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3.330 \(\int \cot ^3(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx\)

Optimal. Leaf size=219 \[ \frac {\left (8 a^2 A-12 a b B-3 A b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{4 \sqrt {a} d}-\frac {(a-i b)^{3/2} (A-i B) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {(a+i b)^{3/2} (A+i B) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {(4 a B+5 A b) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 d}-\frac {a A \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d} \]

[Out]

-(a-I*b)^(3/2)*(A-I*B)*arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/d-(a+I*b)^(3/2)*(A+I*B)*arctanh((a+b*tan(
d*x+c))^(1/2)/(a+I*b)^(1/2))/d+1/4*(8*A*a^2-3*A*b^2-12*B*a*b)*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/d/a^(1/2
)-1/4*(5*A*b+4*B*a)*cot(d*x+c)*(a+b*tan(d*x+c))^(1/2)/d-1/2*a*A*cot(d*x+c)^2*(a+b*tan(d*x+c))^(1/2)/d

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Rubi [A]  time = 0.96, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {3605, 3649, 3653, 3539, 3537, 63, 208, 3634} \[ \frac {\left (8 a^2 A-12 a b B-3 A b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{4 \sqrt {a} d}-\frac {(a-i b)^{3/2} (A-i B) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {(a+i b)^{3/2} (A+i B) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {(4 a B+5 A b) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 d}-\frac {a A \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((8*a^2*A - 3*A*b^2 - 12*a*b*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(4*Sqrt[a]*d) - ((a - I*b)^(3/2)*(A
 - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d - ((a + I*b)^(3/2)*(A + I*B)*ArcTanh[Sqrt[a + b*Tan
[c + d*x]]/Sqrt[a + I*b]])/d - ((5*A*b + 4*a*B)*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(4*d) - (a*A*Cot[c + d*
x]^2*Sqrt[a + b*Tan[c + d*x]])/(2*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 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 ^3(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx &=-\frac {a A \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {1}{2} \int \frac {\cot ^2(c+d x) \left (\frac {1}{2} a (5 A b+4 a B)-2 \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)-\frac {1}{2} b (3 a A-4 b B) \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx\\ &=-\frac {(5 A b+4 a B) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 d}-\frac {a A \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {\int \frac {\cot (c+d x) \left (\frac {1}{4} a \left (8 a^2 A-3 A b^2-12 a b B\right )+2 a \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)+\frac {1}{4} a b (5 A b+4 a B) \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 a}\\ &=-\frac {(5 A b+4 a B) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 d}-\frac {a A \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {\int \frac {2 a \left (2 a A b+a^2 B-b^2 B\right )-2 a \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 a}-\frac {1}{8} \left (8 a^2 A-3 A b^2-12 a b 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 {(5 A b+4 a B) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 d}-\frac {a A \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {1}{2} \left ((a+i b)^2 (i A-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)^2 (i A+B)\right ) \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx-\frac {\left (8 a^2 A-3 A b^2-12 a b B\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{8 d}\\ &=-\frac {(5 A b+4 a B) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 d}-\frac {a A \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {\left ((a-i b)^2 (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)^2 (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 (8 a^2 A-3 A b^2-12 a b 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 )}{4 b d}\\ &=\frac {\left (8 a^2 A-3 A b^2-12 a b B\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{4 \sqrt {a} d}-\frac {(5 A b+4 a B) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 d}-\frac {a A \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {\left ((a+i b)^2 (i A-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)^2 (i A+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 (8 a^2 A-3 A b^2-12 a b B\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{4 \sqrt {a} d}-\frac {(a-i b)^{3/2} (A-i B) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {(a+i b)^{3/2} (A+i B) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {(5 A b+4 a B) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 d}-\frac {a A \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\\ \end {align*}

