3.338 \(\int \cot ^4(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx\)

Optimal. Leaf size=277 \[ \frac {\left (8 a^2 A-18 a b B-11 A b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 d}+\frac {\left (16 a^3 B+40 a^2 A b-30 a b^2 B-5 A b^3\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{8 \sqrt {a} d}-\frac {(a-i b)^{5/2} (B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{5/2} (-B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {a (2 a B+3 A b) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{4 d}-\frac {a A \cot ^3(c+d x) (a+b \tan (c+d x))^{3/2}}{3 d} \]

[Out]

-(a-I*b)^(5/2)*(I*A+B)*arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/d+(a+I*b)^(5/2)*(I*A-B)*arctanh((a+b*tan(
d*x+c))^(1/2)/(a+I*b)^(1/2))/d+1/8*(40*A*a^2*b-5*A*b^3+16*B*a^3-30*B*a*b^2)*arctanh((a+b*tan(d*x+c))^(1/2)/a^(
1/2))/d/a^(1/2)+1/8*(8*A*a^2-11*A*b^2-18*B*a*b)*cot(d*x+c)*(a+b*tan(d*x+c))^(1/2)/d-1/4*a*(3*A*b+2*B*a)*cot(d*
x+c)^2*(a+b*tan(d*x+c))^(1/2)/d-1/3*a*A*cot(d*x+c)^3*(a+b*tan(d*x+c))^(3/2)/d

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Rubi [A]  time = 1.33, antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3605, 3645, 3649, 3653, 3539, 3537, 63, 208, 3634} \[ \frac {\left (40 a^2 A b+16 a^3 B-30 a b^2 B-5 A b^3\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{8 \sqrt {a} d}+\frac {\left (8 a^2 A-18 a b B-11 A b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 d}-\frac {(a-i b)^{5/2} (B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{5/2} (-B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {a (2 a B+3 A b) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{4 d}-\frac {a A \cot ^3(c+d x) (a+b \tan (c+d x))^{3/2}}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]

[Out]

((40*a^2*A*b - 5*A*b^3 + 16*a^3*B - 30*a*b^2*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(8*Sqrt[a]*d) - ((a
- I*b)^(5/2)*(I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d + ((a + I*b)^(5/2)*(I*A - B)*ArcTan
h[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + ((8*a^2*A - 11*A*b^2 - 18*a*b*B)*Cot[c + d*x]*Sqrt[a + b*Tan[c
+ d*x]])/(8*d) - (a*(3*A*b + 2*a*B)*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(4*d) - (a*A*Cot[c + d*x]^3*(a +
b*Tan[c + d*x])^(3/2))/(3*d)

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rule 3537

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c*
d)/f, Subst[Int[(a + (b*x)/d)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &&
NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]

Rule 3539

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c
+ I*d)/2, Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x], x] + Dist[(c - I*d)/2, Int[(a + b*Tan[e + f*x]
)^m*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0]
&& NeQ[c^2 + d^2, 0] &&  !IntegerQ[m]

Rule 3605

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

Rule 3634

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 3645

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*d^2 + c*(c*C - B*d))*(a + b*T
an[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 + d^2)), x] - Dist[1/(d*(n + 1)*(c^2 + d^2)), I
nt[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*d*(b*d*m - a*c*(n + 1)) + (c*C - B*d)*(b*c
*m + a*d*(n + 1)) - d*(n + 1)*((A - C)*(b*c - a*d) + B*(a*c + b*d))*Tan[e + f*x] - b*(d*(B*c - A*d)*(m + n + 1
) - C*(c^2*m - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c -
a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

Rule 3649

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*T
an[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 3653

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*} \int \cot ^4(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx &=-\frac {a A \cot ^3(c+d x) (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {1}{3} \int \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)} \left (\frac {3}{2} a (3 A b+2 a B)-3 \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)-\frac {3}{2} b (a A-2 b B) \tan ^2(c+d x)\right ) \, dx\\ &=-\frac {a (3 A b+2 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{4 d}-\frac {a A \cot ^3(c+d x) (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {1}{6} \int \frac {\cot ^2(c+d x) \left (-\frac {3}{4} a \left (8 a^2 A-11 A b^2-18 a b B\right )-6 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)-\frac {3}{4} b \left (13 a A b+6 a^2 B-8 b^2 B\right ) \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx\\ &=\frac {\left (8 a^2 A-11 A b^2-18 a b B\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 d}-\frac {a (3 A b+2 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{4 d}-\frac {a A \cot ^3(c+d x) (a+b \tan (c+d x))^{3/2}}{3 d}-\frac {\int \frac {\cot (c+d x) \left (\frac {3}{8} a \left (40 a^2 A b-5 A b^3+16 a^3 B-30 a b^2 B\right )-6 a \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \tan (c+d x)-\frac {3}{8} a b \left (8 a^2 A-11 A b^2-18 a b B\right ) \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{6 a}\\ &=\frac {\left (8 a^2 A-11 A b^2-18 a b B\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 d}-\frac {a (3 A b+2 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{4 d}-\frac {a A \cot ^3(c+d x) (a+b \tan (c+d x))^{3/2}}{3 d}-\frac {\int \frac {-6 a \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )-6 a \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{6 a}-\frac {1}{16} \left (40 a^2 A b-5 A b^3+16 a^3 B-30 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\\ &=\frac {\left (8 a^2 A-11 A b^2-18 a b B\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 d}-\frac {a (3 A b+2 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{4 d}-\frac {a A \cot ^3(c+d x) (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {1}{2} \left ((a-i b)^3 (A-i B)\right ) \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx+\frac {1}{2} \left ((a+i b)^3 (A+i B)\right ) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx-\frac {\left (40 a^2 A b-5 A b^3+16 a^3 B-30 a b^2 B\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{16 d}\\ &=\frac {\left (8 a^2 A-11 A b^2-18 a b B\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 d}-\frac {a (3 A b+2 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{4 d}-\frac {a A \cot ^3(c+d x) (a+b \tan (c+d x))^{3/2}}{3 d}-\frac {\left (i (a+i b)^3 (A+i B)\right ) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 d}+\frac {\left ((a-i b)^3 (i A+B)\right ) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 d}-\frac {\left (40 a^2 A b-5 A b^3+16 a^3 B-30 a b^2 B\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{8 b d}\\ &=\frac {\left (40 a^2 A b-5 A b^3+16 a^3 B-30 a b^2 B\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{8 \sqrt {a} d}+\frac {\left (8 a^2 A-11 A b^2-18 a b B\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 d}-\frac {a (3 A b+2 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{4 d}-\frac {a A \cot ^3(c+d x) (a+b \tan (c+d x))^{3/2}}{3 d}-\frac {\left ((a-i b)^3 (A-i B)\right ) \operatorname {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 ((a+i b)^3 (A+i B)\right ) \operatorname {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 (40 a^2 A b-5 A b^3+16 a^3 B-30 a b^2 B\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{8 \sqrt {a} d}-\frac {(a-i b)^{5/2} (i A+B) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{5/2} (i A-B) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {\left (8 a^2 A-11 A b^2-18 a b B\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 d}-\frac {a (3 A b+2 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{4 d}-\frac {a A \cot ^3(c+d x) (a+b \tan (c+d x))^{3/2}}{3 d}\\ \end {align*}

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Mathematica [A]  time = 6.39, size = 548, normalized size = 1.98 \[ -\frac {2 b B \cot ^3(c+d x) (a+b \tan (c+d x))^{3/2}}{3 d}-\frac {2}{3} \left (\frac {3 A b^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{5 d}-\frac {2}{5} \left (\frac {\left (6 A b^2-5 a (a A-2 b B)\right ) \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{4 d}-\frac {\frac {15 a \left (6 a^2 B+13 a A b-8 b^2 B\right ) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{16 d}-\frac {\frac {45 a^2 \left (8 a^2 A-18 a b B-11 A b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{16 d}-\frac {\frac {i \sqrt {a-i b} \left (-\frac {45}{2} a^3 \left (a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right )+\frac {45}{2} i a^3 \left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d (-a+i b)}-\frac {i \sqrt {a+i b} \left (-\frac {45}{2} a^3 \left (a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right )-\frac {45}{2} i a^3 \left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d (-a-i b)}-\frac {45 a^{5/2} \left (16 a^3 B+40 a^2 A b-30 a b^2 B-5 A b^3\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{16 d}}{a}}{2 a}}{3 a}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^4*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]

[Out]

(-2*b*B*Cot[c + d*x]^3*(a + b*Tan[c + d*x])^(3/2))/(3*d) - (2*((3*A*b^2*Cot[c + d*x]^3*Sqrt[a + b*Tan[c + d*x]
])/(5*d) - (2*(((6*A*b^2 - 5*a*(a*A - 2*b*B))*Cot[c + d*x]^3*Sqrt[a + b*Tan[c + d*x]])/(4*d) - ((15*a*(13*a*A*
b + 6*a^2*B - 8*b^2*B)*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(16*d) - (-(((-45*a^(5/2)*(40*a^2*A*b - 5*A*b^
3 + 16*a^3*B - 30*a*b^2*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(16*d) + (I*Sqrt[a - I*b]*(((45*I)/2)*a^
3*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B) - (45*a^3*(a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B))/2)*ArcTanh[Sqrt[a
+ b*Tan[c + d*x]]/Sqrt[a - I*b]])/((-a + I*b)*d) - (I*Sqrt[a + I*b]*(((-45*I)/2)*a^3*(3*a^2*A*b - A*b^3 + a^3
*B - 3*a*b^2*B) - (45*a^3*(a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B))/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a
+ I*b]])/((-a - I*b)*d))/a) + (45*a^2*(8*a^2*A - 11*A*b^2 - 18*a*b*B)*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(
16*d))/(2*a))/(3*a)))/5))/3

