\(\int \frac {A+B \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))} \, dx\) [403]

Optimal result

Integrand size = 33, antiderivative size = 325 \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))} \, dx=-\frac {(b (A-B)-a (A+B)) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}+\frac {(b (A-B)-a (A+B)) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}+\frac {2 b^{5/2} (A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} \left (a^2+b^2\right ) d}+\frac {(a (A-B)+b (A+B)) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a (A-B)+b (A+B)) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {2 A}{3 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 (A b-a B)}{a^2 d \sqrt {\tan (c+d x)}} \]

[Out]

2*b^(5/2)*(A*b-B*a)*arctan(b^(1/2)*tan(d*x+c)^(1/2)/a^(1/2))/a^(5/2)/(a^2+b^2)/d+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)/a^2/d/tan(d*x+c)^(1/2)-2
/3*A/a/d/tan(d*x+c)^(3/2)

Rubi [A] (verified)

Time = 1.36 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.394, Rules used = {3690, 3730, 3734, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211} \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))} \, dx=-\frac {(b (A-B)-a (A+B)) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )}+\frac {(b (A-B)-a (A+B)) \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )}+\frac {(a (A-B)+b (A+B)) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}-\frac {(a (A-B)+b (A+B)) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}+\frac {2 (A b-a B)}{a^2 d \sqrt {\tan (c+d x)}}+\frac {2 b^{5/2} (A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d \left (a^2+b^2\right )}-\frac {2 A}{3 a d \tan ^{\frac {3}{2}}(c+d x)} \]

[In]

Int[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(5/2)*(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*b^(5/2)*(A*b - a*B)*ArcTan[(Sqrt
[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(5/2)*(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) - ((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) - (2*A)/(3*a*d*Tan[c + d*x]^(3/2)) + (2*(A*b - a*B))/(a^2*d*Sq
rt[Tan[c + d*x]])

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 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 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 {2 A}{3 a d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 \int \frac {\frac {3}{2} (A b-a B)+\frac {3}{2} a A \tan (c+d x)+\frac {3}{2} A b \tan ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))} \, dx}{3 a} \\ & = -\frac {2 A}{3 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 (A b-a B)}{a^2 d \sqrt {\tan (c+d x)}}+\frac {4 \int \frac {-\frac {3}{4} \left (a^2 A-A b^2+a b B\right )-\frac {3}{4} a^2 B \tan (c+d x)+\frac {3}{4} b (A b-a B) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{3 a^2} \\ & = -\frac {2 A}{3 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 (A b-a B)}{a^2 d \sqrt {\tan (c+d x)}}+\frac {4 \int \frac {-\frac {3}{4} a^2 (a A+b B)+\frac {3}{4} a^2 (A b-a B) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{3 a^2 \left (a^2+b^2\right )}+\frac {\left (b^3 (A b-a B)\right ) \int \frac {1+\tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{a^2 \left (a^2+b^2\right )} \\ & = -\frac {2 A}{3 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 (A b-a B)}{a^2 d \sqrt {\tan (c+d x)}}+\frac {8 \text {Subst}\left (\int \frac {-\frac {3}{4} a^2 (a A+b B)+\frac {3}{4} a^2 (A b-a B) x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{3 a^2 \left (a^2+b^2\right ) d}+\frac {\left (b^3 (A b-a B)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{a^2 \left (a^2+b^2\right ) d} \\ & = -\frac {2 A}{3 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 (A b-a B)}{a^2 d \sqrt {\tan (c+d x)}}+\frac {\left (2 b^3 (A b-a B)\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a^2 \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 b^{5/2} (A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} \left (a^2+b^2\right ) d}-\frac {2 A}{3 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 (A b-a B)}{a^2 d \sqrt {\tan (c+d x)}}+\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 b^{5/2} (A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} \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 {(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 {2 A}{3 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 (A b-a B)}{a^2 d \sqrt {\tan (c+d x)}}+\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 b^{5/2} (A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} \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 {(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 {2 A}{3 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 (A b-a B)}{a^2 d \sqrt {\tan (c+d x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.63 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.54 \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))} \, dx=\frac {\frac {3 \sqrt [4]{-1} (A-i B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{a-i b}+\frac {6 b^{5/2} (A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} \left (a^2+b^2\right )}+\frac {3 \sqrt [4]{-1} (A+i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{a+i b}-\frac {2 (a A+(-3 A b+3 a B) \tan (c+d x))}{a^2 \tan ^{\frac {3}{2}}(c+d x)}}{3 d} \]

