\(\int \frac {A+B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx\) [401]

Optimal result

Integrand size = 33, antiderivative size = 278 \[ \int \frac {A+B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx=\frac {(b (A-B)-a (A+B)) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {(b (A-B)-a (A+B)) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}+\frac {2 \sqrt {b} (A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} \left (a^2+b^2\right ) d}-\frac {(a (A-B)+b (A+B)) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}+\frac {(a (A-B)+b (A+B)) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3694, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211} \[ \int \frac {A+B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx=\frac {(b (A-B)-a (A+B)) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )}-\frac {(b (A-B)-a (A+B)) \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )}+\frac {2 \sqrt {b} (A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d \left (a^2+b^2\right )}-\frac {(a (A-B)+b (A+B)) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}+\frac {(a (A-B)+b (A+B)) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )} \]

[In]

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

[Out]

((b*(A - B) - a*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - ((b*(A - B) - a*(A
+ B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) + (2*Sqrt[b]*(A*b - a*B)*ArcTan[(Sqrt[b]
*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*(a^2 + b^2)*d) - ((a*(A - B) + b*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c +
d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) + ((a*(A - B) + b*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]
] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^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 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 211

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

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1182

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(-a)*c]

Rule 3615

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[2/f, Subst[I
nt[(b*c + d*x^2)/(b^2 + x^4), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2,
0] && NeQ[c^2 + d^2, 0]

Rule 3694

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

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

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.70 \[ \int \frac {A+B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx=-\frac {2 \sqrt {2} (b (-A+B)+a (A+B)) \left (\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )\right )+\frac {8 \sqrt {b} (-A b+a B) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a}}+\sqrt {2} (a (A-B)+b (A+B)) \left (\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right )}{4 \left (a^2+b^2\right ) d} \]

[In]

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

[Out]

-1/4*(2*Sqrt[2]*(b*(-A + B) + a*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan
[c + d*x]]]) + (8*Sqrt[b]*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/Sqrt[a] + Sqrt[2]*(a*(A
- B) + b*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] +
Tan[c + d*x]]))/((a^2 + b^2)*d)

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(\frac {\frac {2 \left (A b -B a \right ) b \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{\left (a^{2}+b^{2}\right ) \sqrt {a b}}+\frac {\frac {\left (a A +B b \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-A b +B a \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{a^{2}+b^{2}}}{d}\)	\(244\)
default	\(\frac {\frac {2 \left (A b -B a \right ) b \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{\left (a^{2}+b^{2}\right ) \sqrt {a b}}+\frac {\frac {\left (a A +B b \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-A b +B a \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{a^{2}+b^{2}}}{d}\)	\(244\)

[In]

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

[Out]

1/d*(2*(A*b-B*a)*b/(a^2+b^2)/(a*b)^(1/2)*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))+2/(a^2+b^2)*(1/8*(A*a+B*b)*2^(
1/2)*(ln((1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))+2*arctan(1+2^(1/2)*t
an(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2)))+1/8*(-A*b+B*a)*2^(1/2)*(ln((1-2^(1/2)*tan(d*x+c)^(1/2)
+tan(d*x+c))/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))+2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)
*tan(d*x+c)^(1/2)))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2999 vs. \(2 (240) = 480\).

Time = 4.61 (sec) , antiderivative size = 6024, normalized size of antiderivative = 21.67 \[ \int \frac {A+B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.78 \[ \int \frac {A+B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx=-\frac {\frac {8 \, {\left (B a b - A b^{2}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{2} + b^{2}\right )} \sqrt {a b}} - \frac {2 \, \sqrt {2} {\left ({\left (A + B\right )} a - {\left (A - B\right )} b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left ({\left (A + B\right )} a - {\left (A - B\right )} b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + \sqrt {2} {\left ({\left (A - B\right )} a + {\left (A + B\right )} b\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt {2} {\left ({\left (A - B\right )} a + {\left (A + B\right )} b\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{a^{2} + b^{2}}}{4 \, d} \]

[In]

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

[Out]

