\(\int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{\sqrt {a+b \tan (c+d x)}} \, dx\) [348]

Optimal result

Integrand size = 33, antiderivative size = 169 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {(A b-2 a B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {(i A+B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{\sqrt {a-i b} d}-\frac {(i A-B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{\sqrt {a+i b} d}-\frac {A \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {3690, 3734, 3620, 3618, 65, 214, 3715} \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {(A b-2 a B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {(B+i A) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}-\frac {(-B+i A) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}-\frac {A \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a d} \]

[In]

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

[Out]

((A*b - 2*a*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) + ((I*A + B)*ArcTanh[Sqrt[a + b*Tan[c +
d*x]]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d) - ((I*A - B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/(Sqrt[a
+ I*b]*d) - (A*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(a*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 3690

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*(A*b - a*B)*(a + b*Tan[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[b*B*(b*c*(m + 1) + a*d*(n + 1)) + A*(a*(b*c - a*d)*(m + 1) - b^2*d*(
m + n + 2)) - (A*b - a*B)*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(A*b - a*B)*(m + n + 2)*Tan[e + f*x]^2, x], x
], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
&& LtQ[m, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) &&  !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ
[a, 0])))

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

Mathematica [A] (verified)

Time = 3.14 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.19 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {\frac {b (A b-2 a B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2}}+\frac {\left (A \sqrt {-b^2}+b B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {-b^2}}}\right )}{\sqrt {a-\sqrt {-b^2}}}-\frac {\left (A \sqrt {-b^2}-b B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+\sqrt {-b^2}}}\right )}{\sqrt {a+\sqrt {-b^2}}}-\frac {A b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a}}{b d} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(4056\) vs. \(2(143)=286\).

Time = 0.22 (sec) , antiderivative size = 4057, normalized size of antiderivative = 24.01

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(4057\)
default	\(\text {Expression too large to display}\)	\(4057\)

[In]

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

[Out]

-1/4/d*b^2/(a^2+b^2)^(3/2)*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1
/2))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+1/4/d/(a^2+b^2)^(3/2)*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^2+2/d/(a^2+b^2)/(2*(a^2+b^2)^(1/2)-2*a)
^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B*a^2-1/
4/d/(a^2+b^2)*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-2/d/(a^2+b^2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2
+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B*a^2-1/4/d/(a^2+b^2)^(3/2)*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^2-1/d/(a^2+b^2)
^(3/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b
^2)^(1/2)-2*a)^(1/2))*B*a^3+1/d/(a^2+b^2)^(1/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^
(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B*a-1/d*b/(a^2+b^2)^(1/2)/(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-2/d*
b^3/(a^2+b^2)^(3/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/
2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*A+1/4/d*b^2/(a^2+b^2)^(3/2)*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^
2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+1/d*b/(a^2+b^2)^(1/2)/(2*(a^2+b^2)^(1
/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*
A+1/d/b^2/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^
2+b^2)^(1/2)-2*a)^(1/2))*B*a^2-1/4/d/b^2*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+1/4/d/b^2*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/b^2/(2*(a^2+b^2)^(1/2)-2*a)^(1/
2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B*a^2-1/d*b^
2/(a^2+b^2)/(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*b/(a^2+b^2)*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/(a^2+b^2)*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b
^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+1/d*b^2/(a^2+b^2)/(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+2/d*b^3/(a^2+b^2)^(3/2)/(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*t
an(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*A-2*B*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/d/a^(1/2)+1/d*b
/a^(3/2)*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))*A-1/d*A/a*(a+b*tan(d*x+c))^(1/2)/tan(d*x+c)+1/d/b^2/(a^2+b^2)
^(1/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b
^2)^(1/2)-2*a)^(1/2))*B*a^3-1/4/d/b/(a^2+b^2)^(3/2)*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*
tan(d*x+c)-a-(a^2+b^2)^(1/2))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^3-1/4/d*b/(a^2+b^2)^(3/2)*ln((a+b*tan(d*x+c))^
(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a-1/4/d/b/
(a^2+b^2)*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*A*(2*(a^2+b^
2)^(1/2)+2*a)^(1/2)*a^2-1/d/b^2/(a^2+b^2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a
^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B*a^4-1/4/d/b^2/(a^2+b^2)*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^3+1/4/d/b/(a^2+
b^2)*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*A*(2*(a^2+b^2)^(1
/2)+2*a)^(1/2)*a^2+1/4/d/b^2/(a^2+b^2)*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^3+1/d/b^2*(a^2+b^2)^(1/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arc
tan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B*a-1/d/b^2/(a^2+b
^2)^(1/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^
2+b^2)^(1/2)-2*a)^(1/2))*B*a^3+1/4/d/b/(a^2+b^2)^(3/2)*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(
1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^3+1/4/d*b/(a^2+b^2)^(3/2)*ln(b*tan(d*x+c)+a
+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a+1/d/b
/(a^2+b^2)^(1/2)/(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^2-3/d*b/(a^2+b^2)^(3/2)/(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^2+1/d*b^2/(a^2+b^2)^(3/2)/(2*(a
^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*
a)^(1/2))*B*a+3/d*b/(a^2+b^2)^(3/2)/(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^2-1/d*b^2/(a^2+b^2)^(3/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)
*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B*a-1/d/b/(a^2
+b^2)^(1/2)/(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+1/d/b/(a^2+b^2)^(3/2)/(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^4-1/d/b^2*(a^2+b^2)^(1/2)/(2*(a^2+b^
2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1
/2))*B*a+1/d/b^2/(a^2+b^2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+
c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B*a^4-1/d/b/(a^2+b^2)^(3/2)/(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^4-1/d/(a^2+b^2)^(1/2)
/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1
/2)-2*a)^(1/2))*B*a+1/d/(a^2+b^2)^(3/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2
+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B*a^3+1/4/d/(a^2+b^2)*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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1783 vs. \(2 (139) = 278\).