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Mathematica [A]  time = 2.51, size = 195, normalized size = 0.89 \[ \frac {\left (8 a^2 A-12 a b B-3 A b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )-\sqrt {a} \left (4 (a-i b)^{3/2} (A-i B) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )+4 (a+i b)^{3/2} (A+i B) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+\cot (c+d x) \sqrt {a+b \tan (c+d x)} (2 a A \cot (c+d x)+4 a B+5 A b)\right )}{4 \sqrt {a} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((8*a^2*A - 3*A*b^2 - 12*a*b*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]] - Sqrt[a]*(4*(a - I*b)^(3/2)*(A - I*
B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + 4*(a + I*b)^(3/2)*(A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*
x]]/Sqrt[a + I*b]] + Cot[c + d*x]*(5*A*b + 4*a*B + 2*a*A*Cot[c + d*x])*Sqrt[a + b*Tan[c + d*x]]))/(4*Sqrt[a]*d
)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [C]  time = 4.10, size = 102706, normalized size = 468.98 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^3*(a+b*tan(d*x+c))^(3/2)*(A+B*tan(d*x+c)),x)

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cot \left (d x + c\right )^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((B*tan(d*x + c) + A)*(b*tan(d*x + c) + a)^(3/2)*cot(d*x + c)^3, x)

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mupad [B]  time = 8.65, size = 23016, normalized size = 105.10 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^3*(A + B*tan(c + d*x))*(a + b*tan(c + d*x))^(3/2),x)

[Out]

atan(((((932*A^3*a^3*b^12*d^2 + 1344*A^3*a^5*b^10*d^2 - 192*A^3*a^7*b^8*d^2 + 1600*B^3*a^2*b^13*d^2 + 704*B^3*
a^4*b^11*d^2 - 896*B^3*a^6*b^9*d^2 + 348*A^2*B*b^15*d^2 - 604*A^3*a*b^14*d^2 + 1760*A*B^2*a*b^14*d^2 - 3232*A*
B^2*a^3*b^12*d^2 - 4416*A*B^2*a^5*b^10*d^2 + 576*A*B^2*a^7*b^8*d^2 - 3780*A^2*B*a^2*b^13*d^2 - 1440*A^2*B*a^4*
b^11*d^2 + 2688*A^2*B*a^6*b^9*d^2)/(2*d^5) - (((384*A*b^12*d^4 + 1280*B*a*b^11*d^4 - 384*A*a^2*b^10*d^4 - 768*
A*a^4*b^8*d^4 + 1280*B*a^3*b^9*d^4)/(2*d^5) + ((512*b^10*d^4 + 768*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(((
(8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64
 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2
+ 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 +
2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2))/d^4)*((((8*A^2*a^3*d^2 -
8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 +
 A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 +
 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3
*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(1088*A^2*a^3
*b^10*d^2 - 576*A^2*a^5*b^8*d^2 - 1472*B^2*a^3*b^10*d^2 + 320*B^2*a^5*b^8*d^2 - 512*A*B*b^13*d^2 + 668*A^2*a*b
^12*d^2 - 704*B^2*a*b^12*d^2 + 224*A*B*a^2*b^11*d^2 + 2816*A*B*a^4*b^9*d^2))/d^4)*((((8*A^2*a^3*d^2 - 8*B^2*a^
3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6
 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a
^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b
^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2))*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2
 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 +
 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2
*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d
^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(41*A^4*b^16 + 32*B^4*b^16 + 55*A^2*B^2*b^1
6 + 26*A^4*a^2*b^14 + 553*A^4*a^4*b^12 - 304*A^4*a^6*b^10 + 96*A^4*a^8*b^8 - 16*B^4*a^2*b^14 + 1056*B^4*a^4*b^
12 - 16*B^4*a^6*b^10 + 32*B^4*a^8*b^8 + 1078*A^2*B^2*a^2*b^14 - 2953*A^2*B^2*a^4*b^12 + 2368*A^2*B^2*a^6*b^10
- 72*A*B^3*a*b^15 + 144*A^3*B*a*b^15 + 1776*A*B^3*a^3*b^13 - 2376*A*B^3*a^5*b^11 + 192*A*B^3*a^7*b^9 - 1080*A^
3*B*a^3*b^13 + 2120*A^3*B*a^5*b^11 - 704*A^3*B*a^7*b^9))/d^4)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d
^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6
 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a
^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2
*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2)*1i - (((932*A^3*a^3*b^12*d^2 + 1344*A^3*a^5*b^10*d^2 - 192*A^3*a^7*b^8*
d^2 + 1600*B^3*a^2*b^13*d^2 + 704*B^3*a^4*b^11*d^2 - 896*B^3*a^6*b^9*d^2 + 348*A^2*B*b^15*d^2 - 604*A^3*a*b^14
*d^2 + 1760*A*B^2*a*b^14*d^2 - 3232*A*B^2*a^3*b^12*d^2 - 4416*A*B^2*a^5*b^10*d^2 + 576*A*B^2*a^7*b^8*d^2 - 378
0*A^2*B*a^2*b^13*d^2 - 1440*A^2*B*a^4*b^11*d^2 + 2688*A^2*B*a^6*b^9*d^2)/(2*d^5) - (((384*A*b^12*d^4 + 1280*B*