-1/4*(8*(B*a*b - A*b^2)*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b))/((a^2 + b^2)*sqrt(a*b)) - (2*sqrt(2)*((A + B)*a
- (A - B)*b)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 2*sqrt(2)*((A + B)*a - (A - B)*b)*arctan(
-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c)))) + sqrt(2)*((A - B)*a + (A + B)*b)*log(sqrt(2)*sqrt(tan(d*x + c)
) + tan(d*x + c) + 1) - sqrt(2)*((A - B)*a + (A + B)*b)*log(-sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1))/(
a^2 + b^2))/d

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 13.41 (sec) , antiderivative size = 14816, normalized size of antiderivative = 53.29 \[ \int \frac {A+B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx=\text {Too large to display} \]

[In]

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

[Out]

atan(((((32*(13*B^3*a^2*b^4*d^2 + B^3*a^4*b^2*d^2))/d^5 + (((32*(12*B*a*b^7*d^4 + 24*B*a^3*b^5*d^4 + 12*B*a^5*
b^3*d^4))/d^5 - (32*tan(c + d*x)^(1/2)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))
^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*
b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2)
- 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(20*B^2*a^3*b^4*d^2
+ 2*B^2*a^5*b^2*d^2 - 14*B^2*a*b^6*d^2))/d^4)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^
2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2))*(((64*B^4*a^2*b^2*d^4 - B^4*(1
6*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/
2) - (32*tan(c + d*x)^(1/2)*(B^4*b^5 - 2*B^4*a^2*b^3))/d^4)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d
^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*1i - (((32*(13*B^
3*a^2*b^4*d^2 + B^3*a^4*b^2*d^2))/d^5 + (((32*(12*B*a*b^7*d^4 + 24*B*a^3*b^5*d^4 + 12*B*a^5*b^3*d^4))/d^5 + (3
2*tan(c + d*x)^(1/2)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b
*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b
^3*d^4))/d^4)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(
16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(20*B^2*a^3*b^4*d^2 + 2*B^2*a^5*b^2*d^
2 - 14*B^2*a*b^6*d^2))/d^4)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*
B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2))*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4
*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d
*x)^(1/2)*(B^4*b^5 - 2*B^4*a^2*b^3))/d^4)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^
4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*1i)/((((32*(13*B^3*a^2*b^4*d^2 + B^
3*a^4*b^2*d^2))/d^5 + (((32*(12*B*a*b^7*d^4 + 24*B*a^3*b^5*d^4 + 12*B*a^5*b^3*d^4))/d^5 - (32*tan(c + d*x)^(1/
2)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4
+ b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(((6
4*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*
d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(20*B^2*a^3*b^4*d^2 + 2*B^2*a^5*b^2*d^2 - 14*B^2*a*b^6*d
^2))/d^4)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(
a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2))*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*
d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(B^4*b^5
- 2*B^4*a^2*b^3))/d^4)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*
a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (((32*(13*B^3*a^2*b^4*d^2 + B^3*a^4*b^2*d^2))/d^5 +
(((32*(12*B*a*b^7*d^4 + 24*B*a^3*b^5*d^4 + 12*B*a^5*b^3*d^4))/d^5 + (32*tan(c + d*x)^(1/2)*(((64*B^4*a^2*b^2*
d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^
2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(((64*B^4*a^2*b^2*d^4 - B
^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4))
)^(1/2) - (32*tan(c + d*x)^(1/2)*(20*B^2*a^3*b^4*d^2 + 2*B^2*a^5*b^2*d^2 - 14*B^2*a*b^6*d^2))/d^4)*(((64*B^4*a
^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2
*a^2*b^2*d^4)))^(1/2))*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a
*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(B^4*b^5 - 2*B^4*a^2*b^3))/d^
4)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4