[In]

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

[Out]

((3*(-1)^(1/4)*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/(a - I*b) + (6*b^(5/2)*(A*b - a*B)*ArcTan[(Sqr
t[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(5/2)*(a^2 + b^2)) + (3*(-1)^(1/4)*(A + I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan
[c + d*x]]])/(a + I*b) - (2*(a*A + (-3*A*b + 3*a*B)*Tan[c + d*x]))/(a^2*Tan[c + d*x]^(3/2)))/(3*d)

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(\frac {\frac {2 b^{3} \left (A b -B a \right ) \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{a^{2} \left (a^{2}+b^{2}\right ) \sqrt {a b}}-\frac {2 A}{3 a \tan \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 \left (-A b +B a \right )}{a^{2} \sqrt {\tan \left (d x +c \right )}}+\frac {\frac {\left (-a A -B b \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (A b -B a \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{a^{2}+b^{2}}}{d}\)	\(286\)
default	\(\frac {\frac {2 b^{3} \left (A b -B a \right ) \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{a^{2} \left (a^{2}+b^{2}\right ) \sqrt {a b}}-\frac {2 A}{3 a \tan \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 \left (-A b +B a \right )}{a^{2} \sqrt {\tan \left (d x +c \right )}}+\frac {\frac {\left (-a A -B b \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (A b -B a \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{a^{2}+b^{2}}}{d}\)	\(286\)

[In]

int((A+B*tan(d*x+c))/tan(d*x+c)^(5/2)/(a+b*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(2/a^2*b^3*(A*b-B*a)/(a^2+b^2)/(a*b)^(1/2)*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))-2/3/a*A/tan(d*x+c)^(3/2)
-2/a^2*(-A*b+B*a)/tan(d*x+c)^(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)*tan(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. 3129 vs. \(2 (281) = 562\).

Time = 27.01 (sec) , antiderivative size = 6284, normalized size of antiderivative = 19.34 \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.79 \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))} \, dx=-\frac {\frac {24 \, {\left (B a b^{3} - A b^{4}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{4} + a^{2} b^{2}\right )} \sqrt {a b}} + \frac {3 \, {\left (2 \, \sqrt {2} {\left ({\left (A + B\right )} a - {\left (A - B\right )} b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left ({\left (A + B\right )} a - {\left (A - B\right )} b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + \sqrt {2} {\left ({\left (A - B\right )} a + {\left (A + B\right )} b\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt {2} {\left ({\left (A - B\right )} a + {\left (A + B\right )} b\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )}}{a^{2} + b^{2}} + \frac {8 \, {\left (A a + 3 \, {\left (B a - A b\right )} \tan \left (d x + c\right )\right )}}{a^{2} \tan \left (d x + c\right )^{\frac {3}{2}}}}{12 \, d} \]

[In]

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

[Out]

-1/12*(24*(B*a*b^3 - A*b^4)*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b))/((a^4 + a^2*b^2)*sqrt(a*b)) + 3*(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(ta
n(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) + 8*(A*a + 3*(B*a - A*b)*tan(d*x + c))/(a^2*tan(d*x + c)^(3/2)))/d

Giac [F(-1)]

Timed out. \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 15.15 (sec) , antiderivative size = 16111, normalized size of antiderivative = 49.57 \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))} \, dx=\text {Too large to display} \]

[In]

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

[Out]