a*b^11*d^4 - 384*A*a^2*b^10*d^4 - 768*A*a^4*b^8*d^4 + 1280*B*a^3*b^9*d^4)/(2*d^5) - ((512*b^10*d^4 + 768*a^2*b
^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*
B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^
2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2)
)^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*
d^4))^(1/2))/d^4)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 4
8*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2
*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3
*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - ((a
 + b*tan(c + d*x))^(1/2)*(1088*A^2*a^3*b^10*d^2 - 576*A^2*a^5*b^8*d^2 - 1472*B^2*a^3*b^10*d^2 + 320*B^2*a^5*b^
8*d^2 - 512*A*B*b^13*d^2 + 668*A^2*a*b^12*d^2 - 704*B^2*a*b^12*d^2 + 224*A*B*a^2*b^11*d^2 + 2816*A*B*a^4*b^9*d
^2))/d^4)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^
2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3
*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B
^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2))*((((8*A^2*a^
3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A
^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a
^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3
*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(41*A
^4*b^16 + 32*B^4*b^16 + 55*A^2*B^2*b^16 + 26*A^4*a^2*b^14 + 553*A^4*a^4*b^12 - 304*A^4*a^6*b^10 + 96*A^4*a^8*b
^8 - 16*B^4*a^2*b^14 + 1056*B^4*a^4*b^12 - 16*B^4*a^6*b^10 + 32*B^4*a^8*b^8 + 1078*A^2*B^2*a^2*b^14 - 2953*A^2
*B^2*a^4*b^12 + 2368*A^2*B^2*a^6*b^10 - 72*A*B^3*a*b^15 + 144*A^3*B*a*b^15 + 1776*A*B^3*a^3*b^13 - 2376*A*B^3*
a^5*b^11 + 192*A*B^3*a^7*b^9 - 1080*A^3*B*a^3*b^13 + 2120*A^3*B*a^5*b^11 - 704*A^3*B*a^7*b^9))/d^4)*((((8*A^2*
a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*
(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4
*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b
^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2)*1i)/((((932*A^3*a^3*b^12*d^2 + 13
44*A^3*a^5*b^10*d^2 - 192*A^3*a^7*b^8*d^2 + 1600*B^3*a^2*b^13*d^2 + 704*B^3*a^4*b^11*d^2 - 896*B^3*a^6*b^9*d^2
 + 348*A^2*B*b^15*d^2 - 604*A^3*a*b^14*d^2 + 1760*A*B^2*a*b^14*d^2 - 3232*A*B^2*a^3*b^12*d^2 - 4416*A*B^2*a^5*
b^10*d^2 + 576*A*B^2*a^7*b^8*d^2 - 3780*A^2*B*a^2*b^13*d^2 - 1440*A^2*B*a^4*b^11*d^2 + 2688*A^2*B*a^6*b^9*d^2)
/(2*d^5) - (((384*A*b^12*d^4 + 1280*B*a*b^11*d^4 - 384*A*a^2*b^10*d^4 - 768*A*a^4*b^8*d^4 + 1280*B*a^3*b^9*d^4
)/(2*d^5) + ((512*b^10*d^4 + 768*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 1
6*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^
6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 +
6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 +
3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2))/d^4)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 2
4*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A
^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4
 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 -
 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(1088*A^2*a^3*b^10*d^2 - 576*A^2*a^5*b^8*d^2 -
1472*B^2*a^3*b^10*d^2 + 320*B^2*a^5*b^8*d^2 - 512*A*B*b^13*d^2 + 668*A^2*a*b^12*d^2 - 704*B^2*a*b^12*d^2 + 224
*A*B*a^2*b^11*d^2 + 2816*A*B*a^4*b^9*d^2))/d^4)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*
b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a
^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2
*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a
^2*b*d^2)/(4*d^4))^(1/2))*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2
*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3
*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) +
 A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/
2) - ((a + b*tan(c + d*x))^(1/2)*(41*A^4*b^16 + 32*B^4*b^16 + 55*A^2*B^2*b^16 + 26*A^4*a^2*b^14 + 553*A^4*a^4*
b^12 - 304*A^4*a^6*b^10 + 96*A^4*a^8*b^8 - 16*B^4*a^2*b^14 + 1056*B^4*a^4*b^12 - 16*B^4*a^6*b^10 + 32*B^4*a^8*
b^8 + 1078*A^2*B^2*a^2*b^14 - 2953*A^2*B^2*a^4*b^12 + 2368*A^2*B^2*a^6*b^10 - 72*A*B^3*a*b^15 + 144*A^3*B*a*b^
15 + 1776*A*B^3*a^3*b^13 - 2376*A*B^3*a^5*b^11 + 192*A*B^3*a^7*b^9 - 1080*A^3*B*a^3*b^13 + 2120*A^3*B*a^5*b^11
 - 704*A^3*B*a^7*b^9))/d^4)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b
^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 +
 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2)
 + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(
1/2) + (((932*A^3*a^3*b^12*d^2 + 1344*A^3*a^5*b^10*d^2 - 192*A^3*a^7*b^8*d^2 + 1600*B^3*a^2*b^13*d^2 + 704*B^3
