\(\int \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx\) [323]

Optimal result

Integrand size = 33, antiderivative size = 219 \[ \int \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\frac {\left (8 a^2 A+A b^2-4 a b B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{3/2} d}-\frac {\sqrt {a-i b} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {\sqrt {a+i b} (A+i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {(A b+4 a B) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 a d}-\frac {A \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {3689, 3730, 3734, 3620, 3618, 65, 214, 3715} \[ \int \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\frac {\left (8 a^2 A-4 a b B+A b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{3/2} d}-\frac {\sqrt {a-i b} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {\sqrt {a+i b} (A+i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {(4 a B+A b) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 a d}-\frac {A \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d} \]

[In]

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

[Out]

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

Rule 65

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

Rule 214

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

Rule 3618

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

Rule 3620

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

Rule 3689

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

Rule 3715

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

Rule 3730

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

Rule 3734

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

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

Mathematica [A] (verified)

Time = 4.92 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.24 \[ \int \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\frac {\frac {\left (8 a^2 A+A b^2-4 a b B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2}}+\frac {\frac {4 \left (-a A b+A b \sqrt {-b^2}+b^2 B+a \sqrt {-b^2} B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {-b^2}}}\right )}{\sqrt {a-\sqrt {-b^2}}}-\frac {4 \left (a A b+A b \sqrt {-b^2}-b^2 B+a \sqrt {-b^2} B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+\sqrt {-b^2}}}\right )}{\sqrt {a+\sqrt {-b^2}}}-\frac {b \cot (c+d x) (A b+4 a B+2 a A \cot (c+d x)) \sqrt {a+b \tan (c+d x)}}{a}}{b}}{4 d} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1144\) vs. \(2(185)=370\).

Time = 0.25 (sec) , antiderivative size = 1145, normalized size of antiderivative = 5.23

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1145\)
default	\(\text {Expression too large to display}\)	\(1145\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1338 vs. \(2 (179) = 358\).

Time = 3.17 (sec) , antiderivative size = 2691, normalized size of antiderivative = 12.29 \[ \int \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/8*(4*a^2*d*sqrt(-(2*A*B*b + d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)
/d^4) - (A^2 - B^2)*a)/d^2)*log(-(2*(A^3*B + A*B^3)*a + (A^4 - B^4)*b)*sqrt(b*tan(d*x + c) + a) + (B*d^3*sqrt(
-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) + (2*A^2*B*a + (A^3 - A*B^2)*b)*d)
*sqrt(-(2*A*B*b + d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2
- B^2)*a)/d^2))*tan(d*x + c)^2 - 4*a^2*d*sqrt(-(2*A*B*b + d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (
A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2)*log(-(2*(A^3*B + A*B^3)*a + (A^4 - B^4)*b)*sqrt(b*tan(d
*x + c) + a) - (B*d^3*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) + (2*A^
2*B*a + (A^3 - A*B^2)*b)*d)*sqrt(-(2*A*B*b + d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B
^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2))*tan(d*x + c)^2 - 4*a^2*d*sqrt(-(2*A*B*b - d^2*sqrt(-(4*A^2*B^2*a^2
+ 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2)*log(-(2*(A^3*B + A*B^3)*a +
(A^4 - B^4)*b)*sqrt(b*tan(d*x + c) + a) + (B*d^3*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B
^2 + B^4)*b^2)/d^4) - (2*A^2*B*a + (A^3 - A*B^2)*b)*d)*sqrt(-(2*A*B*b - d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B -
A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2))*tan(d*x + c)^2 + 4*a^2*d*sqrt(-(2*A*B*b
- d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2)*l
og(-(2*(A^3*B + A*B^3)*a + (A^4 - B^4)*b)*sqrt(b*tan(d*x + c) + a) - (B*d^3*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B -
A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (2*A^2*B*a + (A^3 - A*B^2)*b)*d)*sqrt(-(2*A*B*b - d^2*sqrt(-(
4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2))*tan(d*x + c)^
2 + (8*A*a^2 - 4*B*a*b + A*b^2)*sqrt(a)*log((b*tan(d*x + c) + 2*sqrt(b*tan(d*x + c) + a)*sqrt(a) + 2*a)/tan(d*
x + c))*tan(d*x + c)^2 - 2*(2*A*a^2 + (4*B*a^2 + A*a*b)*tan(d*x + c))*sqrt(b*tan(d*x + c) + a))/(a^2*d*tan(d*x
+ c)^2), 1/4*(2*a^2*d*sqrt(-(2*A*B*b + d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 +
B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2)*log(-(2*(A^3*B + A*B^3)*a + (A^4 - B^4)*b)*sqrt(b*tan(d*x + c) + a) + (B*
d^3*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) + (2*A^2*B*a + (A^3 - A*B
^2)*b)*d)*sqrt(-(2*A*B*b + d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4
) - (A^2 - B^2)*a)/d^2))*tan(d*x + c)^2 - 2*a^2*d*sqrt(-(2*A*B*b + d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3
)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2)*log(-(2*(A^3*B + A*B^3)*a + (A^4 - B^4)*b)*sqr
t(b*tan(d*x + c) + a) - (B*d^3*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4
) + (2*A^2*B*a + (A^3 - A*B^2)*b)*d)*sqrt(-(2*A*B*b + d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4
- 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2))*tan(d*x + c)^2 - 2*a^2*d*sqrt(-(2*A*B*b - d^2*sqrt(-(4*A^2
*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2)*log(-(2*(A^3*B + A*
B^3)*a + (A^4 - B^4)*b)*sqrt(b*tan(d*x + c) + a) + (B*d^3*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4
- 2*A^2*B^2 + B^4)*b^2)/d^4) - (2*A^2*B*a + (A^3 - A*B^2)*b)*d)*sqrt(-(2*A*B*b - d^2*sqrt(-(4*A^2*B^2*a^2 + 4*
(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2))*tan(d*x + c)^2 + 2*a^2*d*sqrt(-
(2*A*B*b - d^2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*
a)/d^2)*log(-(2*(A^3*B + A*B^3)*a + (A^4 - B^4)*b)*sqrt(b*tan(d*x + c) + a) - (B*d^3*sqrt(-(4*A^2*B^2*a^2 + 4*
(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (2*A^2*B*a + (A^3 - A*B^2)*b)*d)*sqrt(-(2*A*B*b - d^
2*sqrt(-(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4) - (A^2 - B^2)*a)/d^2))*tan(
d*x + c)^2 - (8*A*a^2 - 4*B*a*b + A*b^2)*sqrt(-a)*arctan(sqrt(b*tan(d*x + c) + a)*sqrt(-a)/a)*tan(d*x + c)^2 -
(2*A*a^2 + (4*B*a^2 + A*a*b)*tan(d*x + c))*sqrt(b*tan(d*x + c) + a))/(a^2*d*tan(d*x + c)^2)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