atan(((tan(c + d*x)^(1/2)*(64*A^4*a^14*b^9*d^5 + 32*A^4*a^18*b^5*d^5) + (-((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)*(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)*(512*a^18*b^9*d^9 + 512*
a^20*b^7*d^9 - 512*a^22*b^5*d^9 - 512*a^24*b^3*d^9) - 512*A*a^16*b^10*d^8 - 512*A*a^18*b^8*d^8 + 384*A*a^20*b^
6*d^8 + 256*A*a^22*b^4*d^8 - 128*A*a^24*b^2*d^8) - tan(c + d*x)^(1/2)*(512*A^2*a^15*b^10*d^7 + 448*A^2*a^19*b^
6*d^7 - 128*A^2*a^21*b^4*d^7 - 64*A^2*a^23*b^2*d^7))*(-((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) + 384*A^3*a^15*b^9*d^6
- 32*A^3*a^19*b^5*d^6 - 32*A^3*a^21*b^3*d^6))*(-((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 + (tan(c + d*x)^(1/2)*(64*A
^4*a^14*b^9*d^5 + 32*A^4*a^18*b^5*d^5) + (-((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)
*(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)*(512*a^18*b^9*d^9 + 512*a^20*b^7*d^9 - 512*a^22*b^5*d^9
- 512*a^24*b^3*d^9) + 512*A*a^16*b^10*d^8 + 512*A*a^18*b^8*d^8 - 384*A*a^20*b^6*d^8 - 256*A*a^22*b^4*d^8 + 12
8*A*a^24*b^2*d^8) - tan(c + d*x)^(1/2)*(512*A^2*a^15*b^10*d^7 + 448*A^2*a^19*b^6*d^7 - 128*A^2*a^21*b^4*d^7 -
64*A^2*a^23*b^2*d^7))*(-((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) - 384*A^3*a^15*b^9*d^6 + 32*A^3*a^19*b^5*d^6 + 32*A^3*
a^21*b^3*d^6))*(-((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)/((tan(c + d*x)^(1/2)*(64*A^4*a^14*b^9*d^5 + 32*A^4*a^18*b
^5*d^5) + (-((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)*(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)*(512*a^18*b^9*d^9 + 512*a^20*b^7*d^9 - 512*a^22*b^5*d^9 - 512*a^24*b^3*d^9) + 512*A*a^
16*b^10*d^8 + 512*A*a^18*b^8*d^8 - 384*A*a^20*b^6*d^8 - 256*A*a^22*b^4*d^8 + 128*A*a^24*b^2*d^8) - tan(c + d*x
)^(1/2)*(512*A^2*a^15*b^10*d^7 + 448*A^2*a^19*b^6*d^7 - 128*A^2*a^21*b^4*d^7 - 64*A^2*a^23*b^2*d^7))*(-((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) - 384*A^3*a^15*b^9*d^6 + 32*A^3*a^19*b^5*d^6 + 32*A^3*a^21*b^3*d^6))*(-((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) - (tan(c + d*x)^(1/2)*(64*A^4*a^14*b^9*d^5 + 32*A^4*a^18*b^5*d^5) + (-((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)*(tan(c + d*x)^(1/2)*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 1
6*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)*(512*a^18*
b^9*d^9 + 512*a^20*b^7*d^9 - 512*a^22*b^5*d^9 - 512*a^24*b^3*d^9) - 512*A*a^16*b^10*d^8 - 512*A*a^18*b^8*d^8 +
384*A*a^20*b^6*d^8 + 256*A*a^22*b^4*d^8 - 128*A*a^24*b^2*d^8) - tan(c + d*x)^(1/2)*(512*A^2*a^15*b^10*d^7 + 4
48*A^2*a^19*b^6*d^7 - 128*A^2*a^21*b^4*d^7 - 64*A^2*a^23*b^2*d^7))*(-((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) + 384*A^3
*a^15*b^9*d^6 - 32*A^3*a^19*b^5*d^6 - 32*A^3*a^21*b^3*d^6))*(-((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^5*a^14*b^
8*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)*2i + atan(((tan(c + d*x)^(1/2)*(64*A^4*a^14*b^9*d^5 + 32*A^4*a^18*b^
5*d^5) + (((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)*(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)*(512*a^18*b^9*d^9 + 512*a^20*b^7*d^9 - 512*a^22*b^5*d^9 - 512*a^24*b^3*d^9) - 512*A*a^16*b
^10*d^8 - 512*A*a^18*b^8*d^8 + 384*A*a^20*b^6*d^8 + 256*A*a^22*b^4*d^8 - 128*A*a^24*b^2*d^8) - tan(c + d*x)^(1