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a+b*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a+b*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [C]  time = 6.06, size = 171974, normalized size = 620.84 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^4*(a+b*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x)

[Out]

result too large to display

________________________________________________________________________________________

maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a+b*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

mupad [B]  time = 10.07, size = 33949, normalized size = 122.56 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^4*(A + B*tan(c + d*x))*(a + b*tan(c + d*x))^(5/2),x)

[Out]

(atan((((((a + b*tan(c + d*x))^(1/2)*(153*A^4*b^20 + 128*B^4*b^20 + 231*A^2*B^2*b^20 - 7*A^4*a^2*b^18 + 9895*A
^4*a^4*b^16 - 27465*A^4*a^6*b^14 + 26320*A^4*a^8*b^12 - 832*A^4*a^10*b^10 + 128*A^4*a^12*b^8 - 132*B^4*a^2*b^1
8 + 16380*B^4*a^4*b^16 - 25596*B^4*a^6*b^14 + 21060*B^4*a^8*b^12 - 4032*B^4*a^10*b^10 + 384*B^4*a^12*b^8 + 681
1*A^2*B^2*a^2*b^18 - 61315*A^2*B^2*a^4*b^16 + 184661*A^2*B^2*a^6*b^14 - 121620*A^2*B^2*a^8*b^12 + 23296*A^2*B^
2*a^10*b^10 - 300*A*B^3*a*b^19 + 600*A^3*B*a*b^19 + 17860*A*B^3*a^3*b^17 - 91700*A*B^3*a^5*b^15 + 110172*A*B^3
*a^7*b^13 - 43520*A*B^3*a^9*b^11 + 4352*A*B^3*a^11*b^9 - 12860*A^3*B*a^3*b^17 + 79680*A^3*B*a^5*b^15 - 126700*
A^3*B*a^7*b^13 + 40960*A^3*B*a^9*b^11 - 1280*A^3*B*a^11*b^9))/(64*d^4) + (((3225*A^3*a^3*b^15*d^2 - 1088*A^3*a
^5*b^13*d^2 - 3984*A^3*a^7*b^11*d^2 + 736*A^3*a^9*b^9*d^2 + 1854*B^3*a^2*b^16*d^2 - 3456*B^3*a^4*b^14*d^2 - 29
10*B^3*a^6*b^12*d^2 + 2304*B^3*a^8*b^10*d^2 - 96*B^3*a^10*b^8*d^2 + (295*A^2*B*b^18*d^2)/2 - 407*A^3*a*b^17*d^
2 + 1178*A*B^2*a*b^17*d^2 - 10572*A*B^2*a^3*b^15*d^2 + 1930*A*B^2*a^5*b^13*d^2 + 11472*A*B^2*a^7*b^11*d^2 - 22
08*A*B^2*a^9*b^9*d^2 - 4716*A^2*B*a^2*b^16*d^2 + (22193*A^2*B*a^4*b^14*d^2)/2 + 8568*A^2*B*a^6*b^12*d^2 - 7104
*A^2*B*a^8*b^10*d^2 + 288*A^2*B*a^10*b^8*d^2)/(16*d^5) + ((((a + b*tan(c + d*x))^(1/2)*(320*A^2*a^3*b^12*d^2 -
19456*A^2*a^5*b^10*d^2 + 1280*A^2*a^7*b^8*d^2 - 2320*B^2*a^3*b^12*d^2 + 16896*B^2*a^5*b^10*d^2 - 2304*B^2*a^7
*b^8*d^2 - 2048*A*B*b^15*d^2 + 4764*A^2*a*b^14*d^2 - 4864*B^2*a*b^14*d^2 + 14160*A*B*a^2*b^13*d^2 + 30720*A*B*
a^4*b^11*d^2 - 18432*A*B*a^6*b^9*d^2))/(64*d^4) - (((160*A*b^13*d^4 + 832*B*a*b^12*d^4 - 864*A*a^2*b^11*d^4 -
1024*A*a^4*b^9*d^4 + 448*B*a^3*b^10*d^4 - 384*B*a^5*b^8*d^4)/(16*d^5) - ((2048*b^10*d^4 + 3072*a^2*b^8*d^4)*(a
+ b*tan(c + d*x))^(1/2)*(256*B^2*a^7 + 25*A^2*a*b^6 - 400*A^2*a^3*b^4 + 1600*A^2*a^5*b^2 + 900*B^2*a^3*b^4 -
960*B^2*a^5*b^2 + 1280*A*B*a^6*b + 300*A*B*a^2*b^5 - 2560*A*B*a^4*b^3)^(1/2))/(1024*a*d^5))*(256*B^2*a^7 + 25*
A^2*a*b^6 - 400*A^2*a^3*b^4 + 1600*A^2*a^5*b^2 + 900*B^2*a^3*b^4 - 960*B^2*a^5*b^2 + 1280*A*B*a^6*b + 300*A*B*
a^2*b^5 - 2560*A*B*a^4*b^3)^(1/2))/(16*a*d))*(256*B^2*a^7 + 25*A^2*a*b^6 - 400*A^2*a^3*b^4 + 1600*A^2*a^5*b^2
+ 900*B^2*a^3*b^4 - 960*B^2*a^5*b^2 + 1280*A*B*a^6*b + 300*A*B*a^2*b^5 - 2560*A*B*a^4*b^3)^(1/2))/(16*a*d))*(2
56*B^2*a^7 + 25*A^2*a*b^6 - 400*A^2*a^3*b^4 + 1600*A^2*a^5*b^2 + 900*B^2*a^3*b^4 - 960*B^2*a^5*b^2 + 1280*A*B*
a^6*b + 300*A*B*a^2*b^5 - 2560*A*B*a^4*b^3)^(1/2))/(16*a*d))*(256*B^2*a^7 + 25*A^2*a*b^6 - 400*A^2*a^3*b^4 + 1
600*A^2*a^5*b^2 + 900*B^2*a^3*b^4 - 960*B^2*a^5*b^2 + 1280*A*B*a^6*b + 300*A*B*a^2*b^5 - 2560*A*B*a^4*b^3)^(1/
2)*1i)/(a*d) + ((((a + b*tan(c + d*x))^(1/2)*(153*A^4*b^20 + 128*B^4*b^20 + 231*A^2*B^2*b^20 - 7*A^4*a^2*b^18
+ 9895*A^4*a^4*b^16 - 27465*A^4*a^6*b^14 + 26320*A^4*a^8*b^12 - 832*A^4*a^10*b^10 + 128*A^4*a^12*b^8 - 132*B^4
*a^2*b^18 + 16380*B^4*a^4*b^16 - 25596*B^4*a^6*b^14 + 21060*B^4*a^8*b^12 - 4032*B^4*a^10*b^10 + 384*B^4*a^12*b
^8 + 6811*A^2*B^2*a^2*b^18 - 61315*A^2*B^2*a^4*b^16 + 184661*A^2*B^2*a^6*b^14 - 121620*A^2*B^2*a^8*b^12 + 2329
6*A^2*B^2*a^10*b^10 - 300*A*B^3*a*b^19 + 600*A^3*B*a*b^19 + 17860*A*B^3*a^3*b^17 - 91700*A*B^3*a^5*b^15 + 1101
72*A*B^3*a^7*b^13 - 43520*A*B^3*a^9*b^11 + 4352*A*B^3*a^11*b^9 - 12860*A^3*B*a^3*b^17 + 79680*A^3*B*a^5*b^15 -
126700*A^3*B*a^7*b^13 + 40960*A^3*B*a^9*b^11 - 1280*A^3*B*a^11*b^9))/(64*d^4) - (((3225*A^3*a^3*b^15*d^2 - 10
88*A^3*a^5*b^13*d^2 - 3984*A^3*a^7*b^11*d^2 + 736*A^3*a^9*b^9*d^2 + 1854*B^3*a^2*b^16*d^2 - 3456*B^3*a^4*b^14*
d^2 - 2910*B^3*a^6*b^12*d^2 + 2304*B^3*a^8*b^10*d^2 - 96*B^3*a^10*b^8*d^2 + (295*A^2*B*b^18*d^2)/2 - 407*A^3*a
*b^17*d^2 + 1178*A*B^2*a*b^17*d^2 - 10572*A*B^2*a^3*b^15*d^2 + 1930*A*B^2*a^5*b^13*d^2 + 11472*A*B^2*a^7*b^11*
d^2 - 2208*A*B^2*a^9*b^9*d^2 - 4716*A^2*B*a^2*b^16*d^2 + (22193*A^2*B*a^4*b^14*d^2)/2 + 8568*A^2*B*a^6*b^12*d^
2 - 7104*A^2*B*a^8*b^10*d^2 + 288*A^2*B*a^10*b^8*d^2)/(16*d^5) - ((((a + b*tan(c + d*x))^(1/2)*(320*A^2*a^3*b^
12*d^2 - 19456*A^2*a^5*b^10*d^2 + 1280*A^2*a^7*b^8*d^2 - 2320*B^2*a^3*b^12*d^2 + 16896*B^2*a^5*b^10*d^2 - 2304
*B^2*a^7*b^8*d^2 - 2048*A*B*b^15*d^2 + 4764*A^2*a*b^14*d^2 - 4864*B^2*a*b^14*d^2 + 14160*A*B*a^2*b^13*d^2 + 30
720*A*B*a^4*b^11*d^2 - 18432*A*B*a^6*b^9*d^2))/(64*d^4) + (((160*A*b^13*d^4 + 832*B*a*b^12*d^4 - 864*A*a^2*b^1
1*d^4 - 1024*A*a^4*b^9*d^4 + 448*B*a^3*b^10*d^4 - 384*B*a^5*b^8*d^4)/(16*d^5) + ((2048*b^10*d^4 + 3072*a^2*b^8
*d^4)*(a + b*tan(c + d*x))^(1/2)*(256*B^2*a^7 + 25*A^2*a*b^6 - 400*A^2*a^3*b^4 + 1600*A^2*a^5*b^2 + 900*B^2*a^
3*b^4 - 960*B^2*a^5*b^2 + 1280*A*B*a^6*b + 300*A*B*a^2*b^5 - 2560*A*B*a^4*b^3)^(1/2))/(1024*a*d^5))*(256*B^2*a
^7 + 25*A^2*a*b^6 - 400*A^2*a^3*b^4 + 1600*A^2*a^5*b^2 + 900*B^2*a^3*b^4 - 960*B^2*a^5*b^2 + 1280*A*B*a^6*b +
300*A*B*a^2*b^5 - 2560*A*B*a^4*b^3)^(1/2))/(16*a*d))*(256*B^2*a^7 + 25*A^2*a*b^6 - 400*A^2*a^3*b^4 + 1600*A^2*
a^5*b^2 + 900*B^2*a^3*b^4 - 960*B^2*a^5*b^2 + 1280*A*B*a^6*b + 300*A*B*a^2*b^5 - 2560*A*B*a^4*b^3)^(1/2))/(16*
a*d))*(256*B^2*a^7 + 25*A^2*a*b^6 - 400*A^2*a^3*b^4 + 1600*A^2*a^5*b^2 + 900*B^2*a^3*b^4 - 960*B^2*a^5*b^2 + 1
280*A*B*a^6*b + 300*A*B*a^2*b^5 - 2560*A*B*a^4*b^3)^(1/2))/(16*a*d))*(256*B^2*a^7 + 25*A^2*a*b^6 - 400*A^2*a^3
*b^4 + 1600*A^2*a^5*b^2 + 900*B^2*a^3*b^4 - 960*B^2*a^5*b^2 + 1280*A*B*a^6*b + 300*A*B*a^2*b^5 - 2560*A*B*a^4*
b^3)^(1/2)*1i)/(a*d))/(((55*A^5*b^23)/4 + 20*A*B^4*b^23 + 120*B^5*a*b^22 + (135*A^3*B^2*b^23)/4 + 60*A^5*a^2*b
^21 - (2045*A^5*a^4*b^19)/2 - 2550*A^5*a^6*b^17 - (5125*A^5*a^8*b^15)/4 + 620*A^5*a^10*b^13 + 260*A^5*a^12*b^1
1 - 160*A^5*a^14*b^9 - 19*B^5*a^3*b^20 + 336*B^5*a^5*b^18 + 1818*B^5*a^7*b^16 + 1448*B^5*a^9*b^14 - 399*B^5*a^
11*b^12 - 504*B^5*a^13*b^10 + 1527*A^2*B^3*a^3*b^20 + (2889*A^2*B^3*a^5*b^18)/2 - 2582*A^2*B^3*a^7*b^16 - (136
