}{8} \left (8 a^2 A b-A b^3+16 a^3 B+2 a b^2 B\right )-6 a^2 (a A-b B) \tan (c+d x)-\frac {3}{8} b \left (8 a^2 A+A b^2-2 a b B\right ) \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{6 a^2} \\ & = \frac {\left (8 a^2 A+A b^2-2 a b B\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 a^2 d}-\frac {(A b+6 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 a d}-\frac {A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}-\frac {\int \frac {-6 a^2 (a A-b B)-6 a^2 (A b+a B) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{6 a^2}-\frac {\left (8 a^2 A b-A b^3+16 a^3 B+2 a b^2 B\right ) \int \frac {\cot (c+d x) \left (1+\tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{16 a^2} \\ & = \frac {\left (8 a^2 A+A b^2-2 a b B\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 a^2 d}-\frac {(A b+6 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 a d}-\frac {A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{2} ((a-i b) (A-i B)) \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx+\frac {1}{2} ((a+i b) (A+i B)) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx-\frac {\left (8 a^2 A b-A b^3+16 a^3 B+2 a b^2 B\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{16 a^2 d} \\ & = \frac {\left (8 a^2 A+A b^2-2 a b B\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 a^2 d}-\frac {(A b+6 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 a d}-\frac {A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {(i (a-i b) (A-i B)) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 d}-\frac {((i a-b) (A+i B)) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 d}-\frac {\left (8 a^2 A b-A b^3+16 a^3 B+2 a b^2 B\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{8 a^2 b d} \\ & = \frac {\left (8 a^2 A b-A b^3+16 a^3 B+2 a b^2 B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{8 a^{5/2} d}+\frac {\left (8 a^2 A+A b^2-2 a b B\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 a^2 d}-\frac {(A b+6 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 a d}-\frac {A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}-\frac {((a-i b) (A-i B)) \text {Subst}\left (\int \frac {1}{-1-\frac {i a}{b}+\frac {i x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{b d}-\frac {((a+i b) (A+i B)) \text {Subst}\left (\int \frac {1}{-1+\frac {i a}{b}-\frac {i x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{b d} \\ & = \frac {\left (8 a^2 A b-A b^3+16 a^3 B+2 a b^2 B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{8 a^{5/2} d}-\frac {\sqrt {a-i b} (i A+B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {\sqrt {a+i b} (i A-B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {\left (8 a^2 A+A b^2-2 a b B\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 a^2 d}-\frac {(A b+6 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 a d}-\frac {A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(566\) vs. \(2(279)=558\).

Time = 6.49 (sec) , antiderivative size = 566, normalized size of antiderivative = 2.03 \[ \int \cot ^4(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\frac {2 b^4 \left (\frac {5 A \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{16 a^{5/2} b}+\frac {(A b+a B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} b^4}-\frac {3 (A b+a B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{8 a^{5/2} b^2}-\frac {(a A-b B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{2 a^{3/2} b^3}+\frac {\left (a A b-A b \sqrt {-b^2}-b^2 B-a \sqrt {-b^2} B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {-b^2}}}\right )}{2 b^4 \sqrt {-b^2} \sqrt {a-\sqrt {-b^2}}}-\frac {\left (a A b+A b \sqrt {-b^2}-b^2 B+a \sqrt {-b^2} B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+\sqrt {-b^2}}}\right )}{2 \left (-b^2\right )^{5/2} \sqrt {a+\sqrt {-b^2}}}-\frac {5 A \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{16 a^2 b^2}+\frac {3 (A b+a B) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 a^2 b^3}+\frac {(a A-b B) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{2 a b^4}+\frac {5 A \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{24 a b^3}-\frac {(A b+a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{4 a b^4}-\frac {A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{6 b^4}\right )}{d} \]

[In]

Integrate[Cot[c + d*x]^4*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]

[Out]

(2*b^4*((5*A*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(16*a^(5/2)*b) + ((A*b + a*B)*ArcTanh[Sqrt[a + b*Tan[c
+ d*x]]/Sqrt[a]])/(Sqrt[a]*b^4) - (3*(A*b + a*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(8*a^(5/2)*b^2) -
((a*A - b*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(2*a^(3/2)*b^3) + ((a*A*b - A*b*Sqrt[-b^2] - b^2*B -
a*Sqrt[-b^2]*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/(2*b^4*Sqrt[-b^2]*Sqrt[a - Sqrt[-b^2]]
) - ((a*A*b + A*b*Sqrt[-b^2] - b^2*B + a*Sqrt[-b^2]*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])
/(2*(-b^2)^(5/2)*Sqrt[a + Sqrt[-b^2]]) - (5*A*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(16*a^2*b^2) + (3*(A*b +
a*B)*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(8*a^2*b^3) + ((a*A - b*B)*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/
(2*a*b^4) + (5*A*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(24*a*b^3) - ((A*b + a*B)*Cot[c + d*x]^2*Sqrt[a + b*
Tan[c + d*x]])/(4*a*b^4) - (A*Cot[c + d*x]^3*Sqrt[a + b*Tan[c + d*x]])/(6*b^4)))/d

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1294\) vs. \(2(241)=482\).

Time = 0.23 (sec) , antiderivative size = 1295, normalized size of antiderivative = 4.64

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1295\)
default	\(\text {Expression too large to display}\)	\(1295\)

[In]

int(cot(d*x+c)^4*(a+b*tan(d*x+c))^(1/2)*(A+B*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-1/4/d/b*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*A*(2*(a^2+b^2
)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)+1/4/d/b*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1
/2)+(a^2+b^2)^(1/2))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a-1/4/d*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+
b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-1/d/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arcta
n((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B*(a^2+b^2)^(1/2)+1/
d*b/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)
^(1/2)-2*a)^(1/2))*A+1/d/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a
)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B*a+1/4/d/b*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*
tan(d*x+c)-a-(a^2+b^2)^(1/2))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)-1/4/d/b*ln((a+b*tan(d*x+c))^(1/2
)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a+1/4/d*ln((a+
b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1
/2)+1/d/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+
b^2)^(1/2)-2*a)^(1/2))*B*(a^2+b^2)^(1/2)-1/d*b/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(
1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*A-1/d/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(
a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B*a+1/d/b^2/tan(d*x+c)^3*(a
+b*tan(d*x+c))^(5/2)*A+1/8/d/tan(d*x+c)^3/a^2*(a+b*tan(d*x+c))^(5/2)*A-1/4/d/b/tan(d*x+c)^3/a*(a+b*tan(d*x+c))
^(5/2)*B-2/d/b^2/tan(d*x+c)^3*A*a*(a+b*tan(d*x+c))^(3/2)-1/3/d/tan(d*x+c)^3*A/a*(a+b*tan(d*x+c))^(3/2)+1/d/b^2
/tan(d*x+c)^3*(a+b*tan(d*x+c))^(1/2)*A*a^2-1/8/d/tan(d*x+c)^3*(a+b*tan(d*x+c))^(1/2)*A+1/4/d/b/tan(d*x+c)^3*(a
+b*tan(d*x+c))^(1/2)*B*a+1/d*b/a^(1/2)*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))*A-1/8/d*b^3/a^(5/2)*arctanh((a+
b*tan(d*x+c))^(1/2)/a^(1/2))*A+2/d*a^(1/2)*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))*B+1/4/d*b^2/a^(3/2)*arctanh
((a+b*tan(d*x+c))^(1/2)/a^(1/2))*B

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1376 vs. \(2 (235) = 470\).