*a^4*b^11*d^2 - 896*B^3*a^6*b^9*d^2 + 348*A^2*B*b^15*d^2 - 604*A^3*a*b^14*d^2 + 1760*A*B^2*a*b^14*d^2 - 3232*A
*B^2*a^3*b^12*d^2 - 4416*A*B^2*a^5*b^10*d^2 + 576*A*B^2*a^7*b^8*d^2 - 3780*A^2*B*a^2*b^13*d^2 - 1440*A^2*B*a^4
*b^11*d^2 + 2688*A^2*B*a^6*b^9*d^2)/(2*d^5) - (((384*A*b^12*d^4 + 1280*B*a*b^11*d^4 - 384*A*a^2*b^10*d^4 - 768
*A*a^4*b^8*d^4 + 1280*B*a^3*b^9*d^4)/(2*d^5) - ((512*b^10*d^4 + 768*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((
((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/6
4 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2
 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 +
 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2))/d^4)*((((8*A^2*a^3*d^2 -
 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6
+ A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4
+ 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 -
3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(1088*A^2*a^
3*b^10*d^2 - 576*A^2*a^5*b^8*d^2 - 1472*B^2*a^3*b^10*d^2 + 320*B^2*a^5*b^8*d^2 - 512*A*B*b^13*d^2 + 668*A^2*a*
b^12*d^2 - 704*B^2*a*b^12*d^2 + 224*A*B*a^2*b^11*d^2 + 2816*A*B*a^4*b^9*d^2))/d^4)*((((8*A^2*a^3*d^2 - 8*B^2*a
^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^
6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*
a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*
b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2))*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^
2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6
+ 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^
2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*
d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(41*A^4*b^16 + 32*B^4*b^16 + 55*A^2*B^2*b^
16 + 26*A^4*a^2*b^14 + 553*A^4*a^4*b^12 - 304*A^4*a^6*b^10 + 96*A^4*a^8*b^8 - 16*B^4*a^2*b^14 + 1056*B^4*a^4*b
^12 - 16*B^4*a^6*b^10 + 32*B^4*a^8*b^8 + 1078*A^2*B^2*a^2*b^14 - 2953*A^2*B^2*a^4*b^12 + 2368*A^2*B^2*a^6*b^10
 - 72*A*B^3*a*b^15 + 144*A^3*B*a*b^15 + 1776*A*B^3*a^3*b^13 - 2376*A*B^3*a^5*b^11 + 192*A*B^3*a^7*b^9 - 1080*A
^3*B*a^3*b^13 + 2120*A^3*B*a^5*b^11 - 704*A^3*B*a^7*b^9))/d^4)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*
d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^
6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*
a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^
2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - (15*A^5*b^18 + 24*A*B^4*b^18 + 96*B^5*a*b^17 + 39*A^3*B^2*b^18 + 71*
A^5*a^2*b^16 - 119*A^5*a^4*b^14 - 391*A^5*a^6*b^12 - 216*A^5*a^8*b^10 + 96*B^5*a^3*b^15 + 96*B^5*a^7*b^11 + 96
*B^5*a^9*b^9 + 660*A^2*B^3*a^3*b^15 + 886*A^2*B^3*a^5*b^13 + 200*A^2*B^3*a^7*b^11 - 128*A^2*B^3*a^9*b^9 - 185*
A^3*B^2*a^2*b^16 - 279*A^3*B^2*a^4*b^14 + 89*A^3*B^2*a^6*b^12 + 80*A^3*B^2*a^8*b^10 - 64*A^3*B^2*a^10*b^8 + 6*
A^4*B*a*b^17 - 256*A*B^4*a^2*b^16 - 160*A*B^4*a^4*b^14 + 480*A*B^4*a^6*b^12 + 296*A*B^4*a^8*b^10 - 64*A*B^4*a^
10*b^8 + 102*A^2*B^3*a*b^17 + 564*A^4*B*a^3*b^15 + 886*A^4*B*a^5*b^13 + 104*A^4*B*a^7*b^11 - 224*A^4*B*a^9*b^9
)/d^5))*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*
b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A
^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2
*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2)*2i + atan(((((9
32*A^3*a^3*b^12*d^2 + 1344*A^3*a^5*b^10*d^2 - 192*A^3*a^7*b^8*d^2 + 1600*B^3*a^2*b^13*d^2 + 704*B^3*a^4*b^11*d
^2 - 896*B^3*a^6*b^9*d^2 + 348*A^2*B*b^15*d^2 - 604*A^3*a*b^14*d^2 + 1760*A*B^2*a*b^14*d^2 - 3232*A*B^2*a^3*b^
12*d^2 - 4416*A*B^2*a^5*b^10*d^2 + 576*A*B^2*a^7*b^8*d^2 - 3780*A^2*B*a^2*b^13*d^2 - 1440*A^2*B*a^4*b^11*d^2 +
 2688*A^2*B*a^6*b^9*d^2)/(2*d^5) - (((384*A*b^12*d^4 + 1280*B*a*b^11*d^4 - 384*A*a^2*b^10*d^4 - 768*A*a^4*b^8*
d^4 + 1280*B*a^3*b^9*d^4)/(2*d^5) + ((512*b^10*d^4 + 768*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(-(((8*A^2*a^
3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A
^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a
^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3
*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2))/d^4)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^
3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6
 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a
^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b
^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(1088*A^2*a^3*b^10*d^
2 - 576*A^2*a^5*b^8*d^2 - 1472*B^2*a^3*b^10*d^2 + 320*B^2*a^5*b^8*d^2 - 512*A*B*b^13*d^2 + 668*A^2*a*b^12*d^2
- 704*B^2*a*b^12*d^2 + 224*A*B*a^2*b^11*d^2 + 2816*A*B*a^4*b^9*d^2))/d^4)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 +