03*A^2*B^3*a^9*b^14)/4 - 99*A^2*B^3*a^11*b^12 + 592*A^2*B^3*a^13*b^10 - 64*A^2*B^3*a^15*b^8 - 430*A^3*B^2*a^2*
b^21 - (245*A^3*B^2*a^4*b^19)/2 + 446*A^3*B^2*a^6*b^17 - (10165*A^3*B^2*a^8*b^15)/4 - 4366*A^3*B^2*a^10*b^13 -
1528*A^3*B^2*a^12*b^11 + 192*A^3*B^2*a^14*b^9 + (105*A^4*B*a*b^22)/4 - 490*A*B^4*a^2*b^21 + 900*A*B^4*a^4*b^1
9 + 2996*A*B^4*a^6*b^17 - 1260*A*B^4*a^8*b^15 - 4986*A*B^4*a^10*b^13 - 1788*A*B^4*a^12*b^11 + 352*A*B^4*a^14*b
^9 + (585*A^2*B^3*a*b^22)/4 + 1546*A^4*B*a^3*b^20 + (2217*A^4*B*a^5*b^18)/2 - 4400*A^4*B*a^7*b^16 - (19395*A^4
*B*a^9*b^14)/4 + 300*A^4*B*a^11*b^12 + 1096*A^4*B*a^13*b^10 - 64*A^4*B*a^15*b^8)/d^5 - ((((a + b*tan(c + d*x))
^(1/2)*(153*A^4*b^20 + 128*B^4*b^20 + 231*A^2*B^2*b^20 - 7*A^4*a^2*b^18 + 9895*A^4*a^4*b^16 - 27465*A^4*a^6*b^
14 + 26320*A^4*a^8*b^12 - 832*A^4*a^10*b^10 + 128*A^4*a^12*b^8 - 132*B^4*a^2*b^18 + 16380*B^4*a^4*b^16 - 25596
*B^4*a^6*b^14 + 21060*B^4*a^8*b^12 - 4032*B^4*a^10*b^10 + 384*B^4*a^12*b^8 + 6811*A^2*B^2*a^2*b^18 - 61315*A^2
*B^2*a^4*b^16 + 184661*A^2*B^2*a^6*b^14 - 121620*A^2*B^2*a^8*b^12 + 23296*A^2*B^2*a^10*b^10 - 300*A*B^3*a*b^19
+ 600*A^3*B*a*b^19 + 17860*A*B^3*a^3*b^17 - 91700*A*B^3*a^5*b^15 + 110172*A*B^3*a^7*b^13 - 43520*A*B^3*a^9*b^
11 + 4352*A*B^3*a^11*b^9 - 12860*A^3*B*a^3*b^17 + 79680*A^3*B*a^5*b^15 - 126700*A^3*B*a^7*b^13 + 40960*A^3*B*a
^9*b^11 - 1280*A^3*B*a^11*b^9))/(64*d^4) + (((3225*A^3*a^3*b^15*d^2 - 1088*A^3*a^5*b^13*d^2 - 3984*A^3*a^7*b^1
1*d^2 + 736*A^3*a^9*b^9*d^2 + 1854*B^3*a^2*b^16*d^2 - 3456*B^3*a^4*b^14*d^2 - 2910*B^3*a^6*b^12*d^2 + 2304*B^3
*a^8*b^10*d^2 - 96*B^3*a^10*b^8*d^2 + (295*A^2*B*b^18*d^2)/2 - 407*A^3*a*b^17*d^2 + 1178*A*B^2*a*b^17*d^2 - 10
572*A*B^2*a^3*b^15*d^2 + 1930*A*B^2*a^5*b^13*d^2 + 11472*A*B^2*a^7*b^11*d^2 - 2208*A*B^2*a^9*b^9*d^2 - 4716*A^
2*B*a^2*b^16*d^2 + (22193*A^2*B*a^4*b^14*d^2)/2 + 8568*A^2*B*a^6*b^12*d^2 - 7104*A^2*B*a^8*b^10*d^2 + 288*A^2*
B*a^10*b^8*d^2)/(16*d^5) + ((((a + b*tan(c + d*x))^(1/2)*(320*A^2*a^3*b^12*d^2 - 19456*A^2*a^5*b^10*d^2 + 1280
*A^2*a^7*b^8*d^2 - 2320*B^2*a^3*b^12*d^2 + 16896*B^2*a^5*b^10*d^2 - 2304*B^2*a^7*b^8*d^2 - 2048*A*B*b^15*d^2 +
4764*A^2*a*b^14*d^2 - 4864*B^2*a*b^14*d^2 + 14160*A*B*a^2*b^13*d^2 + 30720*A*B*a^4*b^11*d^2 - 18432*A*B*a^6*b
^9*d^2))/(64*d^4) - (((160*A*b^13*d^4 + 832*B*a*b^12*d^4 - 864*A*a^2*b^11*d^4 - 1024*A*a^4*b^9*d^4 + 448*B*a^3
*b^10*d^4 - 384*B*a^5*b^8*d^4)/(16*d^5) - ((2048*b^10*d^4 + 3072*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(256*
B^2*a^7 + 25*A^2*a*b^6 - 400*A^2*a^3*b^4 + 1600*A^2*a^5*b^2 + 900*B^2*a^3*b^4 - 960*B^2*a^5*b^2 + 1280*A*B*a^6
*b + 300*A*B*a^2*b^5 - 2560*A*B*a^4*b^3)^(1/2))/(1024*a*d^5))*(256*B^2*a^7 + 25*A^2*a*b^6 - 400*A^2*a^3*b^4 +
1600*A^2*a^5*b^2 + 900*B^2*a^3*b^4 - 960*B^2*a^5*b^2 + 1280*A*B*a^6*b + 300*A*B*a^2*b^5 - 2560*A*B*a^4*b^3)^(1
/2))/(16*a*d))*(256*B^2*a^7 + 25*A^2*a*b^6 - 400*A^2*a^3*b^4 + 1600*A^2*a^5*b^2 + 900*B^2*a^3*b^4 - 960*B^2*a^
5*b^2 + 1280*A*B*a^6*b + 300*A*B*a^2*b^5 - 2560*A*B*a^4*b^3)^(1/2))/(16*a*d))*(256*B^2*a^7 + 25*A^2*a*b^6 - 40
0*A^2*a^3*b^4 + 1600*A^2*a^5*b^2 + 900*B^2*a^3*b^4 - 960*B^2*a^5*b^2 + 1280*A*B*a^6*b + 300*A*B*a^2*b^5 - 2560
*A*B*a^4*b^3)^(1/2))/(16*a*d))*(256*B^2*a^7 + 25*A^2*a*b^6 - 400*A^2*a^3*b^4 + 1600*A^2*a^5*b^2 + 900*B^2*a^3*
b^4 - 960*B^2*a^5*b^2 + 1280*A*B*a^6*b + 300*A*B*a^2*b^5 - 2560*A*B*a^4*b^3)^(1/2))/(a*d) + ((((a + b*tan(c +
d*x))^(1/2)*(153*A^4*b^20 + 128*B^4*b^20 + 231*A^2*B^2*b^20 - 7*A^4*a^2*b^18 + 9895*A^4*a^4*b^16 - 27465*A^4*a
^6*b^14 + 26320*A^4*a^8*b^12 - 832*A^4*a^10*b^10 + 128*A^4*a^12*b^8 - 132*B^4*a^2*b^18 + 16380*B^4*a^4*b^16 -
25596*B^4*a^6*b^14 + 21060*B^4*a^8*b^12 - 4032*B^4*a^10*b^10 + 384*B^4*a^12*b^8 + 6811*A^2*B^2*a^2*b^18 - 6131
5*A^2*B^2*a^4*b^16 + 184661*A^2*B^2*a^6*b^14 - 121620*A^2*B^2*a^8*b^12 + 23296*A^2*B^2*a^10*b^10 - 300*A*B^3*a
*b^19 + 600*A^3*B*a*b^19 + 17860*A*B^3*a^3*b^17 - 91700*A*B^3*a^5*b^15 + 110172*A*B^3*a^7*b^13 - 43520*A*B^3*a
^9*b^11 + 4352*A*B^3*a^11*b^9 - 12860*A^3*B*a^3*b^17 + 79680*A^3*B*a^5*b^15 - 126700*A^3*B*a^7*b^13 + 40960*A^
3*B*a^9*b^11 - 1280*A^3*B*a^11*b^9))/(64*d^4) - (((3225*A^3*a^3*b^15*d^2 - 1088*A^3*a^5*b^13*d^2 - 3984*A^3*a^
7*b^11*d^2 + 736*A^3*a^9*b^9*d^2 + 1854*B^3*a^2*b^16*d^2 - 3456*B^3*a^4*b^14*d^2 - 2910*B^3*a^6*b^12*d^2 + 230
4*B^3*a^8*b^10*d^2 - 96*B^3*a^10*b^8*d^2 + (295*A^2*B*b^18*d^2)/2 - 407*A^3*a*b^17*d^2 + 1178*A*B^2*a*b^17*d^2
- 10572*A*B^2*a^3*b^15*d^2 + 1930*A*B^2*a^5*b^13*d^2 + 11472*A*B^2*a^7*b^11*d^2 - 2208*A*B^2*a^9*b^9*d^2 - 47
16*A^2*B*a^2*b^16*d^2 + (22193*A^2*B*a^4*b^14*d^2)/2 + 8568*A^2*B*a^6*b^12*d^2 - 7104*A^2*B*a^8*b^10*d^2 + 288
*A^2*B*a^10*b^8*d^2)/(16*d^5) - ((((a + b*tan(c + d*x))^(1/2)*(320*A^2*a^3*b^12*d^2 - 19456*A^2*a^5*b^10*d^2 +
1280*A^2*a^7*b^8*d^2 - 2320*B^2*a^3*b^12*d^2 + 16896*B^2*a^5*b^10*d^2 - 2304*B^2*a^7*b^8*d^2 - 2048*A*B*b^15*
d^2 + 4764*A^2*a*b^14*d^2 - 4864*B^2*a*b^14*d^2 + 14160*A*B*a^2*b^13*d^2 + 30720*A*B*a^4*b^11*d^2 - 18432*A*B*
a^6*b^9*d^2))/(64*d^4) + (((160*A*b^13*d^4 + 832*B*a*b^12*d^4 - 864*A*a^2*b^11*d^4 - 1024*A*a^4*b^9*d^4 + 448*
B*a^3*b^10*d^4 - 384*B*a^5*b^8*d^4)/(16*d^5) + ((2048*b^10*d^4 + 3072*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*
(256*B^2*a^7 + 25*A^2*a*b^6 - 400*A^2*a^3*b^4 + 1600*A^2*a^5*b^2 + 900*B^2*a^3*b^4 - 960*B^2*a^5*b^2 + 1280*A*
B*a^6*b + 300*A*B*a^2*b^5 - 2560*A*B*a^4*b^3)^(1/2))/(1024*a*d^5))*(256*B^2*a^7 + 25*A^2*a*b^6 - 400*A^2*a^3*b
^4 + 1600*A^2*a^5*b^2 + 900*B^2*a^3*b^4 - 960*B^2*a^5*b^2 + 1280*A*B*a^6*b + 300*A*B*a^2*b^5 - 2560*A*B*a^4*b^
3)^(1/2))/(16*a*d))*(256*B^2*a^7 + 25*A^2*a*b^6 - 400*A^2*a^3*b^4 + 1600*A^2*a^5*b^2 + 900*B^2*a^3*b^4 - 960*B
^2*a^5*b^2 + 1280*A*B*a^6*b + 300*A*B*a^2*b^5 - 2560*A*B*a^4*b^3)^(1/2))/(16*a*d))*(256*B^2*a^7 + 25*A^2*a*b^6
- 400*A^2*a^3*b^4 + 1600*A^2*a^5*b^2 + 900*B^2*a^3*b^4 - 960*B^2*a^5*b^2 + 1280*A*B*a^6*b + 300*A*B*a^2*b^5 -
2560*A*B*a^4*b^3)^(1/2))/(16*a*d))*(256*B^2*a^7 + 25*A^2*a*b^6 - 400*A^2*a^3*b^4 + 1600*A^2*a^5*b^2 + 900*B^2
*a^3*b^4 - 960*B^2*a^5*b^2 + 1280*A*B*a^6*b + 300*A*B*a^2*b^5 - 2560*A*B*a^4*b^3)^(1/2))/(a*d)))*(256*B^2*a^7
+ 25*A^2*a*b^6 - 400*A^2*a^3*b^4 + 1600*A^2*a^5*b^2 + 900*B^2*a^3*b^4 - 960*B^2*a^5*b^2 + 1280*A*B*a^6*b + 300
*A*B*a^2*b^5 - 2560*A*B*a^4*b^3)^(1/2)*1i)/(8*a*d) - atan(((((((1280*A*b^13*d^4 + 6656*B*a*b^12*d^4 - 6912*A*a
^2*b^11*d^4 - 8192*A*a^4*b^9*d^4 + 3584*B*a^3*b^10*d^4 - 3072*B*a^5*b^8*d^4)/(8*d^5) - ((2048*b^10*d^4 + 3072*
a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^
2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 -
d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*