Time = 1.99 (sec) , antiderivative size = 3582, normalized size of antiderivative = 21.20 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{\sqrt {a+b \tan (c+d x)}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/2*(a^2*d*sqrt(-((a^2 + b^2)*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)
/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + 2*A*B*b + (A^2 - B^2)*a)/((a^2 + b^2)*d^2))*log((2*(A^3*B + A*B^3)*a - (A^4
- B^4)*b)*sqrt(b*tan(d*x + c) + a) + ((A*a^3 + B*a^2*b + A*a*b^2 + B*b^3)*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)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + (2*A*B^2*a^2 - (3*A^2*B - B^3)*a*
b + (A^3 - A*B^2)*b^2)*d)*sqrt(-((a^2 + b^2)*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)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + 2*A*B*b + (A^2 - B^2)*a)/((a^2 + b^2)*d^2)))*tan(d*x + c) - a^
2*d*sqrt(-((a^2 + b^2)*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)/((a^4 +
2*a^2*b^2 + b^4)*d^4)) + 2*A*B*b + (A^2 - B^2)*a)/((a^2 + b^2)*d^2))*log((2*(A^3*B + A*B^3)*a - (A^4 - B^4)*b
)*sqrt(b*tan(d*x + c) + a) - ((A*a^3 + B*a^2*b + A*a*b^2 + B*b^3)*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)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + (2*A*B^2*a^2 - (3*A^2*B - B^3)*a*b + (A^3
- A*B^2)*b^2)*d)*sqrt(-((a^2 + b^2)*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)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + 2*A*B*b + (A^2 - B^2)*a)/((a^2 + b^2)*d^2)))*tan(d*x + c) - a^2*d*sqrt
(((a^2 + b^2)*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)/((a^4 + 2*a^2*b^
2 + b^4)*d^4)) - 2*A*B*b - (A^2 - B^2)*a)/((a^2 + b^2)*d^2))*log((2*(A^3*B + A*B^3)*a - (A^4 - B^4)*b)*sqrt(b*
tan(d*x + c) + a) + ((A*a^3 + B*a^2*b + A*a*b^2 + B*b^3)*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)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - (2*A*B^2*a^2 - (3*A^2*B - B^3)*a*b + (A^3 - A*B^2)
*b^2)*d)*sqrt(((a^2 + b^2)*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)/((a
^4 + 2*a^2*b^2 + b^4)*d^4)) - 2*A*B*b - (A^2 - B^2)*a)/((a^2 + b^2)*d^2)))*tan(d*x + c) + a^2*d*sqrt(((a^2 + b
^2)*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)/((a^4 + 2*a^2*b^2 + b^4)*d
^4)) - 2*A*B*b - (A^2 - B^2)*a)/((a^2 + b^2)*d^2))*log((2*(A^3*B + A*B^3)*a - (A^4 - B^4)*b)*sqrt(b*tan(d*x +
c) + a) - ((A*a^3 + B*a^2*b + A*a*b^2 + B*b^3)*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)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - (2*A*B^2*a^2 - (3*A^2*B - B^3)*a*b + (A^3 - A*B^2)*b^2)*d)*s
qrt(((a^2 + b^2)*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)/((a^4 + 2*a^2
*b^2 + b^4)*d^4)) - 2*A*B*b - (A^2 - B^2)*a)/((a^2 + b^2)*d^2)))*tan(d*x + c) - (2*B*a - A*b)*sqrt(a)*log((b*t
an(d*x + c) + 2*sqrt(b*tan(d*x + c) + a)*sqrt(a) + 2*a)/tan(d*x + c))*tan(d*x + c) - 2*sqrt(b*tan(d*x + c) + a