 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*
a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2
+ 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2
- 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2))*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*
A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2
*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 +
 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6
*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(41*A^4*b^16 + 32*B^4*b^16 + 55*A^2*B^2*b^16 + 26
*A^4*a^2*b^14 + 553*A^4*a^4*b^12 - 304*A^4*a^6*b^10 + 96*A^4*a^8*b^8 - 16*B^4*a^2*b^14 + 1056*B^4*a^4*b^12 - 1
6*B^4*a^6*b^10 + 32*B^4*a^8*b^8 + 1078*A^2*B^2*a^2*b^14 - 2953*A^2*B^2*a^4*b^12 + 2368*A^2*B^2*a^6*b^10 - 72*A
*B^3*a*b^15 + 144*A^3*B*a*b^15 + 1776*A*B^3*a^3*b^13 - 2376*A*B^3*a^5*b^11 + 192*A*B^3*a^7*b^9 - 1080*A^3*B*a^
3*b^13 + 2120*A^3*B*a^5*b^11 - 704*A^3*B*a^7*b^9))/d^4)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 -
24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*
A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^
4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2
+ 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2)*1i - (((932*A^3*a^3*b^12*d^2 + 1344*A^3*a^5*b^10*d^2 - 192*A^3*a^7*b^8*d^2 +
 1600*B^3*a^2*b^13*d^2 + 704*B^3*a^4*b^11*d^2 - 896*B^3*a^6*b^9*d^2 + 348*A^2*B*b^15*d^2 - 604*A^3*a*b^14*d^2
+ 1760*A*B^2*a*b^14*d^2 - 3232*A*B^2*a^3*b^12*d^2 - 4416*A*B^2*a^5*b^10*d^2 + 576*A*B^2*a^7*b^8*d^2 - 3780*A^2
*B*a^2*b^13*d^2 - 1440*A^2*B*a^4*b^11*d^2 + 2688*A^2*B*a^6*b^9*d^2)/(2*d^5) - (((384*A*b^12*d^4 + 1280*B*a*b^1
1*d^4 - 384*A*a^2*b^10*d^4 - 768*A*a^4*b^8*d^4 + 1280*B*a^3*b^9*d^4)/(2*d^5) - ((512*b^10*d^4 + 768*a^2*b^8*d^
4)*(a + b*tan(c + d*x))^(1/2)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*
a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^
6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1
/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4)
)^(1/2))/d^4)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A
*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^
4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^
2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - ((a +
b*tan(c + d*x))^(1/2)*(1088*A^2*a^3*b^10*d^2 - 576*A^2*a^5*b^8*d^2 - 1472*B^2*a^3*b^10*d^2 + 320*B^2*a^5*b^8*d
^2 - 512*A*B*b^13*d^2 + 668*A^2*a*b^12*d^2 - 704*B^2*a*b^12*d^2 + 224*A*B*a^2*b^11*d^2 + 2816*A*B*a^4*b^9*d^2)
)/d^4)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*
b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A
^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2
*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2))*(-(((8*A^2*a^3
*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^
4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^
2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*
d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(41*A^
4*b^16 + 32*B^4*b^16 + 55*A^2*B^2*b^16 + 26*A^4*a^2*b^14 + 553*A^4*a^4*b^12 - 304*A^4*a^6*b^10 + 96*A^4*a^8*b^
8 - 16*B^4*a^2*b^14 + 1056*B^4*a^4*b^12 - 16*B^4*a^6*b^10 + 32*B^4*a^8*b^8 + 1078*A^2*B^2*a^2*b^14 - 2953*A^2*
B^2*a^4*b^12 + 2368*A^2*B^2*a^6*b^10 - 72*A*B^3*a*b^15 + 144*A^3*B*a*b^15 + 1776*A*B^3*a^3*b^13 - 2376*A*B^3*a
^5*b^11 + 192*A*B^3*a^7*b^9 - 1080*A^3*B*a^3*b^13 + 2120*A^3*B*a^5*b^11 - 704*A^3*B*a^7*b^9))/d^4)*(-(((8*A^2*
a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*
(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4
*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b
^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2)*1i)/((((932*A^3*a^3*b^12*d^2 + 13
44*A^3*a^5*b^10*d^2 - 192*A^3*a^7*b^8*d^2 + 1600*B^3*a^2*b^13*d^2 + 704*B^3*a^4*b^11*d^2 - 896*B^3*a^6*b^9*d^2
 + 348*A^2*B*b^15*d^2 - 604*A^3*a*b^14*d^2 + 1760*A*B^2*a*b^14*d^2 - 3232*A*B^2*a^3*b^12*d^2 - 4416*A*B^2*a^5*
b^10*d^2 + 576*A*B^2*a^7*b^8*d^2 - 3780*A^2*B*a^2*b^13*d^2 - 1440*A^2*B*a^4*b^11*d^2 + 2688*A^2*B*a^6*b^9*d^2)
/(2*d^5) - (((384*A*b^12*d^4 + 1280*B*a*b^11*d^4 - 384*A*a^2*b^10*d^4 - 768*A*a^4*b^8*d^4 + 1280*B*a^3*b^9*d^4
)/(2*d^5) + ((512*b^10*d^4 + 768*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 +
16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a
^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 +
 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 -
 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2))/d^4)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 -
 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2