b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^
2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B^2*a^5*d
^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2*
b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2))/(4*d^4))*((((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 -
80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b
*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8
+ 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^
8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5*d^
2 + B^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2
- 20*A*B*a^2*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(320*A^2*a^3*b^12*d^2 -
19456*A^2*a^5*b^10*d^2 + 1280*A^2*a^7*b^8*d^2 - 2320*B^2*a^3*b^12*d^2 + 16896*B^2*a^5*b^10*d^2 - 2304*B^2*a^7*
b^8*d^2 - 2048*A*B*b^15*d^2 + 4764*A^2*a*b^14*d^2 - 4864*B^2*a*b^14*d^2 + 14160*A*B*a^2*b^13*d^2 + 30720*A*B*a
^4*b^11*d^2 - 18432*A*B*a^6*b^9*d^2))/(4*d^4))*((((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2
*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^
2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A
^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2
+ 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B^
2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*A
*B*a^2*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) - (25800*A^3*a^3*b^15*d^2 - 8704*A^3*a^5*b^13*d^2 - 31872*A^
3*a^7*b^11*d^2 + 5888*A^3*a^9*b^9*d^2 + 14832*B^3*a^2*b^16*d^2 - 27648*B^3*a^4*b^14*d^2 - 23280*B^3*a^6*b^12*d
^2 + 18432*B^3*a^8*b^10*d^2 - 768*B^3*a^10*b^8*d^2 + 1180*A^2*B*b^18*d^2 - 3256*A^3*a*b^17*d^2 + 9424*A*B^2*a*
b^17*d^2 - 84576*A*B^2*a^3*b^15*d^2 + 15440*A*B^2*a^5*b^13*d^2 + 91776*A*B^2*a^7*b^11*d^2 - 17664*A*B^2*a^9*b^
9*d^2 - 37728*A^2*B*a^2*b^16*d^2 + 88772*A^2*B*a^4*b^14*d^2 + 68544*A^2*B*a^6*b^12*d^2 - 56832*A^2*B*a^8*b^10*
d^2 + 2304*A^2*B*a^10*b^8*d^2)/(8*d^5))*((((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^
2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 -
d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*
b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^
2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B^2*a^5*d
^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2*
b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(153*A^4*b^20 + 128*B^4*b^20 + 231*A^
2*B^2*b^20 - 7*A^4*a^2*b^18 + 9895*A^4*a^4*b^16 - 27465*A^4*a^6*b^14 + 26320*A^4*a^8*b^12 - 832*A^4*a^10*b^10
+ 128*A^4*a^12*b^8 - 132*B^4*a^2*b^18 + 16380*B^4*a^4*b^16 - 25596*B^4*a^6*b^14 + 21060*B^4*a^8*b^12 - 4032*B^
4*a^10*b^10 + 384*B^4*a^12*b^8 + 6811*A^2*B^2*a^2*b^18 - 61315*A^2*B^2*a^4*b^16 + 184661*A^2*B^2*a^6*b^14 - 12
1620*A^2*B^2*a^8*b^12 + 23296*A^2*B^2*a^10*b^10 - 300*A*B^3*a*b^19 + 600*A^3*B*a*b^19 + 17860*A*B^3*a^3*b^17 -
91700*A*B^3*a^5*b^15 + 110172*A*B^3*a^7*b^13 - 43520*A*B^3*a^9*b^11 + 4352*A*B^3*a^11*b^9 - 12860*A^3*B*a^3*b
^17 + 79680*A^3*B*a^5*b^15 - 126700*A^3*B*a^7*b^13 + 40960*A^3*B*a^9*b^11 - 1280*A^3*B*a^11*b^9))/(4*d^4))*(((
(8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 +
40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b
^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^
4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2
*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^
2 + 2*A*B*b^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2)*
1i - (((((1280*A*b^13*d^4 + 6656*B*a*b^12*d^4 - 6912*A*a^2*b^11*d^4 - 8192*A*a^4*b^9*d^4 + 3584*B*a^3*b^10*d^4
- 3072*B*a^5*b^8*d^4)/(8*d^5) + ((2048*b^10*d^4 + 3072*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((((8*B^2*a^5*
d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b
^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2
*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 +
10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^
4 + 10*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b
^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2))/(4*d^4))*(
(((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2
+ 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4
*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*
B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A
^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*
d^2 + 2*A*B*b^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2
) + ((a + b*tan(c + d*x))^(1/2)*(320*A^2*a^3*b^12*d^2 - 19456*A^2*a^5*b^10*d^2 + 1280*A^2*a^7*b^8*d^2 - 2320*B
^2*a^3*b^12*d^2 + 16896*B^2*a^5*b^10*d^2 - 2304*B^2*a^7*b^8*d^2 - 2048*A*B*b^15*d^2 + 4764*A^2*a*b^14*d^2 - 48
64*B^2*a*b^14*d^2 + 14160*A*B*a^2*b^13*d^2 + 30720*A*B*a^4*b^11*d^2 - 18432*A*B*a^6*b^9*d^2))/(4*d^4))*((((8*B
^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*
B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10
+ 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^
2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2
*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 +
2*A*B*b^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) - (2
5800*A^3*a^3*b^15*d^2 - 8704*A^3*a^5*b^13*d^2 - 31872*A^3*a^7*b^11*d^2 + 5888*A^3*a^9*b^9*d^2 + 14832*B^3*a^2*
b^16*d^2 - 27648*B^3*a^4*b^14*d^2 - 23280*B^3*a^6*b^12*d^2 + 18432*B^3*a^8*b^10*d^2 - 768*B^3*a^10*b^8*d^2 + 1
180*A^2*B*b^18*d^2 - 3256*A^3*a*b^17*d^2 + 9424*A*B^2*a*b^17*d^2 - 84576*A*B^2*a^3*b^15*d^2 + 15440*A*B^2*a^5*
b^13*d^2 + 91776*A*B^2*a^7*b^11*d^2 - 17664*A*B^2*a^9*b^9*d^2 - 37728*A^2*B*a^2*b^16*d^2 + 88772*A^2*B*a^4*b^1
4*d^2 + 68544*A^2*B*a^6*b^12*d^2 - 56832*A^2*B*a^8*b^10*d^2 + 2304*A^2*B*a^10*b^8*d^2)/(8*d^5))*((((8*B^2*a^5*
d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b
^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2
*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 +
10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^
4 + 10*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b
^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) + ((a + b*t
an(c + d*x))^(1/2)*(153*A^4*b^20 + 128*B^4*b^20 + 231*A^2*B^2*b^20 - 7*A^4*a^2*b^18 + 9895*A^4*a^4*b^16 - 2746
5*A^4*a^6*b^14 + 26320*A^4*a^8*b^12 - 832*A^4*a^10*b^10 + 128*A^4*a^12*b^8 - 132*B^4*a^2*b^18 + 16380*B^4*a^4*
b^16 - 25596*B^4*a^6*b^14 + 21060*B^4*a^8*b^12 - 4032*B^4*a^10*b^10 + 384*B^4*a^12*b^8 + 6811*A^2*B^2*a^2*b^18
- 61315*A^2*B^2*a^4*b^16 + 184661*A^2*B^2*a^6*b^14 - 121620*A^2*B^2*a^8*b^12 + 23296*A^2*B^2*a^10*b^10 - 300*
A*B^3*a*b^19 + 600*A^3*B*a*b^19 + 17860*A*B^3*a^3*b^17 - 91700*A*B^3*a^5*b^15 + 110172*A*B^3*a^7*b^13 - 43520*
A*B^3*a^9*b^11 + 4352*A*B^3*a^11*b^9 - 12860*A^3*B*a^3*b^17 + 79680*A^3*B*a^5*b^15 - 126700*A^3*B*a^7*b^13 + 4