Timed out. \[ \int \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 9.46 (sec) , antiderivative size = 14195, normalized size of antiderivative = 64.82 \[ \int \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

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

[Out]

(atan(((((((2*A^3*b^14*d^2 + 2*A^3*a^2*b^12*d^2 - 96*A^3*a^4*b^10*d^2 - 96*A^3*a^6*b^8*d^2 - 160*B^3*a^3*b^11*
d^2 - 160*B^3*a^5*b^9*d^2 + 48*A^2*B*a*b^13*d^2 - 192*A*B^2*a^2*b^12*d^2 + 96*A*B^2*a^4*b^10*d^2 + 288*A*B^2*a
^6*b^8*d^2 + 528*A^2*B*a^3*b^11*d^2 + 480*A^2*B*a^5*b^9*d^2)/(8*a^2*d^5) + (((((64*A*a*b^12*d^4 + 448*A*a^3*b^
10*d^4 + 384*A*a^5*b^8*d^4 - 256*B*a^2*b^11*d^4 - 256*B*a^4*b^9*d^4)/(8*a^2*d^5) - ((512*a^2*b^10*d^4 + 768*a^
4*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(64*A^2*a^7 + A^2*a^3*b^4 + 16*A^2*a^5*b^2 + 16*B^2*a^5*b^2 - 64*A*B*a^6
*b - 8*A*B*a^4*b^3)^(1/2))/(64*a^5*d^5))*(64*A^2*a^7 + A^2*a^3*b^4 + 16*A^2*a^5*b^2 + 16*B^2*a^5*b^2 - 64*A*B*
a^6*b - 8*A*B*a^4*b^3)^(1/2))/(8*a^3*d) - ((a + b*tan(c + d*x))^(1/2)*(128*B^2*a^3*b^10*d^2 - 576*A^2*a^5*b^8*
d^2 - 256*A^2*a^3*b^10*d^2 + 320*B^2*a^5*b^8*d^2 - 4*A^2*a*b^12*d^2 + 544*A*B*a^2*b^11*d^2 + 1024*A*B*a^4*b^9*
d^2))/(8*a^2*d^4))*(64*A^2*a^7 + A^2*a^3*b^4 + 16*A^2*a^5*b^2 + 16*B^2*a^5*b^2 - 64*A*B*a^6*b - 8*A*B*a^4*b^3)
^(1/2))/(8*a^3*d))*(64*A^2*a^7 + A^2*a^3*b^4 + 16*A^2*a^5*b^2 + 16*B^2*a^5*b^2 - 64*A*B*a^6*b - 8*A*B*a^4*b^3)
^(1/2))/(8*a^3*d) - ((a + b*tan(c + d*x))^(1/2)*(A^2*B^2*b^14 - A^4*b^14 + 17*A^4*a^2*b^12 + 16*A^4*a^4*b^10 +
96*A^4*a^6*b^8 + 48*B^4*a^2*b^12 + 48*B^4*a^4*b^10 + 32*B^4*a^6*b^8 + 95*A^2*B^2*a^2*b^12 + 448*A^2*B^2*a^4*b
^10 - 8*A*B^3*a*b^13 + 4*A^3*B*a*b^13 - 120*A*B^3*a^3*b^11 + 64*A*B^3*a^5*b^9 - 8*A^3*B*a^3*b^11 - 320*A^3*B*a
^5*b^9))/(8*a^2*d^4))*(64*A^2*a^7 + A^2*a^3*b^4 + 16*A^2*a^5*b^2 + 16*B^2*a^5*b^2 - 64*A*B*a^6*b - 8*A*B*a^4*b
^3)^(1/2)*1i)/(a^3*d) - (((((2*A^3*b^14*d^2 + 2*A^3*a^2*b^12*d^2 - 96*A^3*a^4*b^10*d^2 - 96*A^3*a^6*b^8*d^2 -
160*B^3*a^3*b^11*d^2 - 160*B^3*a^5*b^9*d^2 + 48*A^2*B*a*b^13*d^2 - 192*A*B^2*a^2*b^12*d^2 + 96*A*B^2*a^4*b^10*
d^2 + 288*A*B^2*a^6*b^8*d^2 + 528*A^2*B*a^3*b^11*d^2 + 480*A^2*B*a^5*b^9*d^2)/(8*a^2*d^5) + (((((64*A*a*b^12*d
^4 + 448*A*a^3*b^10*d^4 + 384*A*a^5*b^8*d^4 - 256*B*a^2*b^11*d^4 - 256*B*a^4*b^9*d^4)/(8*a^2*d^5) + ((512*a^2*
b^10*d^4 + 768*a^4*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(64*A^2*a^7 + A^2*a^3*b^4 + 16*A^2*a^5*b^2 + 16*B^2*a^5
*b^2 - 64*A*B*a^6*b - 8*A*B*a^4*b^3)^(1/2))/(64*a^5*d^5))*(64*A^2*a^7 + A^2*a^3*b^4 + 16*A^2*a^5*b^2 + 16*B^2*
a^5*b^2 - 64*A*B*a^6*b - 8*A*B*a^4*b^3)^(1/2))/(8*a^3*d) + ((a + b*tan(c + d*x))^(1/2)*(128*B^2*a^3*b^10*d^2 -
576*A^2*a^5*b^8*d^2 - 256*A^2*a^3*b^10*d^2 + 320*B^2*a^5*b^8*d^2 - 4*A^2*a*b^12*d^2 + 544*A*B*a^2*b^11*d^2 +
1024*A*B*a^4*b^9*d^2))/(8*a^2*d^4))*(64*A^2*a^7 + A^2*a^3*b^4 + 16*A^2*a^5*b^2 + 16*B^2*a^5*b^2 - 64*A*B*a^6*b
- 8*A*B*a^4*b^3)^(1/2))/(8*a^3*d))*(64*A^2*a^7 + A^2*a^3*b^4 + 16*A^2*a^5*b^2 + 16*B^2*a^5*b^2 - 64*A*B*a^6*b
- 8*A*B*a^4*b^3)^(1/2))/(8*a^3*d) + ((a + b*tan(c + d*x))^(1/2)*(A^2*B^2*b^14 - A^4*b^14 + 17*A^4*a^2*b^12 +
16*A^4*a^4*b^10 + 96*A^4*a^6*b^8 + 48*B^4*a^2*b^12 + 48*B^4*a^4*b^10 + 32*B^4*a^6*b^8 + 95*A^2*B^2*a^2*b^12 +