+ b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (64*B^5*a*b^3)/d^5))*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^
4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*2i + atan(((((32*(
13*B^3*a^2*b^4*d^2 + B^3*a^4*b^2*d^2))/d^5 + (((32*(12*B*a*b^7*d^4 + 24*B*a^3*b^5*d^4 + 12*B*a^5*b^3*d^4))/d^5
- (32*tan(c + d*x)^(1/2)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B
^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16
*a^6*b^3*d^4))/d^4)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b
*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(20*B^2*a^3*b^4*d^2 + 2*B^2*a^5
*b^2*d^2 - 14*B^2*a*b^6*d^2))/d^4)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1
/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2))*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4
+ 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*
tan(c + d*x)^(1/2)*(B^4*b^5 - 2*B^4*a^2*b^3))/d^4)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*
a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*1i - (((32*(13*B^3*a^2*b^
4*d^2 + B^3*a^4*b^2*d^2))/d^5 + (((32*(12*B*a*b^7*d^4 + 24*B*a^3*b^5*d^4 + 12*B*a^5*b^3*d^4))/d^5 + (32*tan(c
+ d*x)^(1/2)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(
16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4)
)/d^4)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^
4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(20*B^2*a^3*b^4*d^2 + 2*B^2*a^5*b^2*d^2 - 14
*B^2*a*b^6*d^2))/d^4)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a
*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2))*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4
+ 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^
(1/2)*(B^4*b^5 - 2*B^4*a^2*b^3))/d^4)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))
^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*1i)/((((32*(13*B^3*a^2*b^4*d^2 + B^3*a
^4*b^2*d^2))/d^5 + (((32*(12*B*a*b^7*d^4 + 24*B*a^3*b^5*d^4 + 12*B*a^5*b^3*d^4))/d^5 - (32*tan(c + d*x)^(1/2)*
(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 +
b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(-((64
*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d
^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(20*B^2*a^3*b^4*d^2 + 2*B^2*a^5*b^2*d^2 - 14*B^2*a*b^6*d^
2))/d^4)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(
a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2))*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2
*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(B^4*b^
5 - 2*B^4*a^2*b^3))/d^4)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^
2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (((32*(13*B^3*a^2*b^4*d^2 + B^3*a^4*b^2*d^2))/d^5
+ (((32*(12*B*a*b^7*d^4 + 24*B*a^3*b^5*d^4 + 12*B*a^5*b^3*d^4))/d^5 + (32*tan(c + d*x)^(1/2)*(-((64*B^4*a^2*b
^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2
*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(-((64*B^4*a^2*b^2*d^4
- B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d
^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(20*B^2*a^3*b^4*d^2 + 2*B^2*a^5*b^2*d^2 - 14*B^2*a*b^6*d^2))/d^4)*(-((64*
B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^
4 + 2*a^2*b^2*d^4)))^(1/2))*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8
*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(B^4*b^5 - 2*B^4*a^2*b^
3))/d^4)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(
a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (64*B^5*a*b^3)/d^5))*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 1
6*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*2i - atan(
((((32*(5*A^3*a*b^5 + A^3*a^3*b^3))/d^3 - (((32*(16*A*b^8*d^2 + 28*A*a^2*b^6*d^2 + 8*A*a^4*b^4*d^2 - 4*A*a^6*b
^2*d^2))/d^3 - (32*tan(c + d*x)^(1/2)*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))
^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*
b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2)
- 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(4*A^2*a^3*b^4*d^2