)*A*a)/(a^2*d*tan(d*x + c)), 1/2*(a^2*d*sqrt(-((a^2 + b^2)*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)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + 2*A*B*b + (A^2 - B^2)*a)/((a^2 + b^2)*d^2))*log(
(2*(A^3*B + A*B^3)*a - (A^4 - B^4)*b)*sqrt(b*tan(d*x + c) + a) + ((A*a^3 + B*a^2*b + A*a*b^2 + B*b^3)*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)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + (2*A*
B^2*a^2 - (3*A^2*B - B^3)*a*b + (A^3 - A*B^2)*b^2)*d)*sqrt(-((a^2 + b^2)*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)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + 2*A*B*b + (A^2 - B^2)*a)/((a^2 + b
^2)*d^2)))*tan(d*x + c) - a^2*d*sqrt(-((a^2 + b^2)*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)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + 2*A*B*b + (A^2 - B^2)*a)/((a^2 + b^2)*d^2))*log((2*(A^3*
B + A*B^3)*a - (A^4 - B^4)*b)*sqrt(b*tan(d*x + c) + a) - ((A*a^3 + B*a^2*b + A*a*b^2 + B*b^3)*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)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + (2*A*B^2*a^2
- (3*A^2*B - B^3)*a*b + (A^3 - A*B^2)*b^2)*d)*sqrt(-((a^2 + b^2)*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)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + 2*A*B*b + (A^2 - B^2)*a)/((a^2 + b^2)*d^2)
))*tan(d*x + c) - a^2*d*sqrt(((a^2 + b^2)*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)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - 2*A*B*b - (A^2 - B^2)*a)/((a^2 + b^2)*d^2))*log((2*(A^3*B + A*B^3
)*a - (A^4 - B^4)*b)*sqrt(b*tan(d*x + c) + a) + ((A*a^3 + B*a^2*b + A*a*b^2 + B*b^3)*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)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - (2*A*B^2*a^2 - (3*A^2*
B - B^3)*a*b + (A^3 - A*B^2)*b^2)*d)*sqrt(((a^2 + b^2)*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)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - 2*A*B*b - (A^2 - B^2)*a)/((a^2 + b^2)*d^2)))*tan(d*x
+ c) + a^2*d*sqrt(((a^2 + b^2)*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
)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - 2*A*B*b - (A^2 - B^2)*a)/((a^2 + b^2)*d^2))*log((2*(A^3*B + A*B^3)*a - (A^4
- B^4)*b)*sqrt(b*tan(d*x + c) + a) - ((A*a^3 + B*a^2*b + A*a*b^2 + B*b^3)*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)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - (2*A*B^2*a^2 - (3*A^2*B - B^3)*a
*b + (A^3 - A*B^2)*b^2)*d)*sqrt(((a^2 + b^2)*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)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - 2*A*B*b - (A^2 - B^2)*a)/((a^2 + b^2)*d^2)))*tan(d*x + c) + 2*
(2*B*a - A*b)*sqrt(-a)*arctan(sqrt(b*tan(d*x + c) + a)*sqrt(-a)/a)*tan(d*x + c) - 2*sqrt(b*tan(d*x + c) + a)*A
*a)/(a^2*d*tan(d*x + c))]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 9.96 (sec) , antiderivative size = 9790, normalized size of antiderivative = 57.93 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{\sqrt {a+b \tan (c+d x)}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