448*A^2*B^2*a^4*b^10 - 8*A*B^3*a*b^13 + 4*A^3*B*a*b^13 - 120*A*B^3*a^3*b^11 + 64*A*B^3*a^5*b^9 - 8*A^3*B*a^3*b
^11 - 320*A^3*B*a^5*b^9))/(8*a^2*d^4))*(64*A^2*a^7 + A^2*a^3*b^4 + 16*A^2*a^5*b^2 + 16*B^2*a^5*b^2 - 64*A*B*a^
6*b - 8*A*B*a^4*b^3)^(1/2)*1i)/(a^3*d))/((A^4*B*b^15 + 7*A^5*a*b^14 + A^2*B^3*b^15 + 63*A^5*a^3*b^12 + 56*A^5*
a^5*b^10 - 16*B^5*a^2*b^13 - 48*B^5*a^4*b^11 - 32*B^5*a^6*b^9 - 23*A^2*B^3*a^2*b^13 + 40*A^2*B^3*a^4*b^11 + 64
*A^2*B^3*a^6*b^9 + 55*A^3*B^2*a^3*b^12 + 112*A^3*B^2*a^5*b^10 + 64*A^3*B^2*a^7*b^8 - 8*A*B^4*a^3*b^12 + 56*A*B
^4*a^5*b^10 + 64*A*B^4*a^7*b^8 + 7*A^3*B^2*a*b^14 - 7*A^4*B*a^2*b^13 + 88*A^4*B*a^4*b^11 + 96*A^4*B*a^6*b^9)/(
a^2*d^5) + (((((2*A^3*b^14*d^2 + 2*A^3*a^2*b^12*d^2 - 96*A^3*a^4*b^10*d^2 - 96*A^3*a^6*b^8*d^2 - 160*B^3*a^3*b
^11*d^2 - 160*B^3*a^5*b^9*d^2 + 48*A^2*B*a*b^13*d^2 - 192*A*B^2*a^2*b^12*d^2 + 96*A*B^2*a^4*b^10*d^2 + 288*A*B
^2*a^6*b^8*d^2 + 528*A^2*B*a^3*b^11*d^2 + 480*A^2*B*a^5*b^9*d^2)/(8*a^2*d^5) + (((((64*A*a*b^12*d^4 + 448*A*a^
3*b^10*d^4 + 384*A*a^5*b^8*d^4 - 256*B*a^2*b^11*d^4 - 256*B*a^4*b^9*d^4)/(8*a^2*d^5) - ((512*a^2*b^10*d^4 + 76
8*a^4*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(64*A^2*a^7 + A^2*a^3*b^4 + 16*A^2*a^5*b^2 + 16*B^2*a^5*b^2 - 64*A*B
*a^6*b - 8*A*B*a^4*b^3)^(1/2))/(64*a^5*d^5))*(64*A^2*a^7 + A^2*a^3*b^4 + 16*A^2*a^5*b^2 + 16*B^2*a^5*b^2 - 64*
A*B*a^6*b - 8*A*B*a^4*b^3)^(1/2))/(8*a^3*d) - ((a + b*tan(c + d*x))^(1/2)*(128*B^2*a^3*b^10*d^2 - 576*A^2*a^5*
b^8*d^2 - 256*A^2*a^3*b^10*d^2 + 320*B^2*a^5*b^8*d^2 - 4*A^2*a*b^12*d^2 + 544*A*B*a^2*b^11*d^2 + 1024*A*B*a^4*
b^9*d^2))/(8*a^2*d^4))*(64*A^2*a^7 + A^2*a^3*b^4 + 16*A^2*a^5*b^2 + 16*B^2*a^5*b^2 - 64*A*B*a^6*b - 8*A*B*a^4*
b^3)^(1/2))/(8*a^3*d))*(64*A^2*a^7 + A^2*a^3*b^4 + 16*A^2*a^5*b^2 + 16*B^2*a^5*b^2 - 64*A*B*a^6*b - 8*A*B*a^4*
b^3)^(1/2))/(8*a^3*d) - ((a + b*tan(c + d*x))^(1/2)*(A^2*B^2*b^14 - A^4*b^14 + 17*A^4*a^2*b^12 + 16*A^4*a^4*b^
10 + 96*A^4*a^6*b^8 + 48*B^4*a^2*b^12 + 48*B^4*a^4*b^10 + 32*B^4*a^6*b^8 + 95*A^2*B^2*a^2*b^12 + 448*A^2*B^2*a
^4*b^10 - 8*A*B^3*a*b^13 + 4*A^3*B*a*b^13 - 120*A*B^3*a^3*b^11 + 64*A*B^3*a^5*b^9 - 8*A^3*B*a^3*b^11 - 320*A^3
*B*a^5*b^9))/(8*a^2*d^4))*(64*A^2*a^7 + A^2*a^3*b^4 + 16*A^2*a^5*b^2 + 16*B^2*a^5*b^2 - 64*A*B*a^6*b - 8*A*B*a
^4*b^3)^(1/2))/(a^3*d) + (((((2*A^3*b^14*d^2 + 2*A^3*a^2*b^12*d^2 - 96*A^3*a^4*b^10*d^2 - 96*A^3*a^6*b^8*d^2 -
160*B^3*a^3*b^11*d^2 - 160*B^3*a^5*b^9*d^2 + 48*A^2*B*a*b^13*d^2 - 192*A*B^2*a^2*b^12*d^2 + 96*A*B^2*a^4*b^10
*d^2 + 288*A*B^2*a^6*b^8*d^2 + 528*A^2*B*a^3*b^11*d^2 + 480*A^2*B*a^5*b^9*d^2)/(8*a^2*d^5) + (((((64*A*a*b^12*
d^4 + 448*A*a^3*b^10*d^4 + 384*A*a^5*b^8*d^4 - 256*B*a^2*b^11*d^4 - 256*B*a^4*b^9*d^4)/(8*a^2*d^5) + ((512*a^2
*b^10*d^4 + 768*a^4*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(64*A^2*a^7 + A^2*a^3*b^4 + 16*A^2*a^5*b^2 + 16*B^2*a^
5*b^2 - 64*A*B*a^6*b - 8*A*B*a^4*b^3)^(1/2))/(64*a^5*d^5))*(64*A^2*a^7 + A^2*a^3*b^4 + 16*A^2*a^5*b^2 + 16*B^2
*a^5*b^2 - 64*A*B*a^6*b - 8*A*B*a^4*b^3)^(1/2))/(8*a^3*d) + ((a + b*tan(c + d*x))^(1/2)*(128*B^2*a^3*b^10*d^2
- 576*A^2*a^5*b^8*d^2 - 256*A^2*a^3*b^10*d^2 + 320*B^2*a^5*b^8*d^2 - 4*A^2*a*b^12*d^2 + 544*A*B*a^2*b^11*d^2 +