0960*A^3*B*a^9*b^11 - 1280*A^3*B*a^11*b^9))/(4*d^4))*((((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 -
80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b
*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8
+ 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^
8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5*d^
2 + B^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2
- 20*A*B*a^2*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2)*1i)/((55*A^5*b^23 + 80*A*B^4*b^23 + 480*B^5*a*b^22 + 1
35*A^3*B^2*b^23 + 240*A^5*a^2*b^21 - 4090*A^5*a^4*b^19 - 10200*A^5*a^6*b^17 - 5125*A^5*a^8*b^15 + 2480*A^5*a^1
0*b^13 + 1040*A^5*a^12*b^11 - 640*A^5*a^14*b^9 - 76*B^5*a^3*b^20 + 1344*B^5*a^5*b^18 + 7272*B^5*a^7*b^16 + 579
2*B^5*a^9*b^14 - 1596*B^5*a^11*b^12 - 2016*B^5*a^13*b^10 + 6108*A^2*B^3*a^3*b^20 + 5778*A^2*B^3*a^5*b^18 - 103
28*A^2*B^3*a^7*b^16 - 13603*A^2*B^3*a^9*b^14 - 396*A^2*B^3*a^11*b^12 + 2368*A^2*B^3*a^13*b^10 - 256*A^2*B^3*a^
15*b^8 - 1720*A^3*B^2*a^2*b^21 - 490*A^3*B^2*a^4*b^19 + 1784*A^3*B^2*a^6*b^17 - 10165*A^3*B^2*a^8*b^15 - 17464
*A^3*B^2*a^10*b^13 - 6112*A^3*B^2*a^12*b^11 + 768*A^3*B^2*a^14*b^9 + 105*A^4*B*a*b^22 - 1960*A*B^4*a^2*b^21 +
3600*A*B^4*a^4*b^19 + 11984*A*B^4*a^6*b^17 - 5040*A*B^4*a^8*b^15 - 19944*A*B^4*a^10*b^13 - 7152*A*B^4*a^12*b^1
1 + 1408*A*B^4*a^14*b^9 + 585*A^2*B^3*a*b^22 + 6184*A^4*B*a^3*b^20 + 4434*A^4*B*a^5*b^18 - 17600*A^4*B*a^7*b^1
6 - 19395*A^4*B*a^9*b^14 + 1200*A^4*B*a^11*b^12 + 4384*A^4*B*a^13*b^10 - 256*A^4*B*a^15*b^8)/(4*d^5) + (((((12
80*A*b^13*d^4 + 6656*B*a*b^12*d^4 - 6912*A*a^2*b^11*d^4 - 8192*A*a^4*b^9*d^4 + 3584*B*a^3*b^10*d^4 - 3072*B*a^
5*b^8*d^4)/(8*d^5) - ((2048*b^10*d^4 + 3072*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((((8*B^2*a^5*d^2 - 8*A^2*
a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160
*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 +
2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*
b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B
^2*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2 - 5*A
^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2))/(4*d^4))*((((8*B^2*a^5
*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*
b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^
2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8
+ 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b
^4 + 10*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*
b^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) - ((a + b*
tan(c + d*x))^(1/2)*(320*A^2*a^3*b^12*d^2 - 19456*A^2*a^5*b^10*d^2 + 1280*A^2*a^7*b^8*d^2 - 2320*B^2*a^3*b^12*
d^2 + 16896*B^2*a^5*b^10*d^2 - 2304*B^2*a^7*b^8*d^2 - 2048*A*B*b^15*d^2 + 4764*A^2*a*b^14*d^2 - 4864*B^2*a*b^1
4*d^2 + 14160*A*B*a^2*b^13*d^2 + 30720*A*B*a^4*b^11*d^2 - 18432*A*B*a^6*b^9*d^2))/(4*d^4))*((((8*B^2*a^5*d^2 -
8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^
2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*
a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B
^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 1
0*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^
2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) - (25800*A^3*a^3
*b^15*d^2 - 8704*A^3*a^5*b^13*d^2 - 31872*A^3*a^7*b^11*d^2 + 5888*A^3*a^9*b^9*d^2 + 14832*B^3*a^2*b^16*d^2 - 2
7648*B^3*a^4*b^14*d^2 - 23280*B^3*a^6*b^12*d^2 + 18432*B^3*a^8*b^10*d^2 - 768*B^3*a^10*b^8*d^2 + 1180*A^2*B*b^
18*d^2 - 3256*A^3*a*b^17*d^2 + 9424*A*B^2*a*b^17*d^2 - 84576*A*B^2*a^3*b^15*d^2 + 15440*A*B^2*a^5*b^13*d^2 + 9
1776*A*B^2*a^7*b^11*d^2 - 17664*A*B^2*a^9*b^9*d^2 - 37728*A^2*B*a^2*b^16*d^2 + 88772*A^2*B*a^4*b^14*d^2 + 6854
4*A^2*B*a^6*b^12*d^2 - 56832*A^2*B*a^8*b^10*d^2 + 2304*A^2*B*a^10*b^8*d^2)/(8*d^5))*((((8*B^2*a^5*d^2 - 8*A^2*
a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160
*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 +
2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*
b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B
^2*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2 - 5*A
^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) - ((a + b*tan(c + d*x))
^(1/2)*(153*A^4*b^20 + 128*B^4*b^20 + 231*A^2*B^2*b^20 - 7*A^4*a^2*b^18 + 9895*A^4*a^4*b^16 - 27465*A^4*a^6*b^
14 + 26320*A^4*a^8*b^12 - 832*A^4*a^10*b^10 + 128*A^4*a^12*b^8 - 132*B^4*a^2*b^18 + 16380*B^4*a^4*b^16 - 25596
*B^4*a^6*b^14 + 21060*B^4*a^8*b^12 - 4032*B^4*a^10*b^10 + 384*B^4*a^12*b^8 + 6811*A^2*B^2*a^2*b^18 - 61315*A^2
*B^2*a^4*b^16 + 184661*A^2*B^2*a^6*b^14 - 121620*A^2*B^2*a^8*b^12 + 23296*A^2*B^2*a^10*b^10 - 300*A*B^3*a*b^19
+ 600*A^3*B*a*b^19 + 17860*A*B^3*a^3*b^17 - 91700*A*B^3*a^5*b^15 + 110172*A*B^3*a^7*b^13 - 43520*A*B^3*a^9*b^
11 + 4352*A*B^3*a^11*b^9 - 12860*A^3*B*a^3*b^17 + 79680*A^3*B*a^5*b^15 - 126700*A^3*B*a^7*b^13 + 40960*A^3*B*a
^9*b^11 - 1280*A^3*B*a^11*b^9))/(4*d^4))*((((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b
^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 -
d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4
*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A
^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B^2*a^5*
d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2
*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) + (((((1280*A*b^13*d^4 + 6656*B*a*b^12*d^4 - 6912*A*a^2*b^11*d^4 -
8192*A*a^4*b^9*d^4 + 3584*B*a^3*b^10*d^4 - 3072*B*a^5*b^8*d^4)/(8*d^5) + ((2048*b^10*d^4 + 3072*a^2*b^8*d^4)*
(a + b*tan(c + d*x))^(1/2)*((((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*
B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10
+ A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*
a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8
+ 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B^2*a^5*d^2 + 10*A^2*a
^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2*b^3*d^2 + 10*
A*B*a^4*b*d^2)/(4*d^4))^(1/2))/(4*d^4))*((((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^
2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 -
d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*
b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^
2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B^2*a^5*d
^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2*
b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(320*A^2*a^3*b^12*d^2 - 19456*A^2*a^5
*b^10*d^2 + 1280*A^2*a^7*b^8*d^2 - 2320*B^2*a^3*b^12*d^2 + 16896*B^2*a^5*b^10*d^2 - 2304*B^2*a^7*b^8*d^2 - 204
8*A*B*b^15*d^2 + 4764*A^2*a*b^14*d^2 - 4864*B^2*a*b^14*d^2 + 14160*A*B*a^2*b^13*d^2 + 30720*A*B*a^4*b^11*d^2 -