/2)*(512*A^2*a^15*b^10*d^7 + 448*A^2*a^19*b^6*d^7 - 128*A^2*a^21*b^4*d^7 - 64*A^2*a^23*b^2*d^7))*(((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) + 384*A^3*a^15*b^9*d^6 - 32*A^3*a^19*b^5*d^6 - 32*A^3*a^21*b^3*d^6))*(((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 + (tan(c + d*x)^(1/2)*(64*A^4*a^14*b^9*d^5 + 32*A^4*a^18*b^5*d^5) + (((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)*(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)*(512*a^18*b^9*d^
9 + 512*a^20*b^7*d^9 - 512*a^22*b^5*d^9 - 512*a^24*b^3*d^9) + 512*A*a^16*b^10*d^8 + 512*A*a^18*b^8*d^8 - 384*A
*a^20*b^6*d^8 - 256*A*a^22*b^4*d^8 + 128*A*a^24*b^2*d^8) - tan(c + d*x)^(1/2)*(512*A^2*a^15*b^10*d^7 + 448*A^2
*a^19*b^6*d^7 - 128*A^2*a^21*b^4*d^7 - 64*A^2*a^23*b^2*d^7))*(((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) - 384*A^3*a^15*b
^9*d^6 + 32*A^3*a^19*b^5*d^6 + 32*A^3*a^21*b^3*d^6))*(((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)/((tan(c + d*x)^(1/2)
*(64*A^4*a^14*b^9*d^5 + 32*A^4*a^18*b^5*d^5) + (((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)*(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)*(512*a^18*b^9*d^9 + 512*a^20*b^7*d^9 - 512*a^22*b^5*
d^9 - 512*a^24*b^3*d^9) + 512*A*a^16*b^10*d^8 + 512*A*a^18*b^8*d^8 - 384*A*a^20*b^6*d^8 - 256*A*a^22*b^4*d^8 +
128*A*a^24*b^2*d^8) - tan(c + d*x)^(1/2)*(512*A^2*a^15*b^10*d^7 + 448*A^2*a^19*b^6*d^7 - 128*A^2*a^21*b^4*d^7
- 64*A^2*a^23*b^2*d^7))*(((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) - 384*A^3*a^15*b^9*d^6 + 32*A^3*a^19*b^5*d^6 + 32*A^
3*a^21*b^3*d^6))*(((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) - (tan(c + d*x)^(1/2)*(64*A^4*a^14*b^9*d^5 + 32*A^4*a^18*b^5
*d^5) + (((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)*(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)*(512*a^18*b^9*d^9 + 512*a^20*b^7*d^9 - 512*a^22*b^5*d^9 - 512*a^24*b^3*d^9) - 512*A*a^16*b^
10*d^8 - 512*A*a^18*b^8*d^8 + 384*A*a^20*b^6*d^8 + 256*A*a^22*b^4*d^8 - 128*A*a^24*b^2*d^8) - tan(c + d*x)^(1/
2)*(512*A^2*a^15*b^10*d^7 + 448*A^2*a^19*b^6*d^7 - 128*A^2*a^21*b^4*d^7 - 64*A^2*a^23*b^2*d^7))*(((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) + 384*A^3*a^15*b^9*d^6 - 32*A^3*a^19*b^5*d^6 - 32*A^3*a^21*b^3*d^6))*(((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^5*a^14*b^8*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)*2i + atan((((((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)*((tan(c + d*x)^(1/2)*(512*B^2*a^8*b^8*d^7 - 448*B^2*a^10*b^6*d^7 + 128*B^2*a^12*b^4*d^7 + 64*B^2*a^14*b^2*d
^7) - (((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)*(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)*(512*a^9*b^9*d^9 +
512*a^11*b^7*d^9 - 512*a^13*b^5*d^9 - 512*a^15*b^3*d^9) - 512*B*a^8*b^9*d^8 - 640*B*a^10*b^7*d^8 + 256*B*a^12
*b^5*d^8 + 384*B*a^14*b^3*d^8))*(((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) + 128*B^3*a^7*b^8*d^6 - 32*B^3*a^11*b^4*d^6 -
32*B^3*a^13*b^2*d^6) + tan(c + d*x)^(1/2)*(64*B^4*a^7*b^7*d^5 - 32*B^4*a^9*b^5*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)*1i + ((((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)*((tan(c + d*x)^(1/2)*(512*B^2*a^8*b^8*d^7 - 448*B^2*a^10*b^6*d
^7 + 128*B^2*a^12*b^4*d^7 + 64*B^2*a^14*b^2*d^7) - (((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)*(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)*(512*a^9*b^9*d^9 + 512*a^11*b^7*d^9 - 512*a^13*b^5*d^9 - 512*a^15*b^3*d^9) + 512*B*