1024*A*B*a^4*b^9*d^2))/(8*a^2*d^4))*(64*A^2*a^7 + A^2*a^3*b^4 + 16*A^2*a^5*b^2 + 16*B^2*a^5*b^2 - 64*A*B*a^6*
b - 8*A*B*a^4*b^3)^(1/2))/(8*a^3*d))*(64*A^2*a^7 + A^2*a^3*b^4 + 16*A^2*a^5*b^2 + 16*B^2*a^5*b^2 - 64*A*B*a^6*
b - 8*A*B*a^4*b^3)^(1/2))/(8*a^3*d) + ((a + b*tan(c + d*x))^(1/2)*(A^2*B^2*b^14 - A^4*b^14 + 17*A^4*a^2*b^12 +
16*A^4*a^4*b^10 + 96*A^4*a^6*b^8 + 48*B^4*a^2*b^12 + 48*B^4*a^4*b^10 + 32*B^4*a^6*b^8 + 95*A^2*B^2*a^2*b^12 +
448*A^2*B^2*a^4*b^10 - 8*A*B^3*a*b^13 + 4*A^3*B*a*b^13 - 120*A*B^3*a^3*b^11 + 64*A*B^3*a^5*b^9 - 8*A^3*B*a^3*
b^11 - 320*A^3*B*a^5*b^9))/(8*a^2*d^4))*(64*A^2*a^7 + A^2*a^3*b^4 + 16*A^2*a^5*b^2 + 16*B^2*a^5*b^2 - 64*A*B*a
^6*b - 8*A*B*a^4*b^3)^(1/2))/(a^3*d)))*(64*A^2*a^7 + A^2*a^3*b^4 + 16*A^2*a^5*b^2 + 16*B^2*a^5*b^2 - 64*A*B*a^
6*b - 8*A*B*a^4*b^3)^(1/2)*1i)/(4*a^3*d) - atan(((((((64*A*a*b^12*d^4 + 448*A*a^3*b^10*d^4 + 384*A*a^5*b^8*d^4
- 256*B*a^2*b^11*d^4 - 256*B*a^4*b^9*d^4)/(a^2*d^5) + ((512*a^2*b^10*d^4 + 768*a^4*b^8*d^4)*(a + b*tan(c + d*
x))^(1/2)*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*
d^4)^(1/2)/(4*d^4) + (A^2*a)/(4*d^2) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2))/(a^2*d^4))*((2*A^2*B^2*b^2*d^
4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A^2*a)
/(4*d^2) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(128*B^2*a^3*b^10*d^2 - 576*
A^2*a^5*b^8*d^2 - 256*A^2*a^3*b^10*d^2 + 320*B^2*a^5*b^8*d^2 - 4*A^2*a*b^12*d^2 + 544*A*B*a^2*b^11*d^2 + 1024*
A*B*a^4*b^9*d^2))/(a^2*d^4))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b
*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A^2*a)/(4*d^2) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) + (2*A^3*b^
14*d^2 + 2*A^3*a^2*b^12*d^2 - 96*A^3*a^4*b^10*d^2 - 96*A^3*a^6*b^8*d^2 - 160*B^3*a^3*b^11*d^2 - 160*B^3*a^5*b^
9*d^2 + 48*A^2*B*a*b^13*d^2 - 192*A*B^2*a^2*b^12*d^2 + 96*A*B^2*a^4*b^10*d^2 + 288*A*B^2*a^6*b^8*d^2 + 528*A^2
*B*a^3*b^11*d^2 + 480*A^2*B*a^5*b^9*d^2)/(a^2*d^5))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^
4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A^2*a)/(4*d^2) - (B^2*a)/(4*d^2) - (A*B*b)/(2*
d^2))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(A^2*B^2*b^14 - A^4*b^14 + 17*A^4*a^2*b^12 + 16*A^4*a^4*b^10 + 96*A^
4*a^6*b^8 + 48*B^4*a^2*b^12 + 48*B^4*a^4*b^10 + 32*B^4*a^6*b^8 + 95*A^2*B^2*a^2*b^12 + 448*A^2*B^2*a^4*b^10 -
8*A*B^3*a*b^13 + 4*A^3*B*a*b^13 - 120*A*B^3*a^3*b^11 + 64*A*B^3*a^5*b^9 - 8*A^3*B*a^3*b^11 - 320*A^3*B*a^5*b^9
))/(a^2*d^4))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*
a*b*d^4)^(1/2)/(4*d^4) + (A^2*a)/(4*d^2) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2)*1i - (((((64*A*a*b^12*d^4
+ 448*A*a^3*b^10*d^4 + 384*A*a^5*b^8*d^4 - 256*B*a^2*b^11*d^4 - 256*B*a^4*b^9*d^4)/(a^2*d^5) - ((512*a^2*b^10*
d^4 + 768*a^4*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*
b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A^2*a)/(4*d^2) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^
2))^(1/2))/(a^2*d^4))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 -
4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A^2*a)/(4*d^2) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) - ((a + b*tan(c +
d*x))^(1/2)*(128*B^2*a^3*b^10*d^2 - 576*A^2*a^5*b^8*d^2 - 256*A^2*a^3*b^10*d^2 + 320*B^2*a^5*b^8*d^2 - 4*A^2*a