atan(((((((8*(32*A*a*b^11*d^4 + 16*A*a^3*b^9*d^4 - 64*B*a^2*b^10*d^4 - 48*B*a^4*b^8*d^4))/(a^2*d^5) - (16*(32*
a^2*b^10*d^4 + 48*a^4*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 -
(16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*d
^4 + b^2*d^4)))^(1/2))/(a^2*d^4))*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4
)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) -
(16*(a + b*tan(c + d*x))^(1/2)*(20*A^2*a^3*b^8*d^2 - 36*B^2*a^3*b^8*d^2 - 4*A^2*a*b^10*d^2 + 48*A*B*a^2*b^9*d^
2))/(a^2*d^4))*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2
+ B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) + (8*(4*A^3*b^11*d^2
+ 16*A^3*a^2*b^9*d^2 + 12*B^3*a^3*b^8*d^2 + 12*A^2*B*a*b^10*d^2 - 48*A*B^2*a^2*b^9*d^2 - 36*A^2*B*a^3*b^8*d^2)
)/(a^2*d^5))*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 +
B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) - (16*(a + b*tan(c + d*
x))^(1/2)*(A^2*B^2*b^10 - A^4*b^10 + 2*A^4*a^2*b^8 + 6*B^4*a^2*b^8 - 4*A*B^3*a*b^9 + 4*A^3*B*a*b^9))/(a^2*d^4)
)*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2)
- 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2)*1i - (((((8*(32*A*a*b^11*d^4 + 16*
A*a^3*b^9*d^4 - 64*B*a^2*b^10*d^4 - 48*B*a^4*b^8*d^4))/(a^2*d^5) + (16*(32*a^2*b^10*d^4 + 48*a^4*b^8*d^4)*(a +
b*tan(c + d*x))^(1/2)*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*
A^2*B^2 + B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2))/(a^2*d^4))*(
(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) -
4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(20*
A^2*a^3*b^8*d^2 - 36*B^2*a^3*b^8*d^2 - 4*A^2*a*b^10*d^2 + 48*A*B*a^2*b^9*d^2))/(a^2*d^4))*((((8*A^2*a*d^2 - 8*
B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2
*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) + (8*(4*A^3*b^11*d^2 + 16*A^3*a^2*b^9*d^2 + 12*B^3*a^3*b
^8*d^2 + 12*A^2*B*a*b^10*d^2 - 48*A*B^2*a^2*b^9*d^2 - 36*A^2*B*a^3*b^8*d^2))/(a^2*d^5))*((((8*A^2*a*d^2 - 8*B^
2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a
*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(A^2*B^2*b^10 - A^4*b^10
+ 2*A^4*a^2*b^8 + 6*B^4*a^2*b^8 - 4*A*B^3*a*b^9 + 4*A^3*B*a*b^9))/(a^2*d^4))*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 1
6*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*
B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2)*1i)/((((((8*(32*A*a*b^11*d^4 + 16*A*a^3*b^9*d^4 - 64*B*a^2*b^10*d^4 -
48*B*a^4*b^8*d^4))/(a^2*d^5) - (16*(32*a^2*b^10*d^4 + 48*a^4*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((((8*A^2*a*
d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) - 4*A^2*a*d^2
+ 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2))/(a^2*d^4))*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A