Time = 9.35 (sec) , antiderivative size = 2769, normalized size of antiderivative = 9.92 \[ \int \cot ^4(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

integrate(cot(d*x+c)^4*(a+b*tan(d*x+c))^(1/2)*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

[-1/48*(24*a^3*d*sqrt((2*A*B*b + d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^
2)/d^4) - (A^2 - B^2)*a)/d^2)*log(-(2*(A^3*B + A*B^3)*a + (A^4 - B^4)*b)*sqrt(b*tan(d*x + c) + a) + (A*d^3*sqr
t(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (2*A*B^2*a + (A^2*B - B^3)*b)*
d)*sqrt((2*A*B*b + d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2
- B^2)*a)/d^2))*tan(d*x + c)^3 - 24*a^3*d*sqrt((2*A*B*b + d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b +
(A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2)*log(-(2*(A^3*B + A*B^3)*a + (A^4 - B^4)*b)*sqrt(b*tan(
d*x + c) + a) - (A*d^3*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (2*A
*B^2*a + (A^2*B - B^3)*b)*d)*sqrt((2*A*B*b + d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B
^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2))*tan(d*x + c)^3 - 24*a^3*d*sqrt((2*A*B*b - d^2*sqrt(-(4*A^2*B^2*a^2
+ 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2)*log(-(2*(A^3*B + A*B^3)*a +
(A^4 - B^4)*b)*sqrt(b*tan(d*x + c) + a) + (A*d^3*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B
^2 + B^4)*b^2)/d^4) + (2*A*B^2*a + (A^2*B - B^3)*b)*d)*sqrt((2*A*B*b - d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A
*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2))*tan(d*x + c)^3 + 24*a^3*d*sqrt((2*A*B*b -
d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2)*lo
g(-(2*(A^3*B + A*B^3)*a + (A^4 - B^4)*b)*sqrt(b*tan(d*x + c) + a) - (A*d^3*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A
*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) + (2*A*B^2*a + (A^2*B - B^3)*b)*d)*sqrt((2*A*B*b - d^2*sqrt(-(4*
A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2))*tan(d*x + c)^3
+ 3*(16*B*a^3 + 8*A*a^2*b + 2*B*a*b^2 - A*b^3)*sqrt(a)*log((b*tan(d*x + c) - 2*sqrt(b*tan(d*x + c) + a)*sqrt(a
) + 2*a)/tan(d*x + c))*tan(d*x + c)^3 + 2*(8*A*a^3 - 3*(8*A*a^3 - 2*B*a^2*b + A*a*b^2)*tan(d*x + c)^2 + 2*(6*B
*a^3 + A*a^2*b)*tan(d*x + c))*sqrt(b*tan(d*x + c) + a))/(a^3*d*tan(d*x + c)^3), -1/24*(12*a^3*d*sqrt((2*A*B*b
+ d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2)*l
og(-(2*(A^3*B + A*B^3)*a + (A^4 - B^4)*b)*sqrt(b*tan(d*x + c) + a) + (A*d^3*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B -
A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (2*A*B^2*a + (A^2*B - B^3)*b)*d)*sqrt((2*A*B*b + d^2*sqrt(-(4
*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2))*tan(d*x + c)^3
- 12*a^3*d*sqrt((2*A*B*b + d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^
4) - (A^2 - B^2)*a)/d^2)*log(-(2*(A^3*B + A*B^3)*a + (A^4 - B^4)*b)*sqrt(b*tan(d*x + c) + a) - (A*d^3*sqrt(-(4
*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (2*A*B^2*a + (A^2*B - B^3)*b)*d)*sq
rt((2*A*B*b + d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^
2)*a)/d^2))*tan(d*x + c)^3 - 12*a^3*d*sqrt((2*A*B*b - d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4
- 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2)*log(-(2*(A^3*B + A*B^3)*a + (A^4 - B^4)*b)*sqrt(b*tan(d*x +
c) + a) + (A*d^3*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) + (2*A*B^2*
a + (A^2*B - B^3)*b)*d)*sqrt((2*A*B*b - d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 +
B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2))*tan(d*x + c)^3 + 12*a^3*d*sqrt((2*A*B*b - d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(
A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2)*log(-(2*(A^3*B + A*B^3)*a + (A^4
- B^4)*b)*sqrt(b*tan(d*x + c) + a) - (A*d^3*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 +
B^4)*b^2)/d^4) + (2*A*B^2*a + (A^2*B - B^3)*b)*d)*sqrt((2*A*B*b - d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)
*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2))*tan(d*x + c)^3 + 3*(16*B*a^3 + 8*A*a^2*b + 2*B
*a*b^2 - A*b^3)*sqrt(-a)*arctan(sqrt(b*tan(d*x + c) + a)*sqrt(-a)/a)*tan(d*x + c)^3 + (8*A*a^3 - 3*(8*A*a^3 -
2*B*a^2*b + A*a*b^2)*tan(d*x + c)^2 + 2*(6*B*a^3 + A*a^2*b)*tan(d*x + c))*sqrt(b*tan(d*x + c) + a))/(a^3*d*tan
(d*x + c)^3)]

Sympy [F]

\[ \int \cot ^4(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\int \left (A + B \tan {\left (c + d x \right )}\right ) \sqrt {a + b \tan {\left (c + d x \right )}} \cot ^{4}{\left (c + d x \right )}\, dx \]

[In]

integrate(cot(d*x+c)**4*(a+b*tan(d*x+c))**(1/2)*(A+B*tan(d*x+c)),x)

[Out]

Integral((A + B*tan(c + d*x))*sqrt(a + b*tan(c + d*x))*cot(c + d*x)**4, x)

Maxima [F]

\[ \int \cot ^4(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} \sqrt {b \tan \left (d x + c\right ) + a} \cot \left (d x + c\right )^{4} \,d x } \]

[In]

integrate(cot(d*x+c)^4*(a+b*tan(d*x+c))^(1/2)*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

integrate((B*tan(d*x + c) + A)*sqrt(b*tan(d*x + c) + a)*cot(d*x + c)^4, x)

Giac [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(cot(d*x+c)^4*(a+b*tan(d*x+c))^(1/2)*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 9.61 (sec) , antiderivative size = 16796, normalized size of antiderivative = 60.20 \[ \int \cot ^4(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

int(cot(c + d*x)^4*(A + B*tan(c + d*x))*(a + b*tan(c + d*x))^(1/2),x)