*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b
^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2
 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(1088*A^2*a^3*b^10*d^2 - 576*A^2*a^5*b^8*d^2
- 1472*B^2*a^3*b^10*d^2 + 320*B^2*a^5*b^8*d^2 - 512*A*B*b^13*d^2 + 668*A^2*a*b^12*d^2 - 704*B^2*a*b^12*d^2 + 2
24*A*B*a^2*b^11*d^2 + 2816*A*B*a^4*b^9*d^2))/d^4)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2
*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^
2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*
A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*
B*a^2*b*d^2)/(4*d^4))^(1/2))*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a
*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6
 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/
2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))
^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(41*A^4*b^16 + 32*B^4*b^16 + 55*A^2*B^2*b^16 + 26*A^4*a^2*b^14 + 553*A^4*
a^4*b^12 - 304*A^4*a^6*b^10 + 96*A^4*a^8*b^8 - 16*B^4*a^2*b^14 + 1056*B^4*a^4*b^12 - 16*B^4*a^6*b^10 + 32*B^4*
a^8*b^8 + 1078*A^2*B^2*a^2*b^14 - 2953*A^2*B^2*a^4*b^12 + 2368*A^2*B^2*a^6*b^10 - 72*A*B^3*a*b^15 + 144*A^3*B*
a*b^15 + 1776*A*B^3*a^3*b^13 - 2376*A*B^3*a^5*b^11 + 192*A*B^3*a^7*b^9 - 1080*A^3*B*a^3*b^13 + 2120*A^3*B*a^5*
b^11 - 704*A^3*B*a^7*b^9))/d^4)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^
2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*
b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^
(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^
4))^(1/2) + (((932*A^3*a^3*b^12*d^2 + 1344*A^3*a^5*b^10*d^2 - 192*A^3*a^7*b^8*d^2 + 1600*B^3*a^2*b^13*d^2 + 70
4*B^3*a^4*b^11*d^2 - 896*B^3*a^6*b^9*d^2 + 348*A^2*B*b^15*d^2 - 604*A^3*a*b^14*d^2 + 1760*A*B^2*a*b^14*d^2 - 3
232*A*B^2*a^3*b^12*d^2 - 4416*A*B^2*a^5*b^10*d^2 + 576*A*B^2*a^7*b^8*d^2 - 3780*A^2*B*a^2*b^13*d^2 - 1440*A^2*
B*a^4*b^11*d^2 + 2688*A^2*B*a^6*b^9*d^2)/(2*d^5) - (((384*A*b^12*d^4 + 1280*B*a*b^11*d^4 - 384*A*a^2*b^10*d^4
- 768*A*a^4*b^8*d^4 + 1280*B*a^3*b^9*d^4)/(2*d^5) - ((512*b^10*d^4 + 768*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/
2)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^
2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a
^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3
*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2))/d^4)*(-(((8*A^2*a^
3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A
^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a
^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3
*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(1088
*A^2*a^3*b^10*d^2 - 576*A^2*a^5*b^8*d^2 - 1472*B^2*a^3*b^10*d^2 + 320*B^2*a^5*b^8*d^2 - 512*A*B*b^13*d^2 + 668
*A^2*a*b^12*d^2 - 704*B^2*a*b^12*d^2 + 224*A*B*a^2*b^11*d^2 + 2816*A*B*a^4*b^9*d^2))/d^4)*(-(((8*A^2*a^3*d^2 -
 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6
+ A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4
+ 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 +
3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2))*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A
*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 +
 B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A
^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B
^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(41*A^4*b^16 + 32*B^4*b^16 + 55*A
^2*B^2*b^16 + 26*A^4*a^2*b^14 + 553*A^4*a^4*b^12 - 304*A^4*a^6*b^10 + 96*A^4*a^8*b^8 - 16*B^4*a^2*b^14 + 1056*
B^4*a^4*b^12 - 16*B^4*a^6*b^10 + 32*B^4*a^8*b^8 + 1078*A^2*B^2*a^2*b^14 - 2953*A^2*B^2*a^4*b^12 + 2368*A^2*B^2
*a^6*b^10 - 72*A*B^3*a*b^15 + 144*A^3*B*a*b^15 + 1776*A*B^3*a^3*b^13 - 2376*A*B^3*a^5*b^11 + 192*A*B^3*a^7*b^9
 - 1080*A^3*B*a^3*b^13 + 2120*A^3*B*a^5*b^11 - 704*A^3*B*a^7*b^9))/d^4)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 1
6*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^
6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 +
6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 -
3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - (15*A^5*b^18 + 24*A*B^4*b^18 + 96*B^5*a*b^17 + 39*A^3*B^2*
b^18 + 71*A^5*a^2*b^16 - 119*A^5*a^4*b^14 - 391*A^5*a^6*b^12 - 216*A^5*a^8*b^10 + 96*B^5*a^3*b^15 + 96*B^5*a^7
*b^11 + 96*B^5*a^9*b^9 + 660*A^2*B^3*a^3*b^15 + 886*A^2*B^3*a^5*b^13 + 200*A^2*B^3*a^7*b^11 - 128*A^2*B^3*a^9*
b^9 - 185*A^3*B^2*a^2*b^16 - 279*A^3*B^2*a^4*b^14 + 89*A^3*B^2*a^6*b^12 + 80*A^3*B^2*a^8*b^10 - 64*A^3*B^2*a^1
0*b^8 + 6*A^4*B*a*b^17 - 256*A*B^4*a^2*b^16 - 160*A*B^4*a^4*b^14 + 480*A*B^4*a^6*b^12 + 296*A*B^4*a^8*b^10 - 6