*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b
*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(20*A^2*a^3*b^8*d^2 - 36*B^2*a^3*b^8*d^
2 - 4*A^2*a*b^10*d^2 + 48*A*B*a^2*b^9*d^2))/(a^2*d^4))*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16
*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^4
+ b^2*d^4)))^(1/2) + (8*(4*A^3*b^11*d^2 + 16*A^3*a^2*b^9*d^2 + 12*B^3*a^3*b^8*d^2 + 12*A^2*B*a*b^10*d^2 - 48*A
*B^2*a^2*b^9*d^2 - 36*A^2*B*a^3*b^8*d^2))/(a^2*d^5))*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a
^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^4 +
b^2*d^4)))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(A^2*B^2*b^10 - A^4*b^10 + 2*A^4*a^2*b^8 + 6*B^4*a^2*b^8 - 4
*A*B^3*a*b^9 + 4*A^3*B*a*b^9))/(a^2*d^4))*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16
*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^
(1/2) + (((((8*(32*A*a*b^11*d^4 + 16*A*a^3*b^9*d^4 - 64*B*a^2*b^10*d^4 - 48*B*a^4*b^8*d^4))/(a^2*d^5) + (16*(3
2*a^2*b^10*d^4 + 48*a^4*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4
- (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2
*d^4 + b^2*d^4)))^(1/2))/(a^2*d^4))*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d
^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2)
+ (16*(a + b*tan(c + d*x))^(1/2)*(20*A^2*a^3*b^8*d^2 - 36*B^2*a^3*b^8*d^2 - 4*A^2*a*b^10*d^2 + 48*A*B*a^2*b^9*
d^2))/(a^2*d^4))*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^
2 + B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) + (8*(4*A^3*b^11*d^
2 + 16*A^3*a^2*b^9*d^2 + 12*B^3*a^3*b^8*d^2 + 12*A^2*B*a*b^10*d^2 - 48*A*B^2*a^2*b^9*d^2 - 36*A^2*B*a^3*b^8*d^
2))/(a^2*d^5))*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2
+ B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) + (16*(a + b*tan(c +
d*x))^(1/2)*(A^2*B^2*b^10 - A^4*b^10 + 2*A^4*a^2*b^8 + 6*B^4*a^2*b^8 - 4*A*B^3*a*b^9 + 4*A^3*B*a*b^9))/(a^2*d^
4))*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/
2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) + (16*(A^4*B*b^10 + 2*A^5*a*b^9
+ A^2*B^3*b^10 - 4*A^2*B^3*a^2*b^8 - 2*A*B^4*a*b^9 - 4*A^4*B*a^2*b^8))/(a^2*d^5)))*((((8*A^2*a*d^2 - 8*B^2*a*d
^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2
- 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2)*2i + atan(((((((8*(32*A*a*b^11*d^4 + 16*A*a^3*b^9*d^4 - 64*B*a^
2*b^10*d^4 - 48*B*a^4*b^8*d^4))/(a^2*d^5) - (16*(32*a^2*b^10*d^4 + 48*a^4*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*
(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2)
+ 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2))/(a^2*d^4))*(-(((8*A^2*a*d^2 - 8*B^