[Out]

atan(((((((224*A*a^4*b^11*d^4 - 32*A*a^2*b^13*d^4 + 256*A*a^6*b^9*d^4 + 64*B*a^3*b^12*d^4 + 448*B*a^5*b^10*d^4
+ 384*B*a^7*b^8*d^4)/(a^4*d^5) - ((2048*a^4*b^10*d^4 + 3072*a^6*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((B^2*a)/
(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d
^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2))/(4*a^4*d^4))*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2)
- (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2
)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(16*B^2*a^3*b^12*d^2 - 512*A^2*a^5*b^10*d^2 -
1280*A^2*a^7*b^8*d^2 - 64*A^2*a^3*b^12*d^2 + 1024*B^2*a^5*b^10*d^2 + 2304*B^2*a^7*b^8*d^2 + 4*A^2*a*b^14*d^2
- 16*A*B*a^2*b^13*d^2 + 2048*A*B*a^4*b^11*d^2 + 4096*A*B*a^6*b^9*d^2))/(4*a^4*d^4))*((B^2*a)/(4*d^2) - (A^2*a)
/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*
d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2) - (16*A^3*a^3*b^13*d^2 - 144*A^3*a^5*b^11*d^2 - 160*A^3*a^7*b^9*d^
2 - 2*B^3*a^2*b^14*d^2 - 2*B^3*a^4*b^12*d^2 + 96*B^3*a^6*b^10*d^2 + 96*B^3*a^8*b^8*d^2 - (A^2*B*b^16*d^2)/2 +
2*A*B^2*a*b^15*d^2 + 50*A*B^2*a^3*b^13*d^2 + 528*A*B^2*a^5*b^11*d^2 + 480*A*B^2*a^7*b^9*d^2 - (49*A^2*B*a^2*b^
14*d^2)/2 + 168*A^2*B*a^4*b^12*d^2 - 96*A^2*B*a^6*b^10*d^2 - 288*A^2*B*a^8*b^8*d^2)/(a^4*d^5))*((B^2*a)/(4*d^2
) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4
*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(A^4*b^16 - A^2*B^2*b^16
- 17*A^4*a^2*b^14 + 208*A^4*a^4*b^12 + 192*A^4*a^6*b^10 + 128*A^4*a^8*b^8 - 4*B^4*a^2*b^14 + 68*B^4*a^4*b^12 +
64*B^4*a^6*b^10 + 384*B^4*a^8*b^8 + 5*A^2*B^2*a^2*b^14 + 236*A^2*B^2*a^4*b^12 + 1792*A^2*B^2*a^6*b^10 + 4*A*B
^3*a*b^15 + 12*A*B^3*a^3*b^13 + 1280*A*B^3*a^7*b^9 - 60*A^3*B*a^3*b^13 + 512*A^3*B*a^5*b^11 - 256*A^3*B*a^7*b^
9))/(4*a^4*d^4))*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A
^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2)*1i - (((((224*A*a^4*b^1
1*d^4 - 32*A*a^2*b^13*d^4 + 256*A*a^6*b^9*d^4 + 64*B*a^3*b^12*d^4 + 448*B*a^5*b^10*d^4 + 384*B*a^7*b^8*d^4)/(a
^4*d^5) + ((2048*a^4*b^10*d^4 + 3072*a^6*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2
) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1
/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2))/(4*a^4*d^4))*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B
^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d
^2))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(16*B^2*a^3*b^12*d^2 - 512*A^2*a^5*b^10*d^2 - 1280*A^2*a^7*b^8*d^2 -
64*A^2*a^3*b^12*d^2 + 1024*B^2*a^5*b^10*d^2 + 2304*B^2*a^7*b^8*d^2 + 4*A^2*a*b^14*d^2 - 16*A*B*a^2*b^13*d^2 +
2048*A*B*a^4*b^11*d^2 + 4096*A*B*a^6*b^9*d^2))/(4*a^4*d^4))*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^
2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*
B*b)/(2*d^2))^(1/2) - (16*A^3*a^3*b^13*d^2 - 144*A^3*a^5*b^11*d^2 - 160*A^3*a^7*b^9*d^2 - 2*B^3*a^2*b^14*d^2 -
2*B^3*a^4*b^12*d^2 + 96*B^3*a^6*b^10*d^2 + 96*B^3*a^8*b^8*d^2 - (A^2*B*b^16*d^2)/2 + 2*A*B^2*a*b^15*d^2 + 50*
A*B^2*a^3*b^13*d^2 + 528*A*B^2*a^5*b^11*d^2 + 480*A*B^2*a^7*b^9*d^2 - (49*A^2*B*a^2*b^14*d^2)/2 + 168*A^2*B*a^
4*b^12*d^2 - 96*A^2*B*a^6*b^10*d^2 - 288*A^2*B*a^8*b^8*d^2)/(a^4*d^5))*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2
*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4
*d^4) + (A*B*b)/(2*d^2))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(A^4*b^16 - A^2*B^2*b^16 - 17*A^4*a^2*b^14 + 208*
A^4*a^4*b^12 + 192*A^4*a^6*b^10 + 128*A^4*a^8*b^8 - 4*B^4*a^2*b^14 + 68*B^4*a^4*b^12 + 64*B^4*a^6*b^10 + 384*B
^4*a^8*b^8 + 5*A^2*B^2*a^2*b^14 + 236*A^2*B^2*a^4*b^12 + 1792*A^2*B^2*a^6*b^10 + 4*A*B^3*a*b^15 + 12*A*B^3*a^3
*b^13 + 1280*A*B^3*a^7*b^9 - 60*A^3*B*a^3*b^13 + 512*A^3*B*a^5*b^11 - 256*A^3*B*a^7*b^9))/(4*a^4*d^4))*((B^2*a
)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b
*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2)*1i)/((((((224*A*a^4*b^11*d^4 - 32*A*a^2*b^13*d^
4 + 256*A*a^6*b^9*d^4 + 64*B*a^3*b^12*d^4 + 448*B*a^5*b^10*d^4 + 384*B*a^7*b^8*d^4)/(a^4*d^5) - ((2048*a^4*b^1
0*d^4 + 3072*a^6*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 -
B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2