*b^12*d^2 + 544*A*B*a^2*b^11*d^2 + 1024*A*B*a^4*b^9*d^2))/(a^2*d^4))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2
*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A^2*a)/(4*d^2) - (B^2*a)/(4*d
^2) - (A*B*b)/(2*d^2))^(1/2) + (2*A^3*b^14*d^2 + 2*A^3*a^2*b^12*d^2 - 96*A^3*a^4*b^10*d^2 - 96*A^3*a^6*b^8*d^2
- 160*B^3*a^3*b^11*d^2 - 160*B^3*a^5*b^9*d^2 + 48*A^2*B*a*b^13*d^2 - 192*A*B^2*a^2*b^12*d^2 + 96*A*B^2*a^4*b^
10*d^2 + 288*A*B^2*a^6*b^8*d^2 + 528*A^2*B*a^3*b^11*d^2 + 480*A^2*B*a^5*b^9*d^2)/(a^2*d^5))*((2*A^2*B^2*b^2*d^
4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A^2*a)
/(4*d^2) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(A^2*B^2*b^14 - A^4*b^14 + 1
7*A^4*a^2*b^12 + 16*A^4*a^4*b^10 + 96*A^4*a^6*b^8 + 48*B^4*a^2*b^12 + 48*B^4*a^4*b^10 + 32*B^4*a^6*b^8 + 95*A^
2*B^2*a^2*b^12 + 448*A^2*B^2*a^4*b^10 - 8*A*B^3*a*b^13 + 4*A^3*B*a*b^13 - 120*A*B^3*a^3*b^11 + 64*A*B^3*a^5*b^
9 - 8*A^3*B*a^3*b^11 - 320*A^3*B*a^5*b^9))/(a^2*d^4))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 -
A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A^2*a)/(4*d^2) - (B^2*a)/(4*d^2) - (A*B*b)/(
2*d^2))^(1/2)*1i)/((((((64*A*a*b^12*d^4 + 448*A*a^3*b^10*d^4 + 384*A*a^5*b^8*d^4 - 256*B*a^2*b^11*d^4 - 256*B*
a^4*b^9*d^4)/(a^2*d^5) + ((512*a^2*b^10*d^4 + 768*a^4*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((2*A^2*B^2*b^2*d^4
- B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A^2*a)/(
4*d^2) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2))/(a^2*d^4))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^
2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A^2*a)/(4*d^2) - (B^2*a)/(4*d^2) - (
A*B*b)/(2*d^2))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(128*B^2*a^3*b^10*d^2 - 576*A^2*a^5*b^8*d^2 - 256*A^2*a^3*
b^10*d^2 + 320*B^2*a^5*b^8*d^2 - 4*A^2*a*b^12*d^2 + 544*A*B*a^2*b^11*d^2 + 1024*A*B*a^4*b^9*d^2))/(a^2*d^4))*(
(2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/
(4*d^4) + (A^2*a)/(4*d^2) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) + (2*A^3*b^14*d^2 + 2*A^3*a^2*b^12*d^2 -
96*A^3*a^4*b^10*d^2 - 96*A^3*a^6*b^8*d^2 - 160*B^3*a^3*b^11*d^2 - 160*B^3*a^5*b^9*d^2 + 48*A^2*B*a*b^13*d^2 -
192*A*B^2*a^2*b^12*d^2 + 96*A*B^2*a^4*b^10*d^2 + 288*A*B^2*a^6*b^8*d^2 + 528*A^2*B*a^3*b^11*d^2 + 480*A^2*B*a^
5*b^9*d^2)/(a^2*d^5))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 -
4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A^2*a)/(4*d^2) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) + ((a + b*tan(c +
d*x))^(1/2)*(A^2*B^2*b^14 - A^4*b^14 + 17*A^4*a^2*b^12 + 16*A^4*a^4*b^10 + 96*A^4*a^6*b^8 + 48*B^4*a^2*b^12 +
48*B^4*a^4*b^10 + 32*B^4*a^6*b^8 + 95*A^2*B^2*a^2*b^12 + 448*A^2*B^2*a^4*b^10 - 8*A*B^3*a*b^13 + 4*A^3*B*a*b^1
3 - 120*A*B^3*a^3*b^11 + 64*A*B^3*a^5*b^9 - 8*A^3*B*a^3*b^11 - 320*A^3*B*a^5*b^9))/(a^2*d^4))*((2*A^2*B^2*b^2*
d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A^2*
a)/(4*d^2) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) + (((((64*A*a*b^12*d^4 + 448*A*a^3*b^10*d^4 + 384*A*a^5*
b^8*d^4 - 256*B*a^2*b^11*d^4 - 256*B*a^4*b^9*d^4)/(a^2*d^5) - ((512*a^2*b^10*d^4 + 768*a^4*b^8*d^4)*(a + b*tan
(c + d*x))^(1/2)*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3