2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a
*d^2 + 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(20*A^2*a^3*b^8*d^2 - 36*
B^2*a^3*b^8*d^2 - 4*A^2*a*b^10*d^2 + 48*A*B*a^2*b^9*d^2))/(a^2*d^4))*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b
*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2
)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) + (8*(4*A^3*b^11*d^2 + 16*A^3*a^2*b^9*d^2 + 12*B^3*a^3*b^8*d^2 + 12*A^2*B*a*
b^10*d^2 - 48*A*B^2*a^2*b^9*d^2 - 36*A^2*B*a^3*b^8*d^2))/(a^2*d^5))*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*
d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2)
/(16*(a^2*d^4 + b^2*d^4)))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(A^2*B^2*b^10 - A^4*b^10 + 2*A^4*a^2*b^8 + 6
*B^4*a^2*b^8 - 4*A*B^3*a*b^9 + 4*A^3*B*a*b^9))/(a^2*d^4))*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 -
(16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2)/(16*(a^2*
d^4 + b^2*d^4)))^(1/2)*1i - (((((8*(32*A*a*b^11*d^4 + 16*A*a^3*b^9*d^4 - 64*B*a^2*b^10*d^4 - 48*B*a^4*b^8*d^4)
)/(a^2*d^5) + (16*(32*a^2*b^10*d^4 + 48*a^4*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2
+ 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 +
8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2))/(a^2*d^4))*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 -
(16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2)/(16*(a^2*d
^4 + b^2*d^4)))^(1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(20*A^2*a^3*b^8*d^2 - 36*B^2*a^3*b^8*d^2 - 4*A^2*a*b^10
*d^2 + 48*A*B*a^2*b^9*d^2))/(a^2*d^4))*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b
^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1
/2) + (8*(4*A^3*b^11*d^2 + 16*A^3*a^2*b^9*d^2 + 12*B^3*a^3*b^8*d^2 + 12*A^2*B*a*b^10*d^2 - 48*A*B^2*a^2*b^9*d^
2 - 36*A^2*B*a^3*b^8*d^2))/(a^2*d^5))*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^
2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/
2) + (16*(a + b*tan(c + d*x))^(1/2)*(A^2*B^2*b^10 - A^4*b^10 + 2*A^4*a^2*b^8 + 6*B^4*a^2*b^8 - 4*A*B^3*a*b^9 +
4*A^3*B*a*b^9))/(a^2*d^4))*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^
4 + 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2)*1i)/(((
(((8*(32*A*a*b^11*d^4 + 16*A*a^3*b^9*d^4 - 64*B*a^2*b^10*d^4 - 48*B*a^4*b^8*d^4))/(a^2*d^5) - (16*(32*a^2*b^10
*d^4 + 48*a^4*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2
*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2)/(16*(a^2*d^4 + b^
2*d^4)))^(1/2))/(a^2*d^4))*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4
+ 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) - (16*(a
+ b*tan(c + d*x))^(1/2)*(20*A^2*a^3*b^8*d^2 - 36*B^2*a^3*b^8*d^2 - 4*A^2*a*b^10*d^2 + 48*A*B*a^2*b^9*d^2))/(a