a^8*b^9*d^8 + 640*B*a^10*b^7*d^8 - 256*B*a^12*b^5*d^8 - 384*B*a^14*b^3*d^8))*(((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)
- 128*B^3*a^7*b^8*d^6 + 32*B^3*a^11*b^4*d^6 + 32*B^3*a^13*b^2*d^6) + tan(c + d*x)^(1/2)*(64*B^4*a^7*b^7*d^5 -
32*B^4*a^9*b^5*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)*1i)/(((((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)*((tan(c + d*x)^(1
/2)*(512*B^2*a^8*b^8*d^7 - 448*B^2*a^10*b^6*d^7 + 128*B^2*a^12*b^4*d^7 + 64*B^2*a^14*b^2*d^7) - (((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)*(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)*(512*a^9*b^9*d^9 + 512*a^11*b^7*d^9 -
512*a^13*b^5*d^9 - 512*a^15*b^3*d^9) - 512*B*a^8*b^9*d^8 - 640*B*a^10*b^7*d^8 + 256*B*a^12*b^5*d^8 + 384*B*a^1
4*b^3*d^8))*(((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) + 128*B^3*a^7*b^8*d^6 - 32*B^3*a^11*b^4*d^6 - 32*B^3*a^13*b^2*d^6
) + tan(c + d*x)^(1/2)*(64*B^4*a^7*b^7*d^5 - 32*B^4*a^9*b^5*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) - ((((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)*((tan(c + d*x)^(1/2)*(512*B^2*a^8*b^8*d^7 - 448*B^2*a^10*b^6*d^7 + 128*B^2*a^12*b^4*d
^7 + 64*B^2*a^14*b^2*d^7) - (((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)*(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)*(512*a^9*b^9*d^9 + 512*a^11*b^7*d^9 - 512*a^13*b^5*d^9 - 512*a^15*b^3*d^9) + 512*B*a^8*b^9*d^8 + 640*B*a^1
0*b^7*d^8 - 256*B*a^12*b^5*d^8 - 384*B*a^14*b^3*d^8))*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 3
2*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) - 128*B^3*a^7*b^8*d^6 +
32*B^3*a^11*b^4*d^6 + 32*B^3*a^13*b^2*d^6) + tan(c + d*x)^(1/2)*(64*B^4*a^7*b^7*d^5 - 32*B^4*a^9*b^5*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)))*(((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((((-((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
)*((tan(c + d*x)^(1/2)*(512*B^2*a^8*b^8*d^7 - 448*B^2*a^10*b^6*d^7 + 128*B^2*a^12*b^4*d^7 + 64*B^2*a^14*b^2*d^
7) - (-((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)*(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)*(512*a^9*b^9*d^9
+ 512*a^11*b^7*d^9 - 512*a^13*b^5*d^9 - 512*a^15*b^3*d^9) - 512*B*a^8*b^9*d^8 - 640*B*a^10*b^7*d^8 + 256*B*a^1
2*b^5*d^8 + 384*B*a^14*b^3*d^8))*(-((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) + 128*B^3*a^7*b^8*d^6 - 32*B^3*a^11*b^4*d^6
- 32*B^3*a^13*b^2*d^6) + tan(c + d*x)^(1/2)*(64*B^4*a^7*b^7*d^5 - 32*B^4*a^9*b^5*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)*1i + ((-((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)*((tan(c + d*x)^(1/2)*(512*B^2*a^8*b^8*d^7 - 448*B^2*a^10*b
^6*d^7 + 128*B^2*a^12*b^4*d^7 + 64*B^2*a^14*b^2*d^7) - (-((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)*(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)*(512*a^9*b^9*d^9 + 512*a^11*b^7*d^9 - 512*a^13*b^5*d^9 - 512*a^15*b^3*d^9) +
512*B*a^8*b^9*d^8 + 640*B*a^10*b^7*d^8 - 256*B*a^12*b^5*d^8 - 384*B*a^14*b^3*d^8))*(-((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) - 128*B^3*a^7*b^8*d^6 + 32*B^3*a^11*b^4*d^6 + 32*B^3*a^13*b^2*d^6) + tan(c + d*x)^(1/2)*(64*B^4*a^7*b^7