18432*A*B*a^6*b^9*d^2))/(4*d^4))*((((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2
+ 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A
^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 +
10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*
a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B^2*a^5*d^2 + 1
0*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2*b^3*d^
2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) - (25800*A^3*a^3*b^15*d^2 - 8704*A^3*a^5*b^13*d^2 - 31872*A^3*a^7*b^11*d^
2 + 5888*A^3*a^9*b^9*d^2 + 14832*B^3*a^2*b^16*d^2 - 27648*B^3*a^4*b^14*d^2 - 23280*B^3*a^6*b^12*d^2 + 18432*B^
3*a^8*b^10*d^2 - 768*B^3*a^10*b^8*d^2 + 1180*A^2*B*b^18*d^2 - 3256*A^3*a*b^17*d^2 + 9424*A*B^2*a*b^17*d^2 - 84
576*A*B^2*a^3*b^15*d^2 + 15440*A*B^2*a^5*b^13*d^2 + 91776*A*B^2*a^7*b^11*d^2 - 17664*A*B^2*a^9*b^9*d^2 - 37728
*A^2*B*a^2*b^16*d^2 + 88772*A^2*B*a^4*b^14*d^2 + 68544*A^2*B*a^6*b^12*d^2 - 56832*A^2*B*a^8*b^10*d^2 + 2304*A^
2*B*a^10*b^8*d^2)/(8*d^5))*((((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*
B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10
+ A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*
a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8
+ 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B^2*a^5*d^2 + 10*A^2*a
^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2*b^3*d^2 + 10*
A*B*a^4*b*d^2)/(4*d^4))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(153*A^4*b^20 + 128*B^4*b^20 + 231*A^2*B^2*b^20 -
7*A^4*a^2*b^18 + 9895*A^4*a^4*b^16 - 27465*A^4*a^6*b^14 + 26320*A^4*a^8*b^12 - 832*A^4*a^10*b^10 + 128*A^4*a^1
2*b^8 - 132*B^4*a^2*b^18 + 16380*B^4*a^4*b^16 - 25596*B^4*a^6*b^14 + 21060*B^4*a^8*b^12 - 4032*B^4*a^10*b^10 +
384*B^4*a^12*b^8 + 6811*A^2*B^2*a^2*b^18 - 61315*A^2*B^2*a^4*b^16 + 184661*A^2*B^2*a^6*b^14 - 121620*A^2*B^2*
a^8*b^12 + 23296*A^2*B^2*a^10*b^10 - 300*A*B^3*a*b^19 + 600*A^3*B*a*b^19 + 17860*A*B^3*a^3*b^17 - 91700*A*B^3*
a^5*b^15 + 110172*A*B^3*a^7*b^13 - 43520*A*B^3*a^9*b^11 + 4352*A*B^3*a^11*b^9 - 12860*A^3*B*a^3*b^17 + 79680*A
^3*B*a^5*b^15 - 126700*A^3*B*a^7*b^13 + 40960*A^3*B*a^9*b^11 - 1280*A^3*B*a^11*b^9))/(4*d^4))*((((8*B^2*a^5*d^
2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4
*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B
^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 1
0*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4
+ 10*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5
*d^2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2)))*((((8*B^2*a
^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*
a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*
A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^
8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6
*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*
B*b^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2)*2i - ((a
+ b*tan(c + d*x))^(1/2)*((5*A*a^2*b^3)/8 + (7*B*a^3*b^2)/4 - A*a^4*b) - (a + b*tan(c + d*x))^(3/2)*(4*B*a^2*b
^2 + (5*A*a*b^3)/3 - 2*A*a^3*b) + (a + b*tan(c + d*x))^(5/2)*((11*A*b^3)/8 - A*a^2*b + (9*B*a*b^2)/4))/(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(((((((1280*A*
b^13*d^4 + 6656*B*a*b^12*d^4 - 6912*A*a^2*b^11*d^4 - 8192*A*a^4*b^9*d^4 + 3584*B*a^3*b^10*d^4 - 3072*B*a^5*b^8
*d^4)/(8*d^5) - ((2048*b^10*d^4 + 3072*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(-(((8*B^2*a^5*d^2 - 8*A^2*a^5*
d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B
*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^
2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6
+ 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a
^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2*a
*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2))/(4*d^4))*(-(((8*B^2*a^5*d^
2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4
*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B
^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 1
0*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4
+ 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5
*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) - ((a + b*tan
(c + d*x))^(1/2)*(320*A^2*a^3*b^12*d^2 - 19456*A^2*a^5*b^10*d^2 + 1280*A^2*a^7*b^8*d^2 - 2320*B^2*a^3*b^12*d^2
+ 16896*B^2*a^5*b^10*d^2 - 2304*B^2*a^7*b^8*d^2 - 2048*A*B*b^15*d^2 + 4764*A^2*a*b^14*d^2 - 4864*B^2*a*b^14*d
^2 + 14160*A*B*a^2*b^13*d^2 + 30720*A*B*a^4*b^11*d^2 - 18432*A*B*a^6*b^9*d^2))/(4*d^4))*(-(((8*B^2*a^5*d^2 - 8
*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2
- 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^
10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4
*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*
A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2
+ 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) - (25800*A^3*a^3*b
^15*d^2 - 8704*A^3*a^5*b^13*d^2 - 31872*A^3*a^7*b^11*d^2 + 5888*A^3*a^9*b^9*d^2 + 14832*B^3*a^2*b^16*d^2 - 276
48*B^3*a^4*b^14*d^2 - 23280*B^3*a^6*b^12*d^2 + 18432*B^3*a^8*b^10*d^2 - 768*B^3*a^10*b^8*d^2 + 1180*A^2*B*b^18
*d^2 - 3256*A^3*a*b^17*d^2 + 9424*A*B^2*a*b^17*d^2 - 84576*A*B^2*a^3*b^15*d^2 + 15440*A*B^2*a^5*b^13*d^2 + 917
76*A*B^2*a^7*b^11*d^2 - 17664*A*B^2*a^9*b^9*d^2 - 37728*A^2*B*a^2*b^16*d^2 + 88772*A^2*B*a^4*b^14*d^2 + 68544*
A^2*B*a^6*b^12*d^2 - 56832*A^2*B*a^8*b^10*d^2 + 2304*A^2*B*a^10*b^8*d^2)/(8*d^5))*(-(((8*B^2*a^5*d^2 - 8*A^2*a
^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*
A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2
*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b
^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^
2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^
2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) - ((a + b*tan(c + d*x))^
(1/2)*(153*A^4*b^20 + 128*B^4*b^20 + 231*A^2*B^2*b^20 - 7*A^4*a^2*b^18 + 9895*A^4*a^4*b^16 - 27465*A^4*a^6*b^1
4 + 26320*A^4*a^8*b^12 - 832*A^4*a^10*b^10 + 128*A^4*a^12*b^8 - 132*B^4*a^2*b^18 + 16380*B^4*a^4*b^16 - 25596*
B^4*a^6*b^14 + 21060*B^4*a^8*b^12 - 4032*B^4*a^10*b^10 + 384*B^4*a^12*b^8 + 6811*A^2*B^2*a^2*b^18 - 61315*A^2*
B^2*a^4*b^16 + 184661*A^2*B^2*a^6*b^14 - 121620*A^2*B^2*a^8*b^12 + 23296*A^2*B^2*a^10*b^10 - 300*A*B^3*a*b^19
+ 600*A^3*B*a*b^19 + 17860*A*B^3*a^3*b^17 - 91700*A*B^3*a^5*b^15 + 110172*A*B^3*a^7*b^13 - 43520*A*B^3*a^9*b^1
1 + 4352*A*B^3*a^11*b^9 - 12860*A^3*B*a^3*b^17 + 79680*A^3*B*a^5*b^15 - 126700*A^3*B*a^7*b^13 + 40960*A^3*B*a^
9*b^11 - 1280*A^3*B*a^11*b^9))/(4*d^4))*(-(((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b
^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 -