^2*d^4))*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4
))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) + (8*(4*A^3*b^11*d^2 + 16*
A^3*a^2*b^9*d^2 + 12*B^3*a^3*b^8*d^2 + 12*A^2*B*a*b^10*d^2 - 48*A*B^2*a^2*b^9*d^2 - 36*A^2*B*a^3*b^8*d^2))/(a^
2*d^5))*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4)
)^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) - (16*(a + b*tan(c + d*x))^
(1/2)*(A^2*B^2*b^10 - A^4*b^10 + 2*A^4*a^2*b^8 + 6*B^4*a^2*b^8 - 4*A*B^3*a*b^9 + 4*A^3*B*a*b^9))/(a^2*d^4))*(-
(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) +
4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) + (((((8*(32*A*a*b^11*d^4 + 16*A*a^3*
b^9*d^4 - 64*B*a^2*b^10*d^4 - 48*B*a^4*b^8*d^4))/(a^2*d^5) + (16*(32*a^2*b^10*d^4 + 48*a^4*b^8*d^4)*(a + b*tan
(c + d*x))^(1/2)*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B
^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2))/(a^2*d^4))*(-(((8
*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) + 4*A^
2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(20*A^2*
a^3*b^8*d^2 - 36*B^2*a^3*b^8*d^2 - 4*A^2*a*b^10*d^2 + 48*A*B*a^2*b^9*d^2))/(a^2*d^4))*(-(((8*A^2*a*d^2 - 8*B^2
*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*
d^2 + 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) + (8*(4*A^3*b^11*d^2 + 16*A^3*a^2*b^9*d^2 + 12*B^3*a^3*b^8*
d^2 + 12*A^2*B*a*b^10*d^2 - 48*A*B^2*a^2*b^9*d^2 - 36*A^2*B*a^3*b^8*d^2))/(a^2*d^5))*(-(((8*A^2*a*d^2 - 8*B^2*
a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d
^2 + 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(A^2*B^2*b^10 - A^4*b^10 +
2*A^4*a^2*b^8 + 6*B^4*a^2*b^8 - 4*A*B^3*a*b^9 + 4*A^3*B*a*b^9))/(a^2*d^4))*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16
*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B
*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) + (16*(A^4*B*b^10 + 2*A^5*a*b^9 + A^2*B^3*b^10 - 4*A^2*B^3*a^2*b^8 - 2
*A*B^4*a*b^9 - 4*A^4*B*a^2*b^8))/(a^2*d^5)))*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4
+ 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4
)))^(1/2)*2i - (atan((((A*b - 2*B*a)*((16*(a + b*tan(c + d*x))^(1/2)*(A^2*B^2*b^10 - A^4*b^10 + 2*A^4*a^2*b^8
+ 6*B^4*a^2*b^8 - 4*A*B^3*a*b^9 + 4*A^3*B*a*b^9))/(a^2*d^4) - (((8*(4*A^3*b^11*d^2 + 16*A^3*a^2*b^9*d^2 + 12*B
^3*a^3*b^8*d^2 + 12*A^2*B*a*b^10*d^2 - 48*A*B^2*a^2*b^9*d^2 - 36*A^2*B*a^3*b^8*d^2))/(a^2*d^5) - ((A*b - 2*B*a
)*((16*(a + b*tan(c + d*x))^(1/2)*(20*A^2*a^3*b^8*d^2 - 36*B^2*a^3*b^8*d^2 - 4*A^2*a*b^10*d^2 + 48*A*B*a^2*b^9
*d^2))/(a^2*d^4) - (((8*(32*A*a*b^11*d^4 + 16*A*a^3*b^9*d^4 - 64*B*a^2*b^10*d^4 - 48*B*a^4*b^8*d^4))/(a^2*d^5)