4*A*B^4*a^10*b^8 + 102*A^2*B^3*a*b^17 + 564*A^4*B*a^3*b^15 + 886*A^4*B*a^5*b^13 + 104*A^4*B*a^7*b^11 - 224*A^4
*B*a^9*b^9)/d^5))*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 -
48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^
2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^
3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2)*2i +
 (((3*A*a*b^2)/4 + B*a^2*b)*(a + b*tan(c + d*x))^(1/2) - ((5*A*b^2)/4 + B*a*b)*(a + b*tan(c + d*x))^(3/2))/(d*
(a + b*tan(c + d*x))^2 + a^2*d - 2*a*d*(a + b*tan(c + d*x))) + (atan((((((a + b*tan(c + d*x))^(1/2)*(41*A^4*b^
16 + 32*B^4*b^16 + 55*A^2*B^2*b^16 + 26*A^4*a^2*b^14 + 553*A^4*a^4*b^12 - 304*A^4*a^6*b^10 + 96*A^4*a^8*b^8 -
16*B^4*a^2*b^14 + 1056*B^4*a^4*b^12 - 16*B^4*a^6*b^10 + 32*B^4*a^8*b^8 + 1078*A^2*B^2*a^2*b^14 - 2953*A^2*B^2*
a^4*b^12 + 2368*A^2*B^2*a^6*b^10 - 72*A*B^3*a*b^15 + 144*A^3*B*a*b^15 + 1776*A*B^3*a^3*b^13 - 2376*A*B^3*a^5*b
^11 + 192*A*B^3*a^7*b^9 - 1080*A^3*B*a^3*b^13 + 2120*A^3*B*a^5*b^11 - 704*A^3*B*a^7*b^9))/(8*d^4) + (((466*A^3
*a^3*b^12*d^2 + 672*A^3*a^5*b^10*d^2 - 96*A^3*a^7*b^8*d^2 + 800*B^3*a^2*b^13*d^2 + 352*B^3*a^4*b^11*d^2 - 448*
B^3*a^6*b^9*d^2 + 174*A^2*B*b^15*d^2 - 302*A^3*a*b^14*d^2 + 880*A*B^2*a*b^14*d^2 - 1616*A*B^2*a^3*b^12*d^2 - 2
208*A*B^2*a^5*b^10*d^2 + 288*A*B^2*a^7*b^8*d^2 - 1890*A^2*B*a^2*b^13*d^2 - 720*A^2*B*a^4*b^11*d^2 + 1344*A^2*B
*a^6*b^9*d^2)/(8*d^5) + ((((a + b*tan(c + d*x))^(1/2)*(1088*A^2*a^3*b^10*d^2 - 576*A^2*a^5*b^8*d^2 - 1472*B^2*
a^3*b^10*d^2 + 320*B^2*a^5*b^8*d^2 - 512*A*B*b^13*d^2 + 668*A^2*a*b^12*d^2 - 704*B^2*a*b^12*d^2 + 224*A*B*a^2*
b^11*d^2 + 2816*A*B*a^4*b^9*d^2))/(8*d^4) - (((192*A*b^12*d^4 + 640*B*a*b^11*d^4 - 192*A*a^2*b^10*d^4 - 384*A*
a^4*b^8*d^4 + 640*B*a^3*b^9*d^4)/(8*d^5) - ((512*b^10*d^4 + 768*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(64*A^
2*a^5 + 9*A^2*a*b^4 - 48*A^2*a^3*b^2 + 144*B^2*a^3*b^2 - 192*A*B*a^4*b + 72*A*B*a^2*b^3)^(1/2))/(64*a*d^5))*(6
4*A^2*a^5 + 9*A^2*a*b^4 - 48*A^2*a^3*b^2 + 144*B^2*a^3*b^2 - 192*A*B*a^4*b + 72*A*B*a^2*b^3)^(1/2))/(8*a*d))*(
64*A^2*a^5 + 9*A^2*a*b^4 - 48*A^2*a^3*b^2 + 144*B^2*a^3*b^2 - 192*A*B*a^4*b + 72*A*B*a^2*b^3)^(1/2))/(8*a*d))*
(64*A^2*a^5 + 9*A^2*a*b^4 - 48*A^2*a^3*b^2 + 144*B^2*a^3*b^2 - 192*A*B*a^4*b + 72*A*B*a^2*b^3)^(1/2))/(8*a*d))
*(64*A^2*a^5 + 9*A^2*a*b^4 - 48*A^2*a^3*b^2 + 144*B^2*a^3*b^2 - 192*A*B*a^4*b + 72*A*B*a^2*b^3)^(1/2)*1i)/(a*d
) + ((((a + b*tan(c + d*x))^(1/2)*(41*A^4*b^16 + 32*B^4*b^16 + 55*A^2*B^2*b^16 + 26*A^4*a^2*b^14 + 553*A^4*a^4
*b^12 - 304*A^4*a^6*b^10 + 96*A^4*a^8*b^8 - 16*B^4*a^2*b^14 + 1056*B^4*a^4*b^12 - 16*B^4*a^6*b^10 + 32*B^4*a^8
*b^8 + 1078*A^2*B^2*a^2*b^14 - 2953*A^2*B^2*a^4*b^12 + 2368*A^2*B^2*a^6*b^10 - 72*A*B^3*a*b^15 + 144*A^3*B*a*b
^15 + 1776*A*B^3*a^3*b^13 - 2376*A*B^3*a^5*b^11 + 192*A*B^3*a^7*b^9 - 1080*A^3*B*a^3*b^13 + 2120*A^3*B*a^5*b^1
1 - 704*A^3*B*a^7*b^9))/(8*d^4) - (((466*A^3*a^3*b^12*d^2 + 672*A^3*a^5*b^10*d^2 - 96*A^3*a^7*b^8*d^2 + 800*B^
3*a^2*b^13*d^2 + 352*B^3*a^4*b^11*d^2 - 448*B^3*a^6*b^9*d^2 + 174*A^2*B*b^15*d^2 - 302*A^3*a*b^14*d^2 + 880*A*
B^2*a*b^14*d^2 - 1616*A*B^2*a^3*b^12*d^2 - 2208*A*B^2*a^5*b^10*d^2 + 288*A*B^2*a^7*b^8*d^2 - 1890*A^2*B*a^2*b^
13*d^2 - 720*A^2*B*a^4*b^11*d^2 + 1344*A^2*B*a^6*b^9*d^2)/(8*d^5) - ((((a + b*tan(c + d*x))^(1/2)*(1088*A^2*a^
3*b^10*d^2 - 576*A^2*a^5*b^8*d^2 - 1472*B^2*a^3*b^10*d^2 + 320*B^2*a^5*b^8*d^2 - 512*A*B*b^13*d^2 + 668*A^2*a*
b^12*d^2 - 704*B^2*a*b^12*d^2 + 224*A*B*a^2*b^11*d^2 + 2816*A*B*a^4*b^9*d^2))/(8*d^4) + (((192*A*b^12*d^4 + 64
0*B*a*b^11*d^4 - 192*A*a^2*b^10*d^4 - 384*A*a^4*b^8*d^4 + 640*B*a^3*b^9*d^4)/(8*d^5) + ((512*b^10*d^4 + 768*a^
2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(64*A^2*a^5 + 9*A^2*a*b^4 - 48*A^2*a^3*b^2 + 144*B^2*a^3*b^2 - 192*A*B*a
^4*b + 72*A*B*a^2*b^3)^(1/2))/(64*a*d^5))*(64*A^2*a^5 + 9*A^2*a*b^4 - 48*A^2*a^3*b^2 + 144*B^2*a^3*b^2 - 192*A
*B*a^4*b + 72*A*B*a^2*b^3)^(1/2))/(8*a*d))*(64*A^2*a^5 + 9*A^2*a*b^4 - 48*A^2*a^3*b^2 + 144*B^2*a^3*b^2 - 192*
A*B*a^4*b + 72*A*B*a^2*b^3)^(1/2))/(8*a*d))*(64*A^2*a^5 + 9*A^2*a*b^4 - 48*A^2*a^3*b^2 + 144*B^2*a^3*b^2 - 192
*A*B*a^4*b + 72*A*B*a^2*b^3)^(1/2))/(8*a*d))*(64*A^2*a^5 + 9*A^2*a*b^4 - 48*A^2*a^3*b^2 + 144*B^2*a^3*b^2 - 19
2*A*B*a^4*b + 72*A*B*a^2*b^3)^(1/2)*1i)/(a*d))/((15*A^5*b^18 + 24*A*B^4*b^18 + 96*B^5*a*b^17 + 39*A^3*B^2*b^18
 + 71*A^5*a^2*b^16 - 119*A^5*a^4*b^14 - 391*A^5*a^6*b^12 - 216*A^5*a^8*b^10 + 96*B^5*a^3*b^15 + 96*B^5*a^7*b^1
1 + 96*B^5*a^9*b^9 + 660*A^2*B^3*a^3*b^15 + 886*A^2*B^3*a^5*b^13 + 200*A^2*B^3*a^7*b^11 - 128*A^2*B^3*a^9*b^9