*B*a*b*d^4)^(1/2)/(4*d^4) + (A^2*a)/(4*d^2) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2))/(a^2*d^4))*((2*A^2*B^2
*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) +
(A^2*a)/(4*d^2) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(128*B^2*a^3*b^10*d^2
- 576*A^2*a^5*b^8*d^2 - 256*A^2*a^3*b^10*d^2 + 320*B^2*a^5*b^8*d^2 - 4*A^2*a*b^12*d^2 + 544*A*B*a^2*b^11*d^2
+ 1024*A*B*a^4*b^9*d^2))/(a^2*d^4))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*
B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A^2*a)/(4*d^2) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) + (2
*A^3*b^14*d^2 + 2*A^3*a^2*b^12*d^2 - 96*A^3*a^4*b^10*d^2 - 96*A^3*a^6*b^8*d^2 - 160*B^3*a^3*b^11*d^2 - 160*B^3
*a^5*b^9*d^2 + 48*A^2*B*a*b^13*d^2 - 192*A*B^2*a^2*b^12*d^2 + 96*A*B^2*a^4*b^10*d^2 + 288*A*B^2*a^6*b^8*d^2 +
528*A^2*B*a^3*b^11*d^2 + 480*A^2*B*a^5*b^9*d^2)/(a^2*d^5))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d
^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A^2*a)/(4*d^2) - (B^2*a)/(4*d^2) - (A*B
*b)/(2*d^2))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(A^2*B^2*b^14 - A^4*b^14 + 17*A^4*a^2*b^12 + 16*A^4*a^4*b^10
+ 96*A^4*a^6*b^8 + 48*B^4*a^2*b^12 + 48*B^4*a^4*b^10 + 32*B^4*a^6*b^8 + 95*A^2*B^2*a^2*b^12 + 448*A^2*B^2*a^4*
b^10 - 8*A*B^3*a*b^13 + 4*A^3*B*a*b^13 - 120*A*B^3*a^3*b^11 + 64*A*B^3*a^5*b^9 - 8*A^3*B*a^3*b^11 - 320*A^3*B*
a^5*b^9))/(a^2*d^4))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4
*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A^2*a)/(4*d^2) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) + (A^4*B*b^15 + 7*A
^5*a*b^14 + A^2*B^3*b^15 + 63*A^5*a^3*b^12 + 56*A^5*a^5*b^10 - 16*B^5*a^2*b^13 - 48*B^5*a^4*b^11 - 32*B^5*a^6*
b^9 - 23*A^2*B^3*a^2*b^13 + 40*A^2*B^3*a^4*b^11 + 64*A^2*B^3*a^6*b^9 + 55*A^3*B^2*a^3*b^12 + 112*A^3*B^2*a^5*b
^10 + 64*A^3*B^2*a^7*b^8 - 8*A*B^4*a^3*b^12 + 56*A*B^4*a^5*b^10 + 64*A*B^4*a^7*b^8 + 7*A^3*B^2*a*b^14 - 7*A^4*
B*a^2*b^13 + 88*A^4*B*a^4*b^11 + 96*A^4*B*a^6*b^9)/(a^2*d^5)))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a
^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A^2*a)/(4*d^2) - (B^2*a)/(4*d^2) -
(A*B*b)/(2*d^2))^(1/2)*2i - (((A*b^2)/4 - B*a*b)*(a + b*tan(c + d*x))^(1/2) + ((A*b^2 + 4*B*a*b)*(a + b*tan(c
+ d*x))^(3/2))/(4*a))/(d*(a + b*tan(c + d*x))^2 + a^2*d - 2*a*d*(a + b*tan(c + d*x))) - atan(((((((64*A*a*b^12
*d^4 + 448*A*a^3*b^10*d^4 + 384*A*a^5*b^8*d^4 - 256*B*a^2*b^11*d^4 - 256*B*a^4*b^9*d^4)/(a^2*d^5) + ((512*a^2*
b^10*d^4 + 768*a^4*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4
*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (B^2*a)/(4*d^2) - (A*B*b)/
(2*d^2))^(1/2))/(a^2*d^4))*((A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d
^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) + ((a + b*tan
(c + d*x))^(1/2)*(128*B^2*a^3*b^10*d^2 - 576*A^2*a^5*b^8*d^2 - 256*A^2*a^3*b^10*d^2 + 320*B^2*a^5*b^8*d^2 - 4*
A^2*a*b^12*d^2 + 544*A*B*a^2*b^11*d^2 + 1024*A*B*a^4*b^9*d^2))/(a^2*d^4))*((A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^
4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (B^2*a)
/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) + (2*A^3*b^14*d^2 + 2*A^3*a^2*b^12*d^2 - 96*A^3*a^4*b^10*d^2 - 96*A^3*a^6*b^