*d^2))^(1/2))/(4*a^4*d^4))*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a
^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2) + ((a + b*tan
(c + d*x))^(1/2)*(16*B^2*a^3*b^12*d^2 - 512*A^2*a^5*b^10*d^2 - 1280*A^2*a^7*b^8*d^2 - 64*A^2*a^3*b^12*d^2 + 10
24*B^2*a^5*b^10*d^2 + 2304*B^2*a^7*b^8*d^2 + 4*A^2*a*b^14*d^2 - 16*A*B*a^2*b^13*d^2 + 2048*A*B*a^4*b^11*d^2 +
4096*A*B*a^6*b^9*d^2))/(4*a^4*d^4))*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*
A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2) - (1
6*A^3*a^3*b^13*d^2 - 144*A^3*a^5*b^11*d^2 - 160*A^3*a^7*b^9*d^2 - 2*B^3*a^2*b^14*d^2 - 2*B^3*a^4*b^12*d^2 + 96
*B^3*a^6*b^10*d^2 + 96*B^3*a^8*b^8*d^2 - (A^2*B*b^16*d^2)/2 + 2*A*B^2*a*b^15*d^2 + 50*A*B^2*a^3*b^13*d^2 + 528
*A*B^2*a^5*b^11*d^2 + 480*A*B^2*a^7*b^9*d^2 - (49*A^2*B*a^2*b^14*d^2)/2 + 168*A^2*B*a^4*b^12*d^2 - 96*A^2*B*a^
6*b^10*d^2 - 288*A^2*B*a^8*b^8*d^2)/(a^4*d^5))*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b
^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))
^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(A^4*b^16 - A^2*B^2*b^16 - 17*A^4*a^2*b^14 + 208*A^4*a^4*b^12 + 192*A^4*a
^6*b^10 + 128*A^4*a^8*b^8 - 4*B^4*a^2*b^14 + 68*B^4*a^4*b^12 + 64*B^4*a^6*b^10 + 384*B^4*a^8*b^8 + 5*A^2*B^2*a
^2*b^14 + 236*A^2*B^2*a^4*b^12 + 1792*A^2*B^2*a^6*b^10 + 4*A*B^3*a*b^15 + 12*A*B^3*a^3*b^13 + 1280*A*B^3*a^7*b
^9 - 60*A^3*B*a^3*b^13 + 512*A^3*B*a^5*b^11 - 256*A^3*B*a^7*b^9))/(4*a^4*d^4))*((B^2*a)/(4*d^2) - (A^2*a)/(4*d
^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^
(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2) + (((((224*A*a^4*b^11*d^4 - 32*A*a^2*b^13*d^4 + 256*A*a^6*b^9*d^4 + 64*
B*a^3*b^12*d^4 + 448*B*a^5*b^10*d^4 + 384*B*a^7*b^8*d^4)/(a^4*d^5) + ((2048*a^4*b^10*d^4 + 3072*a^6*b^8*d^4)*(
a + b*tan(c + d*x))^(1/2)*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^
2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2))/(4*a^4*d^4))*
((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*
B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(16*B^2*a^
3*b^12*d^2 - 512*A^2*a^5*b^10*d^2 - 1280*A^2*a^7*b^8*d^2 - 64*A^2*a^3*b^12*d^2 + 1024*B^2*a^5*b^10*d^2 + 2304*
B^2*a^7*b^8*d^2 + 4*A^2*a*b^14*d^2 - 16*A*B*a^2*b^13*d^2 + 2048*A*B*a^4*b^11*d^2 + 4096*A*B*a^6*b^9*d^2))/(4*a
^4*d^4))*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d
^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2) - (16*A^3*a^3*b^13*d^2 - 144*A^
3*a^5*b^11*d^2 - 160*A^3*a^7*b^9*d^2 - 2*B^3*a^2*b^14*d^2 - 2*B^3*a^4*b^12*d^2 + 96*B^3*a^6*b^10*d^2 + 96*B^3*
a^8*b^8*d^2 - (A^2*B*b^16*d^2)/2 + 2*A*B^2*a*b^15*d^2 + 50*A*B^2*a^3*b^13*d^2 + 528*A*B^2*a^5*b^11*d^2 + 480*A
*B^2*a^7*b^9*d^2 - (49*A^2*B*a^2*b^14*d^2)/2 + 168*A^2*B*a^4*b^12*d^2 - 96*A^2*B*a^6*b^10*d^2 - 288*A^2*B*a^8*
b^8*d^2)/(a^4*d^5))*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4
- A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2) + ((a + b*tan(c + d*
x))^(1/2)*(A^4*b^16 - A^2*B^2*b^16 - 17*A^4*a^2*b^14 + 208*A^4*a^4*b^12 + 192*A^4*a^6*b^10 + 128*A^4*a^8*b^8 -
4*B^4*a^2*b^14 + 68*B^4*a^4*b^12 + 64*B^4*a^6*b^10 + 384*B^4*a^8*b^8 + 5*A^2*B^2*a^2*b^14 + 236*A^2*B^2*a^4*b
^12 + 1792*A^2*B^2*a^6*b^10 + 4*A*B^3*a*b^15 + 12*A*B^3*a^3*b^13 + 1280*A*B^3*a^7*b^9 - 60*A^3*B*a^3*b^13 + 51
2*A^3*B*a^5*b^11 - 256*A^3*B*a^7*b^9))/(4*a^4*d^4))*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 -
B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*
d^2))^(1/2) + (12*A^5*a^4*b^13 - (A^3*B^2*b^17)/4 - (A^5*a^2*b^15)/4 - (A^5*b^17)/4 + 44*A^5*a^6*b^11 + 32*A^5
*a^8*b^9 + 7*B^5*a^3*b^14 + 63*B^5*a^5*b^12 + 56*B^5*a^7*b^10 + (79*A^2*B^3*a^3*b^14)/4 + 67*A^2*B^3*a^5*b^12
+ 112*A^2*B^3*a^7*b^10 + 64*A^2*B^3*a^9*b^8 - (17*A^3*B^2*a^2*b^15)/4 + 20*A^3*B^2*a^4*b^13 - 40*A^3*B^2*a^6*b
^11 - 64*A^3*B^2*a^8*b^9 + (3*A^4*B*a*b^16)/4 - 4*A*B^4*a^2*b^15 + 8*A*B^4*a^4*b^13 - 84*A*B^4*a^6*b^11 - 96*A
*B^4*a^8*b^9 + (3*A^2*B^3*a*b^16)/4 + (51*A^4*B*a^3*b^14)/4 + 4*A^4*B*a^5*b^12 + 56*A^4*B*a^7*b^10 + 64*A^4*B*
a^9*b^8)/(a^4*d^5)))*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4
- A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2)*2i + atan(((((((224
*A*a^4*b^11*d^4 - 32*A*a^2*b^13*d^4 + 256*A*a^6*b^9*d^4 + 64*B*a^3*b^12*d^4 + 448*B*a^5*b^10*d^4 + 384*B*a^7*b