+ 2*A^2*a^5*b^2*d^2 - 30*A^2*a*b^6*d^2))/d^4)*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b
^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2))*(-((64*A^4*a^2*b^2*d^4 - A^4*
(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(
1/2) + (96*A^4*b^5*tan(c + d*x)^(1/2))/d^4)*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2
*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*1i - (((32*(5*A^3*a*b^5 + A^3*a^
3*b^3))/d^3 - (((32*(16*A*b^8*d^2 + 28*A*a^2*b^6*d^2 + 8*A*a^4*b^4*d^2 - 4*A*a^6*b^2*d^2))/d^3 + (32*tan(c + d
*x)^(1/2)*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*
(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d
^4)*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d
^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(4*A^2*a^3*b^4*d^2 + 2*A^2*a^5*b^2*d^2 - 30*A^2
*a*b^6*d^2))/d^4)*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d
^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2))*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 3
2*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (96*A^4*b^5*tan(c + d
*x)^(1/2))/d^4)*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2
)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*1i)/((((32*(5*A^3*a*b^5 + A^3*a^3*b^3))/d^3 - (((32*(16*A*b^
8*d^2 + 28*A*a^2*b^6*d^2 + 8*A*a^4*b^4*d^2 - 4*A*a^6*b^2*d^2))/d^3 - (32*tan(c + d*x)^(1/2)*(-((64*A^4*a^2*b^2
*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b
^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(-((64*A^4*a^2*b^2*d^4 -
A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4
)))^(1/2) - (32*tan(c + d*x)^(1/2)*(4*A^2*a^3*b^4*d^2 + 2*A^2*a^5*b^2*d^2 - 30*A^2*a*b^6*d^2))/d^4)*(-((64*A^4
*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 +
2*a^2*b^2*d^4)))^(1/2))*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^
2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (96*A^4*b^5*tan(c + d*x)^(1/2))/d^4)*(-((64*A^4*a
^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2
*a^2*b^2*d^4)))^(1/2) + (((32*(5*A^3*a*b^5 + A^3*a^3*b^3))/d^3 - (((32*(16*A*b^8*d^2 + 28*A*a^2*b^6*d^2 + 8*A*
a^4*b^4*d^2 - 4*A*a^6*b^2*d^2))/d^3 + (32*tan(c + d*x)^(1/2)*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4
*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 1
6*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 +
32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1
/2)*(4*A^2*a^3*b^4*d^2 + 2*A^2*a^5*b^2*d^2 - 30*A^2*a*b^6*d^2))/d^4)*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4
+ 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2))*(-((64
*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d
^4 + 2*a^2*b^2*d^4)))^(1/2) - (96*A^4*b^5*tan(c + d*x)^(1/2))/d^4)*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 +
16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)))*(-((64*
A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^
4 + 2*a^2*b^2*d^4)))^(1/2)*2i - atan(((((32*(5*A^3*a*b^5 + A^3*a^3*b^3))/d^3 - (((32*(16*A*b^8*d^2 + 28*A*a^2*
b^6*d^2 + 8*A*a^4*b^4*d^2 - 4*A*a^6*b^2*d^2))/d^3 - (32*tan(c + d*x)^(1/2)*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4
*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(1
6*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 +
16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan
(c + d*x)^(1/2)*(4*A^2*a^3*b^4*d^2 + 2*A^2*a^5*b^2*d^2 - 30*A^2*a*b^6*d^2))/d^4)*(((64*A^4*a^2*b^2*d^4 - A^4*(
16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1
/2))*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d
^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (96*A^4*b^5*tan(c + d*x)^(1/2))/d^4)*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a
^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*
1i - (((32*(5*A^3*a*b^5 + A^3*a^3*b^3))/d^3 - (((32*(16*A*b^8*d^2 + 28*A*a^2*b^6*d^2 + 8*A*a^4*b^4*d^2 - 4*A*a
^6*b^2*d^2))/d^3 + (32*tan(c + d*x)^(1/2)*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^
4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a