8*d^2 - 160*B^3*a^3*b^11*d^2 - 160*B^3*a^5*b^9*d^2 + 48*A^2*B*a*b^13*d^2 - 192*A*B^2*a^2*b^12*d^2 + 96*A*B^2*a
^4*b^10*d^2 + 288*A*B^2*a^6*b^8*d^2 + 528*A^2*B*a^3*b^11*d^2 + 480*A^2*B*a^5*b^9*d^2)/(a^2*d^5))*((A^2*a)/(4*d
^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^
(1/2)/(4*d^4) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(A^2*B^2*b^14 - A^4*b^1
4 + 17*A^4*a^2*b^12 + 16*A^4*a^4*b^10 + 96*A^4*a^6*b^8 + 48*B^4*a^2*b^12 + 48*B^4*a^4*b^10 + 32*B^4*a^6*b^8 +
95*A^2*B^2*a^2*b^12 + 448*A^2*B^2*a^4*b^10 - 8*A*B^3*a*b^13 + 4*A^3*B*a*b^13 - 120*A*B^3*a^3*b^11 + 64*A*B^3*a
^5*b^9 - 8*A^3*B*a^3*b^11 - 320*A^3*B*a^5*b^9))/(a^2*d^4))*((A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4
- 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (B^2*a)/(4*d^2) - (A*B
*b)/(2*d^2))^(1/2)*1i - (((((64*A*a*b^12*d^4 + 448*A*a^3*b^10*d^4 + 384*A*a^5*b^8*d^4 - 256*B*a^2*b^11*d^4 - 2
56*B*a^4*b^9*d^4)/(a^2*d^5) - ((512*a^2*b^10*d^4 + 768*a^4*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((A^2*a)/(4*d^2
) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1
/2)/(4*d^4) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2))/(a^2*d^4))*((A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4
*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (B^2*a)/(4*d^2
) - (A*B*b)/(2*d^2))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(128*B^2*a^3*b^10*d^2 - 576*A^2*a^5*b^8*d^2 - 256*A^2
*a^3*b^10*d^2 + 320*B^2*a^5*b^8*d^2 - 4*A^2*a*b^12*d^2 + 544*A*B*a^2*b^11*d^2 + 1024*A*B*a^4*b^9*d^2))/(a^2*d^
4))*((A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 -
4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) + (2*A^3*b^14*d^2 + 2*A^3*a^2*b^12*d
^2 - 96*A^3*a^4*b^10*d^2 - 96*A^3*a^6*b^8*d^2 - 160*B^3*a^3*b^11*d^2 - 160*B^3*a^5*b^9*d^2 + 48*A^2*B*a*b^13*d
^2 - 192*A*B^2*a^2*b^12*d^2 + 96*A*B^2*a^4*b^10*d^2 + 288*A*B^2*a^6*b^8*d^2 + 528*A^2*B*a^3*b^11*d^2 + 480*A^2
*B*a^5*b^9*d^2)/(a^2*d^5))*((A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d
^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) - ((a + b*tan
(c + d*x))^(1/2)*(A^2*B^2*b^14 - A^4*b^14 + 17*A^4*a^2*b^12 + 16*A^4*a^4*b^10 + 96*A^4*a^6*b^8 + 48*B^4*a^2*b^
12 + 48*B^4*a^4*b^10 + 32*B^4*a^6*b^8 + 95*A^2*B^2*a^2*b^12 + 448*A^2*B^2*a^4*b^10 - 8*A*B^3*a*b^13 + 4*A^3*B*
a*b^13 - 120*A*B^3*a^3*b^11 + 64*A*B^3*a^5*b^9 - 8*A^3*B*a^3*b^11 - 320*A^3*B*a^5*b^9))/(a^2*d^4))*((A^2*a)/(4
*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4
)^(1/2)/(4*d^4) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2)*1i)/((((((64*A*a*b^12*d^4 + 448*A*a^3*b^10*d^4 + 38
4*A*a^5*b^8*d^4 - 256*B*a^2*b^11*d^4 - 256*B*a^4*b^9*d^4)/(a^2*d^5) + ((512*a^2*b^10*d^4 + 768*a^4*b^8*d^4)*(a
+ b*tan(c + d*x))^(1/2)*((A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4
+ 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2))/(a^2*d^4))*((A
^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B
*a*b*d^4)^(1/2)/(4*d^4) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(128*B^2*a^3*
b^10*d^2 - 576*A^2*a^5*b^8*d^2 - 256*A^2*a^3*b^10*d^2 + 320*B^2*a^5*b^8*d^2 - 4*A^2*a*b^12*d^2 + 544*A*B*a^2*b
^11*d^2 + 1024*A*B*a^4*b^9*d^2))/(a^2*d^4))*((A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^