d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4
*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A
^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*
d^2 - 10*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2
*b^3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2)*1i - (((((1280*A*b^13*d^4 + 6656*B*a*b^12*d^4 - 6912*A*a^2*b^11*d^
4 - 8192*A*a^4*b^9*d^4 + 3584*B*a^3*b^10*d^4 - 3072*B*a^5*b^8*d^4)/(8*d^5) + ((2048*b^10*d^4 + 3072*a^2*b^8*d^
4)*(a + b*tan(c + d*x))^(1/2)*(-(((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 1
6*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*
a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*
A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2
*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A
^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2 -
10*A*B*a^4*b*d^2)/(4*d^4))^(1/2))/(4*d^4))*(-(((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a
^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/
64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4
*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 +
10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*
a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B
*a^2*b^3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(320*A^2*a^3*b^12*d^2 - 19456*A^
2*a^5*b^10*d^2 + 1280*A^2*a^7*b^8*d^2 - 2320*B^2*a^3*b^12*d^2 + 16896*B^2*a^5*b^10*d^2 - 2304*B^2*a^7*b^8*d^2
- 2048*A*B*b^15*d^2 + 4764*A^2*a*b^14*d^2 - 4864*B^2*a*b^14*d^2 + 14160*A*B*a^2*b^13*d^2 + 30720*A*B*a^4*b^11*
d^2 - 18432*A*B*a^6*b^9*d^2))/(4*d^4))*(-(((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^
2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 -
d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*
b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^
2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d
^2 - 10*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*
b^3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) - (25800*A^3*a^3*b^15*d^2 - 8704*A^3*a^5*b^13*d^2 - 31872*A^3*a^7*b
^11*d^2 + 5888*A^3*a^9*b^9*d^2 + 14832*B^3*a^2*b^16*d^2 - 27648*B^3*a^4*b^14*d^2 - 23280*B^3*a^6*b^12*d^2 + 18
432*B^3*a^8*b^10*d^2 - 768*B^3*a^10*b^8*d^2 + 1180*A^2*B*b^18*d^2 - 3256*A^3*a*b^17*d^2 + 9424*A*B^2*a*b^17*d^
2 - 84576*A*B^2*a^3*b^15*d^2 + 15440*A*B^2*a^5*b^13*d^2 + 91776*A*B^2*a^7*b^11*d^2 - 17664*A*B^2*a^9*b^9*d^2 -
37728*A^2*B*a^2*b^16*d^2 + 88772*A^2*B*a^4*b^14*d^2 + 68544*A^2*B*a^6*b^12*d^2 - 56832*A^2*B*a^8*b^10*d^2 + 2
304*A^2*B*a^10*b^8*d^2)/(8*d^5))*(-(((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2
+ 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A
^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 +
10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*
a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 1
0*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^
2 - 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(153*A^4*b^20 + 128*B^4*b^20 + 231*A^2*B^2*
b^20 - 7*A^4*a^2*b^18 + 9895*A^4*a^4*b^16 - 27465*A^4*a^6*b^14 + 26320*A^4*a^8*b^12 - 832*A^4*a^10*b^10 + 128*
A^4*a^12*b^8 - 132*B^4*a^2*b^18 + 16380*B^4*a^4*b^16 - 25596*B^4*a^6*b^14 + 21060*B^4*a^8*b^12 - 4032*B^4*a^10
*b^10 + 384*B^4*a^12*b^8 + 6811*A^2*B^2*a^2*b^18 - 61315*A^2*B^2*a^4*b^16 + 184661*A^2*B^2*a^6*b^14 - 121620*A
^2*B^2*a^8*b^12 + 23296*A^2*B^2*a^10*b^10 - 300*A*B^3*a*b^19 + 600*A^3*B*a*b^19 + 17860*A*B^3*a^3*b^17 - 91700
*A*B^3*a^5*b^15 + 110172*A*B^3*a^7*b^13 - 43520*A*B^3*a^9*b^11 + 4352*A*B^3*a^11*b^9 - 12860*A^3*B*a^3*b^17 +
79680*A^3*B*a^5*b^15 - 126700*A^3*B*a^7*b^13 + 40960*A^3*B*a^9*b^11 - 1280*A^3*B*a^11*b^9))/(4*d^4))*(-(((8*B^
2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B
^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 +
2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2
*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*
a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2
*A*B*b^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2)*1i)/(
(55*A^5*b^23 + 80*A*B^4*b^23 + 480*B^5*a*b^22 + 135*A^3*B^2*b^23 + 240*A^5*a^2*b^21 - 4090*A^5*a^4*b^19 - 1020
0*A^5*a^6*b^17 - 5125*A^5*a^8*b^15 + 2480*A^5*a^10*b^13 + 1040*A^5*a^12*b^11 - 640*A^5*a^14*b^9 - 76*B^5*a^3*b
^20 + 1344*B^5*a^5*b^18 + 7272*B^5*a^7*b^16 + 5792*B^5*a^9*b^14 - 1596*B^5*a^11*b^12 - 2016*B^5*a^13*b^10 + 61
08*A^2*B^3*a^3*b^20 + 5778*A^2*B^3*a^5*b^18 - 10328*A^2*B^3*a^7*b^16 - 13603*A^2*B^3*a^9*b^14 - 396*A^2*B^3*a^
11*b^12 + 2368*A^2*B^3*a^13*b^10 - 256*A^2*B^3*a^15*b^8 - 1720*A^3*B^2*a^2*b^21 - 490*A^3*B^2*a^4*b^19 + 1784*
A^3*B^2*a^6*b^17 - 10165*A^3*B^2*a^8*b^15 - 17464*A^3*B^2*a^10*b^13 - 6112*A^3*B^2*a^12*b^11 + 768*A^3*B^2*a^1
4*b^9 + 105*A^4*B*a*b^22 - 1960*A*B^4*a^2*b^21 + 3600*A*B^4*a^4*b^19 + 11984*A*B^4*a^6*b^17 - 5040*A*B^4*a^8*b
^15 - 19944*A*B^4*a^10*b^13 - 7152*A*B^4*a^12*b^11 + 1408*A*B^4*a^14*b^9 + 585*A^2*B^3*a*b^22 + 6184*A^4*B*a^3
*b^20 + 4434*A^4*B*a^5*b^18 - 17600*A^4*B*a^7*b^16 - 19395*A^4*B*a^9*b^14 + 1200*A^4*B*a^11*b^12 + 4384*A^4*B*
a^13*b^10 - 256*A^4*B*a^15*b^8)/(4*d^5) + (((((1280*A*b^13*d^4 + 6656*B*a*b^12*d^4 - 6912*A*a^2*b^11*d^4 - 819
2*A*a^4*b^9*d^4 + 3584*B*a^3*b^10*d^4 - 3072*B*a^5*b^8*d^4)/(8*d^5) - ((2048*b^10*d^4 + 3072*a^2*b^8*d^4)*(a +
b*tan(c + d*x))^(1/2)*(-(((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b
^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 +
A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6
*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 +
20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A^2*a^3*
b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2 - 10*A*B
*a^4*b*d^2)/(4*d^4))^(1/2))/(4*d^4))*(-(((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*
d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^
4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^
6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*
B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2
- 10*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^
3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(320*A^2*a^3*b^12*d^2 - 19456*A^2*a^5*b
^10*d^2 + 1280*A^2*a^7*b^8*d^2 - 2320*B^2*a^3*b^12*d^2 + 16896*B^2*a^5*b^10*d^2 - 2304*B^2*a^7*b^8*d^2 - 2048*
A*B*b^15*d^2 + 4764*A^2*a*b^14*d^2 - 4864*B^2*a*b^14*d^2 + 14160*A*B*a^2*b^13*d^2 + 30720*A*B*a^4*b^11*d^2 - 1
8432*A*B*a^6*b^9*d^2))/(4*d^4))*(-(((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 +
16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^
4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 1