^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/
2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(4*A^2*a^3*b^4*d^
2 + 2*A^2*a^5*b^2*d^2 - 30*A^2*a*b^6*d^2))/d^4)*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*
b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2))*(((64*A^4*a^2*b^2*d^4 - A^4*
(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(
1/2) - (96*A^4*b^5*tan(c + d*x)^(1/2))/d^4)*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*
d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*1i)/((((32*(5*A^3*a*b^5 + A^3*a^3
*b^3))/d^3 - (((32*(16*A*b^8*d^2 + 28*A*a^2*b^6*d^2 + 8*A*a^4*b^4*d^2 - 4*A*a^6*b^2*d^2))/d^3 - (32*tan(c + d*
x)^(1/2)*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a
^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4
)*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4
+ b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(4*A^2*a^3*b^4*d^2 + 2*A^2*a^5*b^2*d^2 - 30*A^2*a*
b^6*d^2))/d^4)*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/
(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2))*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2
*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (96*A^4*b^5*tan(c + d*x)^(
1/2))/d^4)*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*
(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (((32*(5*A^3*a*b^5 + A^3*a^3*b^3))/d^3 - (((32*(16*A*b^8*d^2 + 2
8*A*a^2*b^6*d^2 + 8*A*a^4*b^4*d^2 - 4*A*a^6*b^2*d^2))/d^3 + (32*tan(c + d*x)^(1/2)*(((64*A^4*a^2*b^2*d^4 - A^4
*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^
(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^
4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) +
(32*tan(c + d*x)^(1/2)*(4*A^2*a^3*b^4*d^2 + 2*A^2*a^5*b^2*d^2 - 30*A^2*a*b^6*d^2))/d^4)*(((64*A^4*a^2*b^2*d^4
- A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d
^4)))^(1/2))*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(1
6*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (96*A^4*b^5*tan(c + d*x)^(1/2))/d^4)*(((64*A^4*a^2*b^2*d^4 - A
^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4))
)^(1/2)))*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(
a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*2i - (A*b^3*atan(((A*b^3*((96*A^4*b^5*tan(c + d*x)^(1/2))/d^4 + (A*
b^3*((32*(5*A^3*a*b^5 + A^3*a^3*b^3))/d^3 + (A*b^3*((32*tan(c + d*x)^(1/2)*(4*A^2*a^3*b^4*d^2 + 2*A^2*a^5*b^2*
d^2 - 30*A^2*a*b^6*d^2))/d^4 - (A*b^3*((32*(16*A*b^8*d^2 + 28*A*a^2*b^6*d^2 + 8*A*a^4*b^4*d^2 - 4*A*a^6*b^2*d^
2))/d^3 - (32*A*b^3*tan(c + d*x)^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/(d^4*(
- a*b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2))))/(- a*b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2)))/(- a*b
^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2)))/(- a*b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2))*1i)/(- a*b^7*
d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2) + (A*b^3*((96*A^4*b^5*tan(c + d*x)^(1/2))/d^4 - (A*b^3*((32*(5*A^3*a*
b^5 + A^3*a^3*b^3))/d^3 - (A*b^3*((32*tan(c + d*x)^(1/2)*(4*A^2*a^3*b^4*d^2 + 2*A^2*a^5*b^2*d^2 - 30*A^2*a*b^6
*d^2))/d^4 + (A*b^3*((32*(16*A*b^8*d^2 + 28*A*a^2*b^6*d^2 + 8*A*a^4*b^4*d^2 - 4*A*a^6*b^2*d^2))/d^3 + (32*A*b^
3*tan(c + d*x)^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/(d^4*(- a*b^7*d^2 - 2*a^
3*b^5*d^2 - a^5*b^3*d^2)^(1/2))))/(- a*b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2)))/(- a*b^7*d^2 - 2*a^3*b^5
*d^2 - a^5*b^3*d^2)^(1/2)))/(- a*b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2))*1i)/(- a*b^7*d^2 - 2*a^3*b^5*d^
2 - a^5*b^3*d^2)^(1/2))/((A*b^3*((96*A^4*b^5*tan(c + d*x)^(1/2))/d^4 + (A*b^3*((32*(5*A^3*a*b^5 + A^3*a^3*b^3)
)/d^3 + (A*b^3*((32*tan(c + d*x)^(1/2)*(4*A^2*a^3*b^4*d^2 + 2*A^2*a^5*b^2*d^2 - 30*A^2*a*b^6*d^2))/d^4 - (A*b^
3*((32*(16*A*b^8*d^2 + 28*A*a^2*b^6*d^2 + 8*A*a^4*b^4*d^2 - 4*A*a^6*b^2*d^2))/d^3 - (32*A*b^3*tan(c + d*x)^(1/
2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/(d^4*(- a*b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^