2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1
/2) + (2*A^3*b^14*d^2 + 2*A^3*a^2*b^12*d^2 - 96*A^3*a^4*b^10*d^2 - 96*A^3*a^6*b^8*d^2 - 160*B^3*a^3*b^11*d^2 -
160*B^3*a^5*b^9*d^2 + 48*A^2*B*a*b^13*d^2 - 192*A*B^2*a^2*b^12*d^2 + 96*A*B^2*a^4*b^10*d^2 + 288*A*B^2*a^6*b^
8*d^2 + 528*A^2*B*a^3*b^11*d^2 + 480*A^2*B*a^5*b^9*d^2)/(a^2*d^5))*((A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4
*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (B^2*a)/(4*d^2
) - (A*B*b)/(2*d^2))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(A^2*B^2*b^14 - A^4*b^14 + 17*A^4*a^2*b^12 + 16*A^4*a
^4*b^10 + 96*A^4*a^6*b^8 + 48*B^4*a^2*b^12 + 48*B^4*a^4*b^10 + 32*B^4*a^6*b^8 + 95*A^2*B^2*a^2*b^12 + 448*A^2*
B^2*a^4*b^10 - 8*A*B^3*a*b^13 + 4*A^3*B*a*b^13 - 120*A*B^3*a^3*b^11 + 64*A*B^3*a^5*b^9 - 8*A^3*B*a^3*b^11 - 32
0*A^3*B*a^5*b^9))/(a^2*d^4))*((A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2
*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) + (((((64*A
*a*b^12*d^4 + 448*A*a^3*b^10*d^4 + 384*A*a^5*b^8*d^4 - 256*B*a^2*b^11*d^4 - 256*B*a^4*b^9*d^4)/(a^2*d^5) - ((5
12*a^2*b^10*d^4 + 768*a^4*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*
d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (B^2*a)/(4*d^2) - (
A*B*b)/(2*d^2))^(1/2))/(a^2*d^4))*((A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^
4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) - ((a
+ b*tan(c + d*x))^(1/2)*(128*B^2*a^3*b^10*d^2 - 576*A^2*a^5*b^8*d^2 - 256*A^2*a^3*b^10*d^2 + 320*B^2*a^5*b^8*d
^2 - 4*A^2*a*b^12*d^2 + 544*A*B*a^2*b^11*d^2 + 1024*A*B*a^4*b^9*d^2))/(a^2*d^4))*((A^2*a)/(4*d^2) - (2*A^2*B^2
*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) -
(B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) + (2*A^3*b^14*d^2 + 2*A^3*a^2*b^12*d^2 - 96*A^3*a^4*b^10*d^2 - 96*A^3
*a^6*b^8*d^2 - 160*B^3*a^3*b^11*d^2 - 160*B^3*a^5*b^9*d^2 + 48*A^2*B*a*b^13*d^2 - 192*A*B^2*a^2*b^12*d^2 + 96*
A*B^2*a^4*b^10*d^2 + 288*A*B^2*a^6*b^8*d^2 + 528*A^2*B*a^3*b^11*d^2 + 480*A^2*B*a^5*b^9*d^2)/(a^2*d^5))*((A^2*
a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*
b*d^4)^(1/2)/(4*d^4) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(A^2*B^2*b^14 -
A^4*b^14 + 17*A^4*a^2*b^12 + 16*A^4*a^4*b^10 + 96*A^4*a^6*b^8 + 48*B^4*a^2*b^12 + 48*B^4*a^4*b^10 + 32*B^4*a^6
*b^8 + 95*A^2*B^2*a^2*b^12 + 448*A^2*B^2*a^4*b^10 - 8*A*B^3*a*b^13 + 4*A^3*B*a*b^13 - 120*A*B^3*a^3*b^11 + 64*
A*B^3*a^5*b^9 - 8*A^3*B*a^3*b^11 - 320*A^3*B*a^5*b^9))/(a^2*d^4))*((A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*
b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (B^2*a)/(4*d^2)
- (A*B*b)/(2*d^2))^(1/2) + (A^4*B*b^15 + 7*A^5*a*b^14 + A^2*B^3*b^15 + 63*A^5*a^3*b^12 + 56*A^5*a^5*b^10 - 16
*B^5*a^2*b^13 - 48*B^5*a^4*b^11 - 32*B^5*a^6*b^9 - 23*A^2*B^3*a^2*b^13 + 40*A^2*B^3*a^4*b^11 + 64*A^2*B^3*a^6*
b^9 + 55*A^3*B^2*a^3*b^12 + 112*A^3*B^2*a^5*b^10 + 64*A^3*B^2*a^7*b^8 - 8*A*B^4*a^3*b^12 + 56*A*B^4*a^5*b^10 +
64*A*B^4*a^7*b^8 + 7*A^3*B^2*a*b^14 - 7*A^4*B*a^2*b^13 + 88*A^4*B*a^4*b^11 + 96*A^4*B*a^6*b^9)/(a^2*d^5)))*((
A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*
B*a*b*d^4)^(1/2)/(4*d^4) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2)*2i