^8*d^4)/(a^4*d^5) - ((2048*a^4*b^10*d^4 + 3072*a^6*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((2*A^2*B^2*b^2*d^4 - B
^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d
^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2))/(4*a^4*d^4))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2
*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A
*B*b)/(2*d^2))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(16*B^2*a^3*b^12*d^2 - 512*A^2*a^5*b^10*d^2 - 1280*A^2*a^7*
b^8*d^2 - 64*A^2*a^3*b^12*d^2 + 1024*B^2*a^5*b^10*d^2 + 2304*B^2*a^7*b^8*d^2 + 4*A^2*a*b^14*d^2 - 16*A*B*a^2*b
^13*d^2 + 2048*A*B*a^4*b^11*d^2 + 4096*A*B*a^6*b^9*d^2))/(4*a^4*d^4))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^
2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*
d^2) + (A*B*b)/(2*d^2))^(1/2) - (16*A^3*a^3*b^13*d^2 - 144*A^3*a^5*b^11*d^2 - 160*A^3*a^7*b^9*d^2 - 2*B^3*a^2*
b^14*d^2 - 2*B^3*a^4*b^12*d^2 + 96*B^3*a^6*b^10*d^2 + 96*B^3*a^8*b^8*d^2 - (A^2*B*b^16*d^2)/2 + 2*A*B^2*a*b^15
*d^2 + 50*A*B^2*a^3*b^13*d^2 + 528*A*B^2*a^5*b^11*d^2 + 480*A*B^2*a^7*b^9*d^2 - (49*A^2*B*a^2*b^14*d^2)/2 + 16
8*A^2*B*a^4*b^12*d^2 - 96*A^2*B*a^6*b^10*d^2 - 288*A^2*B*a^8*b^8*d^2)/(a^4*d^5))*((2*A^2*B^2*b^2*d^4 - B^4*b^2
*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) +
(B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(A^4*b^16 - A^2*B^2*b^16 - 17*A^4*a^2*b
^14 + 208*A^4*a^4*b^12 + 192*A^4*a^6*b^10 + 128*A^4*a^8*b^8 - 4*B^4*a^2*b^14 + 68*B^4*a^4*b^12 + 64*B^4*a^6*b^
10 + 384*B^4*a^8*b^8 + 5*A^2*B^2*a^2*b^14 + 236*A^2*B^2*a^4*b^12 + 1792*A^2*B^2*a^6*b^10 + 4*A*B^3*a*b^15 + 12
*A*B^3*a^3*b^13 + 1280*A*B^3*a^7*b^9 - 60*A^3*B*a^3*b^13 + 512*A^3*B*a^5*b^11 - 256*A^3*B*a^7*b^9))/(4*a^4*d^4
))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1
/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2)*1i - (((((224*A*a^4*b^11*d^4 - 32*A*a
^2*b^13*d^4 + 256*A*a^6*b^9*d^4 + 64*B*a^3*b^12*d^4 + 448*B*a^5*b^10*d^4 + 384*B*a^7*b^8*d^4)/(a^4*d^5) + ((20
48*a^4*b^10*d^4 + 3072*a^6*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a
^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) +
(A*B*b)/(2*d^2))^(1/2))/(4*a^4*d^4))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A
*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2) - (
(a + b*tan(c + d*x))^(1/2)*(16*B^2*a^3*b^12*d^2 - 512*A^2*a^5*b^10*d^2 - 1280*A^2*a^7*b^8*d^2 - 64*A^2*a^3*b^1
2*d^2 + 1024*B^2*a^5*b^10*d^2 + 2304*B^2*a^7*b^8*d^2 + 4*A^2*a*b^14*d^2 - 16*A*B*a^2*b^13*d^2 + 2048*A*B*a^4*b
^11*d^2 + 4096*A*B*a^6*b^9*d^2))/(4*a^4*d^4))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*
d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^
(1/2) - (16*A^3*a^3*b^13*d^2 - 144*A^3*a^5*b^11*d^2 - 160*A^3*a^7*b^9*d^2 - 2*B^3*a^2*b^14*d^2 - 2*B^3*a^4*b^1
2*d^2 + 96*B^3*a^6*b^10*d^2 + 96*B^3*a^8*b^8*d^2 - (A^2*B*b^16*d^2)/2 + 2*A*B^2*a*b^15*d^2 + 50*A*B^2*a^3*b^13
*d^2 + 528*A*B^2*a^5*b^11*d^2 + 480*A*B^2*a^7*b^9*d^2 - (49*A^2*B*a^2*b^14*d^2)/2 + 168*A^2*B*a^4*b^12*d^2 - 9
6*A^2*B*a^6*b^10*d^2 - 288*A^2*B*a^8*b^8*d^2)/(a^4*d^5))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4
- A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b
)/(2*d^2))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(A^4*b^16 - A^2*B^2*b^16 - 17*A^4*a^2*b^14 + 208*A^4*a^4*b^12 +
192*A^4*a^6*b^10 + 128*A^4*a^8*b^8 - 4*B^4*a^2*b^14 + 68*B^4*a^4*b^12 + 64*B^4*a^6*b^10 + 384*B^4*a^8*b^8 + 5
*A^2*B^2*a^2*b^14 + 236*A^2*B^2*a^4*b^12 + 1792*A^2*B^2*a^6*b^10 + 4*A*B^3*a*b^15 + 12*A*B^3*a^3*b^13 + 1280*A
*B^3*a^7*b^9 - 60*A^3*B*a^3*b^13 + 512*A^3*B*a^5*b^11 - 256*A^3*B*a^7*b^9))/(4*a^4*d^4))*((2*A^2*B^2*b^2*d^4 -
B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4
*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2)*1i)/((((((224*A*a^4*b^11*d^4 - 32*A*a^2*b^13*d^4 + 256*A*a^6*
b^9*d^4 + 64*B*a^3*b^12*d^4 + 448*B*a^5*b^10*d^4 + 384*B*a^7*b^8*d^4)/(a^4*d^5) - ((2048*a^4*b^10*d^4 + 3072*a
^6*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4
*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2))/
(4*a^4*d^4))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a
*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2) + ((a + b*tan(c + d*x))^(1/