3*d^2)^(1/2))))/(- a*b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2)))/(- a*b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2
)^(1/2)))/(- a*b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2)))/(- a*b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2
) - (A*b^3*((96*A^4*b^5*tan(c + d*x)^(1/2))/d^4 - (A*b^3*((32*(5*A^3*a*b^5 + A^3*a^3*b^3))/d^3 - (A*b^3*((32*t
an(c + d*x)^(1/2)*(4*A^2*a^3*b^4*d^2 + 2*A^2*a^5*b^2*d^2 - 30*A^2*a*b^6*d^2))/d^4 + (A*b^3*((32*(16*A*b^8*d^2
+ 28*A*a^2*b^6*d^2 + 8*A*a^4*b^4*d^2 - 4*A*a^6*b^2*d^2))/d^3 + (32*A*b^3*tan(c + d*x)^(1/2)*(16*b^9*d^4 + 16*a
^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/(d^4*(- a*b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2))))/(- a*
b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2)))/(- a*b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2)))/(- a*b^7*d^
2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2)))/(- a*b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2)))*2i)/(- a*b^7*d^2
- 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2) - (B*a*b*atan(((B*a*b*((32*tan(c + d*x)^(1/2)*(B^4*b^5 - 2*B^4*a^2*b^3))/
d^4 - (B*a*b*((32*(13*B^3*a^2*b^4*d^2 + B^3*a^4*b^2*d^2))/d^5 + (B*a*b*((32*tan(c + d*x)^(1/2)*(20*B^2*a^3*b^4
*d^2 + 2*B^2*a^5*b^2*d^2 - 14*B^2*a*b^6*d^2))/d^4 + (B*a*b*((32*(12*B*a*b^7*d^4 + 24*B*a^3*b^5*d^4 + 12*B*a^5*
b^3*d^4))/d^5 - (32*B*a*b*tan(c + d*x)^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/
(d^4*(- a*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2))))/(- a*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2)))/(- a
*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2)))/(- a*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2))*1i)/(- a*b^5*d^
2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2) + (B*a*b*((32*tan(c + d*x)^(1/2)*(B^4*b^5 - 2*B^4*a^2*b^3))/d^4 + (B*a*b*
((32*(13*B^3*a^2*b^4*d^2 + B^3*a^4*b^2*d^2))/d^5 - (B*a*b*((32*tan(c + d*x)^(1/2)*(20*B^2*a^3*b^4*d^2 + 2*B^2*
a^5*b^2*d^2 - 14*B^2*a*b^6*d^2))/d^4 - (B*a*b*((32*(12*B*a*b^7*d^4 + 24*B*a^3*b^5*d^4 + 12*B*a^5*b^3*d^4))/d^5
+ (32*B*a*b*tan(c + d*x)^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/(d^4*(- a*b^5
*d^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2))))/(- a*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2)))/(- a*b^5*d^2 - a^
5*b*d^2 - 2*a^3*b^3*d^2)^(1/2)))/(- a*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2))*1i)/(- a*b^5*d^2 - a^5*b*d^2
- 2*a^3*b^3*d^2)^(1/2))/((64*B^5*a*b^3)/d^5 - (B*a*b*((32*tan(c + d*x)^(1/2)*(B^4*b^5 - 2*B^4*a^2*b^3))/d^4 -
(B*a*b*((32*(13*B^3*a^2*b^4*d^2 + B^3*a^4*b^2*d^2))/d^5 + (B*a*b*((32*tan(c + d*x)^(1/2)*(20*B^2*a^3*b^4*d^2
+ 2*B^2*a^5*b^2*d^2 - 14*B^2*a*b^6*d^2))/d^4 + (B*a*b*((32*(12*B*a*b^7*d^4 + 24*B*a^3*b^5*d^4 + 12*B*a^5*b^3*d
^4))/d^5 - (32*B*a*b*tan(c + d*x)^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/(d^4*
(- a*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2))))/(- a*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2)))/(- a*b^5*
d^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2)))/(- a*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2)))/(- a*b^5*d^2 - a^5*
b*d^2 - 2*a^3*b^3*d^2)^(1/2) + (B*a*b*((32*tan(c + d*x)^(1/2)*(B^4*b^5 - 2*B^4*a^2*b^3))/d^4 + (B*a*b*((32*(13
*B^3*a^2*b^4*d^2 + B^3*a^4*b^2*d^2))/d^5 - (B*a*b*((32*tan(c + d*x)^(1/2)*(20*B^2*a^3*b^4*d^2 + 2*B^2*a^5*b^2*
d^2 - 14*B^2*a*b^6*d^2))/d^4 - (B*a*b*((32*(12*B*a*b^7*d^4 + 24*B*a^3*b^5*d^4 + 12*B*a^5*b^3*d^4))/d^5 + (32*B
*a*b*tan(c + d*x)^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/(d^4*(- a*b^5*d^2 - a
^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2))))/(- a*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2)))/(- a*b^5*d^2 - a^5*b*d^2
- 2*a^3*b^3*d^2)^(1/2)))/(- a*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2)))/(- a*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^
3*d^2)^(1/2)))*2i)/(- a*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2)