- 185*A^3*B^2*a^2*b^16 - 279*A^3*B^2*a^4*b^14 + 89*A^3*B^2*a^6*b^12 + 80*A^3*B^2*a^8*b^10 - 64*A^3*B^2*a^10*b^
8 + 6*A^4*B*a*b^17 - 256*A*B^4*a^2*b^16 - 160*A*B^4*a^4*b^14 + 480*A*B^4*a^6*b^12 + 296*A*B^4*a^8*b^10 - 64*A*
B^4*a^10*b^8 + 102*A^2*B^3*a*b^17 + 564*A^4*B*a^3*b^15 + 886*A^4*B*a^5*b^13 + 104*A^4*B*a^7*b^11 - 224*A^4*B*a
^9*b^9)/d^5 - ((((a + b*tan(c + d*x))^(1/2)*(41*A^4*b^16 + 32*B^4*b^16 + 55*A^2*B^2*b^16 + 26*A^4*a^2*b^14 + 5
53*A^4*a^4*b^12 - 304*A^4*a^6*b^10 + 96*A^4*a^8*b^8 - 16*B^4*a^2*b^14 + 1056*B^4*a^4*b^12 - 16*B^4*a^6*b^10 +
32*B^4*a^8*b^8 + 1078*A^2*B^2*a^2*b^14 - 2953*A^2*B^2*a^4*b^12 + 2368*A^2*B^2*a^6*b^10 - 72*A*B^3*a*b^15 + 144
*A^3*B*a*b^15 + 1776*A*B^3*a^3*b^13 - 2376*A*B^3*a^5*b^11 + 192*A*B^3*a^7*b^9 - 1080*A^3*B*a^3*b^13 + 2120*A^3
*B*a^5*b^11 - 704*A^3*B*a^7*b^9))/(8*d^4) + (((466*A^3*a^3*b^12*d^2 + 672*A^3*a^5*b^10*d^2 - 96*A^3*a^7*b^8*d^
2 + 800*B^3*a^2*b^13*d^2 + 352*B^3*a^4*b^11*d^2 - 448*B^3*a^6*b^9*d^2 + 174*A^2*B*b^15*d^2 - 302*A^3*a*b^14*d^
2 + 880*A*B^2*a*b^14*d^2 - 1616*A*B^2*a^3*b^12*d^2 - 2208*A*B^2*a^5*b^10*d^2 + 288*A*B^2*a^7*b^8*d^2 - 1890*A^
2*B*a^2*b^13*d^2 - 720*A^2*B*a^4*b^11*d^2 + 1344*A^2*B*a^6*b^9*d^2)/(8*d^5) + ((((a + b*tan(c + d*x))^(1/2)*(1
088*A^2*a^3*b^10*d^2 - 576*A^2*a^5*b^8*d^2 - 1472*B^2*a^3*b^10*d^2 + 320*B^2*a^5*b^8*d^2 - 512*A*B*b^13*d^2 +
668*A^2*a*b^12*d^2 - 704*B^2*a*b^12*d^2 + 224*A*B*a^2*b^11*d^2 + 2816*A*B*a^4*b^9*d^2))/(8*d^4) - (((192*A*b^1
2*d^4 + 640*B*a*b^11*d^4 - 192*A*a^2*b^10*d^4 - 384*A*a^4*b^8*d^4 + 640*B*a^3*b^9*d^4)/(8*d^5) - ((512*b^10*d^
4 + 768*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(64*A^2*a^5 + 9*A^2*a*b^4 - 48*A^2*a^3*b^2 + 144*B^2*a^3*b^2 -
 192*A*B*a^4*b + 72*A*B*a^2*b^3)^(1/2))/(64*a*d^5))*(64*A^2*a^5 + 9*A^2*a*b^4 - 48*A^2*a^3*b^2 + 144*B^2*a^3*b
^2 - 192*A*B*a^4*b + 72*A*B*a^2*b^3)^(1/2))/(8*a*d))*(64*A^2*a^5 + 9*A^2*a*b^4 - 48*A^2*a^3*b^2 + 144*B^2*a^3*
b^2 - 192*A*B*a^4*b + 72*A*B*a^2*b^3)^(1/2))/(8*a*d))*(64*A^2*a^5 + 9*A^2*a*b^4 - 48*A^2*a^3*b^2 + 144*B^2*a^3
*b^2 - 192*A*B*a^4*b + 72*A*B*a^2*b^3)^(1/2))/(8*a*d))*(64*A^2*a^5 + 9*A^2*a*b^4 - 48*A^2*a^3*b^2 + 144*B^2*a^
3*b^2 - 192*A*B*a^4*b + 72*A*B*a^2*b^3)^(1/2))/(a*d) + ((((a + b*tan(c + d*x))^(1/2)*(41*A^4*b^16 + 32*B^4*b^1
6 + 55*A^2*B^2*b^16 + 26*A^4*a^2*b^14 + 553*A^4*a^4*b^12 - 304*A^4*a^6*b^10 + 96*A^4*a^8*b^8 - 16*B^4*a^2*b^14
 + 1056*B^4*a^4*b^12 - 16*B^4*a^6*b^10 + 32*B^4*a^8*b^8 + 1078*A^2*B^2*a^2*b^14 - 2953*A^2*B^2*a^4*b^12 + 2368
*A^2*B^2*a^6*b^10 - 72*A*B^3*a*b^15 + 144*A^3*B*a*b^15 + 1776*A*B^3*a^3*b^13 - 2376*A*B^3*a^5*b^11 + 192*A*B^3
*a^7*b^9 - 1080*A^3*B*a^3*b^13 + 2120*A^3*B*a^5*b^11 - 704*A^3*B*a^7*b^9))/(8*d^4) - (((466*A^3*a^3*b^12*d^2 +
 672*A^3*a^5*b^10*d^2 - 96*A^3*a^7*b^8*d^2 + 800*B^3*a^2*b^13*d^2 + 352*B^3*a^4*b^11*d^2 - 448*B^3*a^6*b^9*d^2
 + 174*A^2*B*b^15*d^2 - 302*A^3*a*b^14*d^2 + 880*A*B^2*a*b^14*d^2 - 1616*A*B^2*a^3*b^12*d^2 - 2208*A*B^2*a^5*b
^10*d^2 + 288*A*B^2*a^7*b^8*d^2 - 1890*A^2*B*a^2*b^13*d^2 - 720*A^2*B*a^4*b^11*d^2 + 1344*A^2*B*a^6*b^9*d^2)/(
8*d^5) - ((((a + b*tan(c + d*x))^(1/2)*(1088*A^2*a^3*b^10*d^2 - 576*A^2*a^5*b^8*d^2 - 1472*B^2*a^3*b^10*d^2 +
320*B^2*a^5*b^8*d^2 - 512*A*B*b^13*d^2 + 668*A^2*a*b^12*d^2 - 704*B^2*a*b^12*d^2 + 224*A*B*a^2*b^11*d^2 + 2816
*A*B*a^4*b^9*d^2))/(8*d^4) + (((192*A*b^12*d^4 + 640*B*a*b^11*d^4 - 192*A*a^2*b^10*d^4 - 384*A*a^4*b^8*d^4 + 6
40*B*a^3*b^9*d^4)/(8*d^5) + ((512*b^10*d^4 + 768*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(64*A^2*a^5 + 9*A^2*a
*b^4 - 48*A^2*a^3*b^2 + 144*B^2*a^3*b^2 - 192*A*B*a^4*b + 72*A*B*a^2*b^3)^(1/2))/(64*a*d^5))*(64*A^2*a^5 + 9*A
^2*a*b^4 - 48*A^2*a^3*b^2 + 144*B^2*a^3*b^2 - 192*A*B*a^4*b + 72*A*B*a^2*b^3)^(1/2))/(8*a*d))*(64*A^2*a^5 + 9*
A^2*a*b^4 - 48*A^2*a^3*b^2 + 144*B^2*a^3*b^2 - 192*A*B*a^4*b + 72*A*B*a^2*b^3)^(1/2))/(8*a*d))*(64*A^2*a^5 + 9
*A^2*a*b^4 - 48*A^2*a^3*b^2 + 144*B^2*a^3*b^2 - 192*A*B*a^4*b + 72*A*B*a^2*b^3)^(1/2))/(8*a*d))*(64*A^2*a^5 +
9*A^2*a*b^4 - 48*A^2*a^3*b^2 + 144*B^2*a^3*b^2 - 192*A*B*a^4*b + 72*A*B*a^2*b^3)^(1/2))/(a*d)))*(64*A^2*a^5 +
9*A^2*a*b^4 - 48*A^2*a^3*b^2 + 144*B^2*a^3*b^2 - 192*A*B*a^4*b + 72*A*B*a^2*b^3)^(1/2)*1i)/(4*a*d)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (A + B \tan {\left (c + d x \right )}\right ) \left (a + b \tan {\left (c + d x \right )}\right )^{\frac {3}{2}} \cot ^{3}{\left (c + d x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**3*(a+b*tan(d*x+c))**(3/2)*(A+B*tan(d*x+c)),x)

[Out]

Integral((A + B*tan(c + d*x))*(a + b*tan(c + d*x))**(3/2)*cot(c + d*x)**3, x)

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