2)*(16*B^2*a^3*b^12*d^2 - 512*A^2*a^5*b^10*d^2 - 1280*A^2*a^7*b^8*d^2 - 64*A^2*a^3*b^12*d^2 + 1024*B^2*a^5*b^1
0*d^2 + 2304*B^2*a^7*b^8*d^2 + 4*A^2*a*b^14*d^2 - 16*A*B*a^2*b^13*d^2 + 2048*A*B*a^4*b^11*d^2 + 4096*A*B*a^6*b
^9*d^2))/(4*a^4*d^4))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 -
4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2) - (16*A^3*a^3*b^13
*d^2 - 144*A^3*a^5*b^11*d^2 - 160*A^3*a^7*b^9*d^2 - 2*B^3*a^2*b^14*d^2 - 2*B^3*a^4*b^12*d^2 + 96*B^3*a^6*b^10*
d^2 + 96*B^3*a^8*b^8*d^2 - (A^2*B*b^16*d^2)/2 + 2*A*B^2*a*b^15*d^2 + 50*A*B^2*a^3*b^13*d^2 + 528*A*B^2*a^5*b^1
1*d^2 + 480*A*B^2*a^7*b^9*d^2 - (49*A^2*B*a^2*b^14*d^2)/2 + 168*A^2*B*a^4*b^12*d^2 - 96*A^2*B*a^6*b^10*d^2 - 2
88*A^2*B*a^8*b^8*d^2)/(a^4*d^5))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3
*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2) - ((a +
b*tan(c + d*x))^(1/2)*(A^4*b^16 - A^2*B^2*b^16 - 17*A^4*a^2*b^14 + 208*A^4*a^4*b^12 + 192*A^4*a^6*b^10 + 128*
A^4*a^8*b^8 - 4*B^4*a^2*b^14 + 68*B^4*a^4*b^12 + 64*B^4*a^6*b^10 + 384*B^4*a^8*b^8 + 5*A^2*B^2*a^2*b^14 + 236*
A^2*B^2*a^4*b^12 + 1792*A^2*B^2*a^6*b^10 + 4*A*B^3*a*b^15 + 12*A*B^3*a^3*b^13 + 1280*A*B^3*a^7*b^9 - 60*A^3*B*
a^3*b^13 + 512*A^3*B*a^5*b^11 - 256*A^3*B*a^7*b^9))/(4*a^4*d^4))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2
*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2)
+ (A*B*b)/(2*d^2))^(1/2) + (((((224*A*a^4*b^11*d^4 - 32*A*a^2*b^13*d^4 + 256*A*a^6*b^9*d^4 + 64*B*a^3*b^12*d^4
+ 448*B*a^5*b^10*d^4 + 384*B*a^7*b^8*d^4)/(a^4*d^5) + ((2048*a^4*b^10*d^4 + 3072*a^6*b^8*d^4)*(a + b*tan(c +
d*x))^(1/2)*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*
b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2))/(4*a^4*d^4))*((2*A^2*B^2*b^
2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^
2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(16*B^2*a^3*b^12*d^2 - 5
12*A^2*a^5*b^10*d^2 - 1280*A^2*a^7*b^8*d^2 - 64*A^2*a^3*b^12*d^2 + 1024*B^2*a^5*b^10*d^2 + 2304*B^2*a^7*b^8*d^
2 + 4*A^2*a*b^14*d^2 - 16*A*B*a^2*b^13*d^2 + 2048*A*B*a^4*b^11*d^2 + 4096*A*B*a^6*b^9*d^2))/(4*a^4*d^4))*((2*A
^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d
^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2) - (16*A^3*a^3*b^13*d^2 - 144*A^3*a^5*b^11*d^2
- 160*A^3*a^7*b^9*d^2 - 2*B^3*a^2*b^14*d^2 - 2*B^3*a^4*b^12*d^2 + 96*B^3*a^6*b^10*d^2 + 96*B^3*a^8*b^8*d^2 -
(A^2*B*b^16*d^2)/2 + 2*A*B^2*a*b^15*d^2 + 50*A*B^2*a^3*b^13*d^2 + 528*A*B^2*a^5*b^11*d^2 + 480*A*B^2*a^7*b^9*d
^2 - (49*A^2*B*a^2*b^14*d^2)/2 + 168*A^2*B*a^4*b^12*d^2 - 96*A^2*B*a^6*b^10*d^2 - 288*A^2*B*a^8*b^8*d^2)/(a^4*
d^5))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)
^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(A^4
*b^16 - A^2*B^2*b^16 - 17*A^4*a^2*b^14 + 208*A^4*a^4*b^12 + 192*A^4*a^6*b^10 + 128*A^4*a^8*b^8 - 4*B^4*a^2*b^1
4 + 68*B^4*a^4*b^12 + 64*B^4*a^6*b^10 + 384*B^4*a^8*b^8 + 5*A^2*B^2*a^2*b^14 + 236*A^2*B^2*a^4*b^12 + 1792*A^2
*B^2*a^6*b^10 + 4*A*B^3*a*b^15 + 12*A*B^3*a^3*b^13 + 1280*A*B^3*a^7*b^9 - 60*A^3*B*a^3*b^13 + 512*A^3*B*a^5*b^
11 - 256*A^3*B*a^7*b^9))/(4*a^4*d^4))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*
A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2) +
(12*A^5*a^4*b^13 - (A^3*B^2*b^17)/4 - (A^5*a^2*b^15)/4 - (A^5*b^17)/4 + 44*A^5*a^6*b^11 + 32*A^5*a^8*b^9 + 7*B
^5*a^3*b^14 + 63*B^5*a^5*b^12 + 56*B^5*a^7*b^10 + (79*A^2*B^3*a^3*b^14)/4 + 67*A^2*B^3*a^5*b^12 + 112*A^2*B^3*
a^7*b^10 + 64*A^2*B^3*a^9*b^8 - (17*A^3*B^2*a^2*b^15)/4 + 20*A^3*B^2*a^4*b^13 - 40*A^3*B^2*a^6*b^11 - 64*A^3*B
^2*a^8*b^9 + (3*A^4*B*a*b^16)/4 - 4*A*B^4*a^2*b^15 + 8*A*B^4*a^4*b^13 - 84*A*B^4*a^6*b^11 - 96*A*B^4*a^8*b^9 +
(3*A^2*B^3*a*b^16)/4 + (51*A^4*B*a^3*b^14)/4 + 4*A^4*B*a^5*b^12 + 56*A^4*B*a^7*b^10 + 64*A^4*B*a^9*b^8)/(a^4*
d^5)))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4
)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2)*2i + ((a + b*tan(c + d*x))^(1/2)*
(A*a^2*b - (A*b^3)/8 + (B*a*b^2)/4) + ((a + b*tan(c + d*x))^(5/2)*(A*b^3 + 8*A*a^2*b - 2*B*a*b^2))/(8*a^2) - (
(A*b^3 + 6*A*a^2*b)*(a + b*tan(c + d*x))^(3/2))/(3*a))/(d*(a + b*tan(c + d*x))^3 - a^3*d - 3*a*d*(a + b*tan(c
+ d*x))^2 + 3*a^2*d*(a + b*tan(c + d*x))) + (atan(-(((((a + b*tan(c + d*x))^(1/2)*(A^4*b^16 - A^2*B^2*b^16 - 1