- (8*(A*b - 2*B*a)*(32*a^2*b^10*d^4 + 48*a^4*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/(a^2*d^5*(a^3)^(1/2)))*(A*b
- 2*B*a))/(2*d*(a^3)^(1/2))))/(2*d*(a^3)^(1/2)))*(A*b - 2*B*a))/(2*d*(a^3)^(1/2)))*1i)/(2*d*(a^3)^(1/2)) + ((
A*b - 2*B*a)*((16*(a + b*tan(c + d*x))^(1/2)*(A^2*B^2*b^10 - A^4*b^10 + 2*A^4*a^2*b^8 + 6*B^4*a^2*b^8 - 4*A*B^
3*a*b^9 + 4*A^3*B*a*b^9))/(a^2*d^4) + (((8*(4*A^3*b^11*d^2 + 16*A^3*a^2*b^9*d^2 + 12*B^3*a^3*b^8*d^2 + 12*A^2*
B*a*b^10*d^2 - 48*A*B^2*a^2*b^9*d^2 - 36*A^2*B*a^3*b^8*d^2))/(a^2*d^5) + ((A*b - 2*B*a)*((16*(a + b*tan(c + d*
x))^(1/2)*(20*A^2*a^3*b^8*d^2 - 36*B^2*a^3*b^8*d^2 - 4*A^2*a*b^10*d^2 + 48*A*B*a^2*b^9*d^2))/(a^2*d^4) + (((8*
(32*A*a*b^11*d^4 + 16*A*a^3*b^9*d^4 - 64*B*a^2*b^10*d^4 - 48*B*a^4*b^8*d^4))/(a^2*d^5) + (8*(A*b - 2*B*a)*(32*
a^2*b^10*d^4 + 48*a^4*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/(a^2*d^5*(a^3)^(1/2)))*(A*b - 2*B*a))/(2*d*(a^3)^(1
/2))))/(2*d*(a^3)^(1/2)))*(A*b - 2*B*a))/(2*d*(a^3)^(1/2)))*1i)/(2*d*(a^3)^(1/2)))/((16*(A^4*B*b^10 + 2*A^5*a*
b^9 + A^2*B^3*b^10 - 4*A^2*B^3*a^2*b^8 - 2*A*B^4*a*b^9 - 4*A^4*B*a^2*b^8))/(a^2*d^5) - ((A*b - 2*B*a)*((16*(a
+ b*tan(c + d*x))^(1/2)*(A^2*B^2*b^10 - A^4*b^10 + 2*A^4*a^2*b^8 + 6*B^4*a^2*b^8 - 4*A*B^3*a*b^9 + 4*A^3*B*a*b
^9))/(a^2*d^4) - (((8*(4*A^3*b^11*d^2 + 16*A^3*a^2*b^9*d^2 + 12*B^3*a^3*b^8*d^2 + 12*A^2*B*a*b^10*d^2 - 48*A*B
^2*a^2*b^9*d^2 - 36*A^2*B*a^3*b^8*d^2))/(a^2*d^5) - ((A*b - 2*B*a)*((16*(a + b*tan(c + d*x))^(1/2)*(20*A^2*a^3
*b^8*d^2 - 36*B^2*a^3*b^8*d^2 - 4*A^2*a*b^10*d^2 + 48*A*B*a^2*b^9*d^2))/(a^2*d^4) - (((8*(32*A*a*b^11*d^4 + 16
*A*a^3*b^9*d^4 - 64*B*a^2*b^10*d^4 - 48*B*a^4*b^8*d^4))/(a^2*d^5) - (8*(A*b - 2*B*a)*(32*a^2*b^10*d^4 + 48*a^4
*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/(a^2*d^5*(a^3)^(1/2)))*(A*b - 2*B*a))/(2*d*(a^3)^(1/2))))/(2*d*(a^3)^(1/
2)))*(A*b - 2*B*a))/(2*d*(a^3)^(1/2))))/(2*d*(a^3)^(1/2)) + ((A*b - 2*B*a)*((16*(a + b*tan(c + d*x))^(1/2)*(A^
2*B^2*b^10 - A^4*b^10 + 2*A^4*a^2*b^8 + 6*B^4*a^2*b^8 - 4*A*B^3*a*b^9 + 4*A^3*B*a*b^9))/(a^2*d^4) + (((8*(4*A^
3*b^11*d^2 + 16*A^3*a^2*b^9*d^2 + 12*B^3*a^3*b^8*d^2 + 12*A^2*B*a*b^10*d^2 - 48*A*B^2*a^2*b^9*d^2 - 36*A^2*B*a
^3*b^8*d^2))/(a^2*d^5) + ((A*b - 2*B*a)*((16*(a + b*tan(c + d*x))^(1/2)*(20*A^2*a^3*b^8*d^2 - 36*B^2*a^3*b^8*d
^2 - 4*A^2*a*b^10*d^2 + 48*A*B*a^2*b^9*d^2))/(a^2*d^4) + (((8*(32*A*a*b^11*d^4 + 16*A*a^3*b^9*d^4 - 64*B*a^2*b
^10*d^4 - 48*B*a^4*b^8*d^4))/(a^2*d^5) + (8*(A*b - 2*B*a)*(32*a^2*b^10*d^4 + 48*a^4*b^8*d^4)*(a + b*tan(c + d*
x))^(1/2))/(a^2*d^5*(a^3)^(1/2)))*(A*b - 2*B*a))/(2*d*(a^3)^(1/2))))/(2*d*(a^3)^(1/2)))*(A*b - 2*B*a))/(2*d*(a
^3)^(1/2))))/(2*d*(a^3)^(1/2))))*(A*b - 2*B*a)*1i)/(d*(a^3)^(1/2)) + (A*b*(a + b*tan(c + d*x))^(1/2))/(a*(a*d
- d*(a + b*tan(c + d*x))))