0*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a
^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10
*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2
- 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) - (25800*A^3*a^3*b^15*d^2 - 8704*A^3*a^5*b^13*d^2 - 31872*A^3*a^7*b^11*d^2
+ 5888*A^3*a^9*b^9*d^2 + 14832*B^3*a^2*b^16*d^2 - 27648*B^3*a^4*b^14*d^2 - 23280*B^3*a^6*b^12*d^2 + 18432*B^3
*a^8*b^10*d^2 - 768*B^3*a^10*b^8*d^2 + 1180*A^2*B*b^18*d^2 - 3256*A^3*a*b^17*d^2 + 9424*A*B^2*a*b^17*d^2 - 845
76*A*B^2*a^3*b^15*d^2 + 15440*A*B^2*a^5*b^13*d^2 + 91776*A*B^2*a^7*b^11*d^2 - 17664*A*B^2*a^9*b^9*d^2 - 37728*
A^2*B*a^2*b^16*d^2 + 88772*A^2*B*a^4*b^14*d^2 + 68544*A^2*B*a^6*b^12*d^2 - 56832*A^2*B*a^8*b^10*d^2 + 2304*A^2
*B*a^10*b^8*d^2)/(8*d^5))*(-(((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*
B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10
+ A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*
a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8
+ 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A^2*a
^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2 - 10*
A*B*a^4*b*d^2)/(4*d^4))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(153*A^4*b^20 + 128*B^4*b^20 + 231*A^2*B^2*b^20 -
7*A^4*a^2*b^18 + 9895*A^4*a^4*b^16 - 27465*A^4*a^6*b^14 + 26320*A^4*a^8*b^12 - 832*A^4*a^10*b^10 + 128*A^4*a^1
2*b^8 - 132*B^4*a^2*b^18 + 16380*B^4*a^4*b^16 - 25596*B^4*a^6*b^14 + 21060*B^4*a^8*b^12 - 4032*B^4*a^10*b^10 +
384*B^4*a^12*b^8 + 6811*A^2*B^2*a^2*b^18 - 61315*A^2*B^2*a^4*b^16 + 184661*A^2*B^2*a^6*b^14 - 121620*A^2*B^2*
a^8*b^12 + 23296*A^2*B^2*a^10*b^10 - 300*A*B^3*a*b^19 + 600*A^3*B*a*b^19 + 17860*A*B^3*a^3*b^17 - 91700*A*B^3*
a^5*b^15 + 110172*A*B^3*a^7*b^13 - 43520*A*B^3*a^9*b^11 + 4352*A*B^3*a^11*b^9 - 12860*A^3*B*a^3*b^17 + 79680*A
^3*B*a^5*b^15 - 126700*A^3*B*a^7*b^13 + 40960*A^3*B*a^9*b^11 - 1280*A^3*B*a^11*b^9))/(4*d^4))*(-(((8*B^2*a^5*d
^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^
4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*
B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 +
10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4
+ 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^
5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) + (((((1280*
A*b^13*d^4 + 6656*B*a*b^12*d^4 - 6912*A*a^2*b^11*d^4 - 8192*A*a^4*b^9*d^4 + 3584*B*a^3*b^10*d^4 - 3072*B*a^5*b
^8*d^4)/(8*d^5) + ((2048*b^10*d^4 + 3072*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(-(((8*B^2*a^5*d^2 - 8*A^2*a^
5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A
*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*
A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^
6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2
*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2
*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2))/(4*d^4))*(-(((8*B^2*a^5*
d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b
^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2
*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 +
10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^
4 + 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b
^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) + ((a + b*t
an(c + d*x))^(1/2)*(320*A^2*a^3*b^12*d^2 - 19456*A^2*a^5*b^10*d^2 + 1280*A^2*a^7*b^8*d^2 - 2320*B^2*a^3*b^12*d
^2 + 16896*B^2*a^5*b^10*d^2 - 2304*B^2*a^7*b^8*d^2 - 2048*A*B*b^15*d^2 + 4764*A^2*a*b^14*d^2 - 4864*B^2*a*b^14
*d^2 + 14160*A*B*a^2*b^13*d^2 + 30720*A*B*a^4*b^11*d^2 - 18432*A*B*a^6*b^9*d^2))/(4*d^4))*(-(((8*B^2*a^5*d^2 -
8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^
2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*
a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B
^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 1
0*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^
2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) - (25800*A^3*a^3
*b^15*d^2 - 8704*A^3*a^5*b^13*d^2 - 31872*A^3*a^7*b^11*d^2 + 5888*A^3*a^9*b^9*d^2 + 14832*B^3*a^2*b^16*d^2 - 2
7648*B^3*a^4*b^14*d^2 - 23280*B^3*a^6*b^12*d^2 + 18432*B^3*a^8*b^10*d^2 - 768*B^3*a^10*b^8*d^2 + 1180*A^2*B*b^
18*d^2 - 3256*A^3*a*b^17*d^2 + 9424*A*B^2*a*b^17*d^2 - 84576*A*B^2*a^3*b^15*d^2 + 15440*A*B^2*a^5*b^13*d^2 + 9
1776*A*B^2*a^7*b^11*d^2 - 17664*A*B^2*a^9*b^9*d^2 - 37728*A^2*B*a^2*b^16*d^2 + 88772*A^2*B*a^4*b^14*d^2 + 6854
4*A^2*B*a^6*b^12*d^2 - 56832*A^2*B*a^8*b^10*d^2 + 2304*A^2*B*a^10*b^8*d^2)/(8*d^5))*(-(((8*B^2*a^5*d^2 - 8*A^2
*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 16
0*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 +
2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4
*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*
B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*
A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) + ((a + b*tan(c + d*x)
)^(1/2)*(153*A^4*b^20 + 128*B^4*b^20 + 231*A^2*B^2*b^20 - 7*A^4*a^2*b^18 + 9895*A^4*a^4*b^16 - 27465*A^4*a^6*b
^14 + 26320*A^4*a^8*b^12 - 832*A^4*a^10*b^10 + 128*A^4*a^12*b^8 - 132*B^4*a^2*b^18 + 16380*B^4*a^4*b^16 - 2559
6*B^4*a^6*b^14 + 21060*B^4*a^8*b^12 - 4032*B^4*a^10*b^10 + 384*B^4*a^12*b^8 + 6811*A^2*B^2*a^2*b^18 - 61315*A^
2*B^2*a^4*b^16 + 184661*A^2*B^2*a^6*b^14 - 121620*A^2*B^2*a^8*b^12 + 23296*A^2*B^2*a^10*b^10 - 300*A*B^3*a*b^1
9 + 600*A^3*B*a*b^19 + 17860*A*B^3*a^3*b^17 - 91700*A*B^3*a^5*b^15 + 110172*A*B^3*a^7*b^13 - 43520*A*B^3*a^9*b
^11 + 4352*A*B^3*a^11*b^9 - 12860*A^3*B*a^3*b^17 + 79680*A^3*B*a^5*b^15 - 126700*A^3*B*a^7*b^13 + 40960*A^3*B*
a^9*b^11 - 1280*A^3*B*a^11*b^9))/(4*d^4))*(-(((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3
*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64
- d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a
^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10
*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^
5*d^2 - 10*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a
^2*b^3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2)))*(-(((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B
^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2
)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10
*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^
2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 -
B^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20
*A*B*a^2*b^3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2)*2i

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**4*(a+b*tan(d*x+c))**(5/2)*(A+B*tan(d*x+c)),x)

[Out]

Timed out

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