7*A^4*a^2*b^14 + 208*A^4*a^4*b^12 + 192*A^4*a^6*b^10 + 128*A^4*a^8*b^8 - 4*B^4*a^2*b^14 + 68*B^4*a^4*b^12 + 64
*B^4*a^6*b^10 + 384*B^4*a^8*b^8 + 5*A^2*B^2*a^2*b^14 + 236*A^2*B^2*a^4*b^12 + 1792*A^2*B^2*a^6*b^10 + 4*A*B^3*
a*b^15 + 12*A*B^3*a^3*b^13 + 1280*A*B^3*a^7*b^9 - 60*A^3*B*a^3*b^13 + 512*A^3*B*a^5*b^11 - 256*A^3*B*a^7*b^9))
/(64*a^4*d^4) + (((16*A^3*a^3*b^13*d^2 - 144*A^3*a^5*b^11*d^2 - 160*A^3*a^7*b^9*d^2 - 2*B^3*a^2*b^14*d^2 - 2*B
^3*a^4*b^12*d^2 + 96*B^3*a^6*b^10*d^2 + 96*B^3*a^8*b^8*d^2 - (A^2*B*b^16*d^2)/2 + 2*A*B^2*a*b^15*d^2 + 50*A*B^
2*a^3*b^13*d^2 + 528*A*B^2*a^5*b^11*d^2 + 480*A*B^2*a^7*b^9*d^2 - (49*A^2*B*a^2*b^14*d^2)/2 + 168*A^2*B*a^4*b^
12*d^2 - 96*A^2*B*a^6*b^10*d^2 - 288*A^2*B*a^8*b^8*d^2)/(16*a^4*d^5) - ((((a + b*tan(c + d*x))^(1/2)*(16*B^2*a
^3*b^12*d^2 - 512*A^2*a^5*b^10*d^2 - 1280*A^2*a^7*b^8*d^2 - 64*A^2*a^3*b^12*d^2 + 1024*B^2*a^5*b^10*d^2 + 2304
*B^2*a^7*b^8*d^2 + 4*A^2*a*b^14*d^2 - 16*A*B*a^2*b^13*d^2 + 2048*A*B*a^4*b^11*d^2 + 4096*A*B*a^6*b^9*d^2))/(64
*a^4*d^4) + (((224*A*a^4*b^11*d^4 - 32*A*a^2*b^13*d^4 + 256*A*a^6*b^9*d^4 + 64*B*a^3*b^12*d^4 + 448*B*a^5*b^10
*d^4 + 384*B*a^7*b^8*d^4)/(16*a^4*d^5) - ((2048*a^4*b^10*d^4 + 3072*a^6*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(2
56*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b
- 4*A*B*a^6*b^5)^(1/2))/(1024*a^9*d^5))*(256*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*
a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A*B*a^6*b^5)^(1/2))/(16*a^5*d))*(256*B^2*a^11 + A^2*a^5*b^6 - 16
*A^2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A*B*a^6*b^5)^(1/2))/(16*a^
5*d))*(256*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B
*a^10*b - 4*A*B*a^6*b^5)^(1/2))/(16*a^5*d))*(256*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*
B^2*a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A*B*a^6*b^5)^(1/2)*1i)/(a^5*d) + ((((a + b*tan(c + d*x))^(1/
2)*(A^4*b^16 - A^2*B^2*b^16 - 17*A^4*a^2*b^14 + 208*A^4*a^4*b^12 + 192*A^4*a^6*b^10 + 128*A^4*a^8*b^8 - 4*B^4*
a^2*b^14 + 68*B^4*a^4*b^12 + 64*B^4*a^6*b^10 + 384*B^4*a^8*b^8 + 5*A^2*B^2*a^2*b^14 + 236*A^2*B^2*a^4*b^12 + 1
792*A^2*B^2*a^6*b^10 + 4*A*B^3*a*b^15 + 12*A*B^3*a^3*b^13 + 1280*A*B^3*a^7*b^9 - 60*A^3*B*a^3*b^13 + 512*A^3*B
*a^5*b^11 - 256*A^3*B*a^7*b^9))/(64*a^4*d^4) - (((16*A^3*a^3*b^13*d^2 - 144*A^3*a^5*b^11*d^2 - 160*A^3*a^7*b^9
*d^2 - 2*B^3*a^2*b^14*d^2 - 2*B^3*a^4*b^12*d^2 + 96*B^3*a^6*b^10*d^2 + 96*B^3*a^8*b^8*d^2 - (A^2*B*b^16*d^2)/2
+ 2*A*B^2*a*b^15*d^2 + 50*A*B^2*a^3*b^13*d^2 + 528*A*B^2*a^5*b^11*d^2 + 480*A*B^2*a^7*b^9*d^2 - (49*A^2*B*a^2
*b^14*d^2)/2 + 168*A^2*B*a^4*b^12*d^2 - 96*A^2*B*a^6*b^10*d^2 - 288*A^2*B*a^8*b^8*d^2)/(16*a^4*d^5) + ((((a +
b*tan(c + d*x))^(1/2)*(16*B^2*a^3*b^12*d^2 - 512*A^2*a^5*b^10*d^2 - 1280*A^2*a^7*b^8*d^2 - 64*A^2*a^3*b^12*d^2
+ 1024*B^2*a^5*b^10*d^2 + 2304*B^2*a^7*b^8*d^2 + 4*A^2*a*b^14*d^2 - 16*A*B*a^2*b^13*d^2 + 2048*A*B*a^4*b^11*d
^2 + 4096*A*B*a^6*b^9*d^2))/(64*a^4*d^4) - (((224*A*a^4*b^11*d^4 - 32*A*a^2*b^13*d^4 + 256*A*a^6*b^9*d^4 + 64*
B*a^3*b^12*d^4 + 448*B*a^5*b^10*d^4 + 384*B*a^7*b^8*d^4)/(16*a^4*d^5) + ((2048*a^4*b^10*d^4 + 3072*a^6*b^8*d^4
)*(a + b*tan(c + d*x))^(1/2)*(256*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 6
4*B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A*B*a^6*b^5)^(1/2))/(1024*a^9*d^5))*(256*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^
7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A*B*a^6*b^5)^(1/2))/(16*a^5*d))*(
256*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b
- 4*A*B*a^6*b^5)^(1/2))/(16*a^5*d))*(256*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7
*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A*B*a^6*b^5)^(1/2))/(16*a^5*d))*(256*B^2*a^11 + A^2*a^5*b^6 - 16*A^
2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A*B*a^6*b^5)^(1/2)*1i)/(a^5*d
))/((12*A^5*a^4*b^13 - (A^3*B^2*b^17)/4 - (A^5*a^2*b^15)/4 - (A^5*b^17)/4 + 44*A^5*a^6*b^11 + 32*A^5*a^8*b^9 +
7*B^5*a^3*b^14 + 63*B^5*a^5*b^12 + 56*B^5*a^7*b^10 + (79*A^2*B^3*a^3*b^14)/4 + 67*A^2*B^3*a^5*b^12 + 112*A^2*
B^3*a^7*b^10 + 64*A^2*B^3*a^9*b^8 - (17*A^3*B^2*a^2*b^15)/4 + 20*A^3*B^2*a^4*b^13 - 40*A^3*B^2*a^6*b^11 - 64*A
^3*B^2*a^8*b^9 + (3*A^4*B*a*b^16)/4 - 4*A*B^4*a^2*b^15 + 8*A*B^4*a^4*b^13 - 84*A*B^4*a^6*b^11 - 96*A*B^4*a^8*b
^9 + (3*A^2*B^3*a*b^16)/4 + (51*A^4*B*a^3*b^14)/4 + 4*A^4*B*a^5*b^12 + 56*A^4*B*a^7*b^10 + 64*A^4*B*a^9*b^8)/(
a^4*d^5) - ((((a + b*tan(c + d*x))^(1/2)*(A^4*b^16 - A^2*B^2*b^16 - 17*A^4*a^2*b^14 + 208*A^4*a^4*b^12 + 192*A
^4*a^6*b^10 + 128*A^4*a^8*b^8 - 4*B^4*a^2*b^14 + 68*B^4*a^4*b^12 + 64*B^4*a^6*b^10 + 384*B^4*a^8*b^8 + 5*A^2*B
^2*a^2*b^14 + 236*A^2*B^2*a^4*b^12 + 1792*A^2*B^2*a^6*b^10 + 4*A*B^3*a*b^15 + 12*A*B^3*a^3*b^13 + 1280*A*B^3*a
^7*b^9 - 60*A^3*B*a^3*b^13 + 512*A^3*B*a^5*b^11 - 256*A^3*B*a^7*b^9))/(64*a^4*d^4) + (((16*A^3*a^3*b^13*d^2 -
144*A^3*a^5*b^11*d^2 - 160*A^3*a^7*b^9*d^2 - 2*B^3*a^2*b^14*d^2 - 2*B^3*a^4*b^12*d^2 + 96*B^3*a^6*b^10*d^2 + 9
6*B^3*a^8*b^8*d^2 - (A^2*B*b^16*d^2)/2 + 2*A*B^2*a*b^15*d^2 + 50*A*B^2*a^3*b^13*d^2 + 528*A*B^2*a^5*b^11*d^2 +
480*A*B^2*a^7*b^9*d^2 - (49*A^2*B*a^2*b^14*d^2)/2 + 168*A^2*B*a^4*b^12*d^2 - 96*A^2*B*a^6*b^10*d^2 - 288*A^2*
B*a^8*b^8*d^2)/(16*a^4*d^5) - ((((a + b*tan(c + d*x))^(1/2)*(16*B^2*a^3*b^12*d^2 - 512*A^2*a^5*b^10*d^2 - 1280
*A^2*a^7*b^8*d^2 - 64*A^2*a^3*b^12*d^2 + 1024*B^2*a^5*b^10*d^2 + 2304*B^2*a^7*b^8*d^2 + 4*A^2*a*b^14*d^2 - 16*
A*B*a^2*b^13*d^2 + 2048*A*B*a^4*b^11*d^2 + 4096*A*B*a^6*b^9*d^2))/(64*a^4*d^4) + (((224*A*a^4*b^11*d^4 - 32*A*
a^2*b^13*d^4 + 256*A*a^6*b^9*d^4 + 64*B*a^3*b^12*d^4 + 448*B*a^5*b^10*d^4 + 384*B*a^7*b^8*d^4)/(16*a^4*d^5) -
((2048*a^4*b^10*d^4 + 3072*a^6*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(256*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^
4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A*B*a^6*b^5)^(1/2))/(1024*a^9*d^5))*(
256*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b
- 4*A*B*a^6*b^5)^(1/2))/(16*a^5*d))*(256*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7
*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A*B*a^6*b^5)^(1/2))/(16*a^5*d))*(256*B^2*a^11 + A^2*a^5*b^6 - 16*A^
2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A*B*a^6*b^5)^(1/2))/(16*a^5*d
))*(256*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^
10*b - 4*A*B*a^6*b^5)^(1/2))/(a^5*d) + ((((a + b*tan(c + d*x))^(1/2)*(A^4*b^16 - A^2*B^2*b^16 - 17*A^4*a^2*b^1
4 + 208*A^4*a^4*b^12 + 192*A^4*a^6*b^10 + 128*A^4*a^8*b^8 - 4*B^4*a^2*b^14 + 68*B^4*a^4*b^12 + 64*B^4*a^6*b^10
+ 384*B^4*a^8*b^8 + 5*A^2*B^2*a^2*b^14 + 236*A^2*B^2*a^4*b^12 + 1792*A^2*B^2*a^6*b^10 + 4*A*B^3*a*b^15 + 12*A
*B^3*a^3*b^13 + 1280*A*B^3*a^7*b^9 - 60*A^3*B*a^3*b^13 + 512*A^3*B*a^5*b^11 - 256*A^3*B*a^7*b^9))/(64*a^4*d^4)
- (((16*A^3*a^3*b^13*d^2 - 144*A^3*a^5*b^11*d^2 - 160*A^3*a^7*b^9*d^2 - 2*B^3*a^2*b^14*d^2 - 2*B^3*a^4*b^12*d
^2 + 96*B^3*a^6*b^10*d^2 + 96*B^3*a^8*b^8*d^2 - (A^2*B*b^16*d^2)/2 + 2*A*B^2*a*b^15*d^2 + 50*A*B^2*a^3*b^13*d^
2 + 528*A*B^2*a^5*b^11*d^2 + 480*A*B^2*a^7*b^9*d^2 - (49*A^2*B*a^2*b^14*d^2)/2 + 168*A^2*B*a^4*b^12*d^2 - 96*A
^2*B*a^6*b^10*d^2 - 288*A^2*B*a^8*b^8*d^2)/(16*a^4*d^5) + ((((a + b*tan(c + d*x))^(1/2)*(16*B^2*a^3*b^12*d^2 -
512*A^2*a^5*b^10*d^2 - 1280*A^2*a^7*b^8*d^2 - 64*A^2*a^3*b^12*d^2 + 1024*B^2*a^5*b^10*d^2 + 2304*B^2*a^7*b^8*
d^2 + 4*A^2*a*b^14*d^2 - 16*A*B*a^2*b^13*d^2 + 2048*A*B*a^4*b^11*d^2 + 4096*A*B*a^6*b^9*d^2))/(64*a^4*d^4) - (
((224*A*a^4*b^11*d^4 - 32*A*a^2*b^13*d^4 + 256*A*a^6*b^9*d^4 + 64*B*a^3*b^12*d^4 + 448*B*a^5*b^10*d^4 + 384*B*
a^7*b^8*d^4)/(16*a^4*d^5) + ((2048*a^4*b^10*d^4 + 3072*a^6*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(256*B^2*a^11 +
A^2*a^5*b^6 - 16*A^2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A*B*a^6*b
^5)^(1/2))/(1024*a^9*d^5))*(256*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 64*
B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A*B*a^6*b^5)^(1/2))/(16*a^5*d))*(256*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^4
+ 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A*B*a^6*b^5)^(1/2))/(16*a^5*d))*(256*B^
2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A
*B*a^6*b^5)^(1/2))/(16*a^5*d))*(256*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 +
64*B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A*B*a^6*b^5)^(1/2))/(a^5*d)))*(256*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^
4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A*B*a^6*b^5)^(1/2)*1i)/(8*a^5*d)