*d^5 - 32*B^4*a^9*b^5*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)*1i)/(((-((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)*((tan(c
+ d*x)^(1/2)*(512*B^2*a^8*b^8*d^7 - 448*B^2*a^10*b^6*d^7 + 128*B^2*a^12*b^4*d^7 + 64*B^2*a^14*b^2*d^7) - (-((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)*(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)*(512*a^9*b^9*d^9 + 512*a^11
*b^7*d^9 - 512*a^13*b^5*d^9 - 512*a^15*b^3*d^9) - 512*B*a^8*b^9*d^8 - 640*B*a^10*b^7*d^8 + 256*B*a^12*b^5*d^8
+ 384*B*a^14*b^3*d^8))*(-((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) + 128*B^3*a^7*b^8*d^6 - 32*B^3*a^11*b^4*d^6 - 32*B^3*
a^13*b^2*d^6) + tan(c + d*x)^(1/2)*(64*B^4*a^7*b^7*d^5 - 32*B^4*a^9*b^5*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
) - ((-((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)*((tan(c + d*x)^(1/2)*(512*B^2*a^8*b^8*d^7 - 448*B^2*a^10*b^6*d^7 + 128*
B^2*a^12*b^4*d^7 + 64*B^2*a^14*b^2*d^7) - (-((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)*(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)*(512*a^9*b^9*d^9 + 512*a^11*b^7*d^9 - 512*a^13*b^5*d^9 - 512*a^15*b^3*d^9) + 512*B*a^8*b^9
*d^8 + 640*B*a^10*b^7*d^8 - 256*B*a^12*b^5*d^8 - 384*B*a^14*b^3*d^8))*(-((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) - 128*
B^3*a^7*b^8*d^6 + 32*B^3*a^11*b^4*d^6 + 32*B^3*a^13*b^2*d^6) + tan(c + d*x)^(1/2)*(64*B^4*a^7*b^7*d^5 - 32*B^4
*a^9*b^5*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)))*(-((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 - ((2*A)/(3*a) - (2*A*
b*tan(c + d*x))/a^2)/(d*tan(c + d*x)^(3/2)) - (B*b^5*atan(((B*b^5*(tan(c + d*x)^(1/2)*(64*B^4*a^7*b^7*d^5 - 32
*B^4*a^9*b^5*d^5) + (B*b^5*(32*B^3*a^11*b^4*d^6 - 128*B^3*a^7*b^8*d^6 + 32*B^3*a^13*b^2*d^6 + (B*b^5*(tan(c +
d*x)^(1/2)*(512*B^2*a^8*b^8*d^7 - 448*B^2*a^10*b^6*d^7 + 128*B^2*a^12*b^4*d^7 + 64*B^2*a^14*b^2*d^7) - (B*b^5*
(512*B*a^8*b^9*d^8 + 640*B*a^10*b^7*d^8 - 256*B*a^12*b^5*d^8 - 384*B*a^14*b^3*d^8 + (B*b^5*tan(c + d*x)^(1/2)*
(512*a^9*b^9*d^9 + 512*a^11*b^7*d^9 - 512*a^13*b^5*d^9 - 512*a^15*b^3*d^9))/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a
^7*b^5*d^2)^(1/2)))/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/2)))/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^7
*b^5*d^2)^(1/2)))/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/2))*1i)/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^
7*b^5*d^2)^(1/2) + (B*b^5*(tan(c + d*x)^(1/2)*(64*B^4*a^7*b^7*d^5 - 32*B^4*a^9*b^5*d^5) + (B*b^5*(128*B^3*a^7*
b^8*d^6 - 32*B^3*a^11*b^4*d^6 - 32*B^3*a^13*b^2*d^6 + (B*b^5*(tan(c + d*x)^(1/2)*(512*B^2*a^8*b^8*d^7 - 448*B^
2*a^10*b^6*d^7 + 128*B^2*a^12*b^4*d^7 + 64*B^2*a^14*b^2*d^7) - (B*b^5*(256*B*a^12*b^5*d^8 - 640*B*a^10*b^7*d^8
- 512*B*a^8*b^9*d^8 + 384*B*a^14*b^3*d^8 + (B*b^5*tan(c + d*x)^(1/2)*(512*a^9*b^9*d^9 + 512*a^11*b^7*d^9 - 51
2*a^13*b^5*d^9 - 512*a^15*b^3*d^9))/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/2)))/(- a^3*b^9*d^2 - 2*a
^5*b^7*d^2 - a^7*b^5*d^2)^(1/2)))/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/2)))/(- a^3*b^9*d^2 - 2*a^5
*b^7*d^2 - a^7*b^5*d^2)^(1/2))*1i)/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/2))/((B*b^5*(tan(c + d*x)^
(1/2)*(64*B^4*a^7*b^7*d^5 - 32*B^4*a^9*b^5*d^5) + (B*b^5*(32*B^3*a^11*b^4*d^6 - 128*B^3*a^7*b^8*d^6 + 32*B^3*a
^13*b^2*d^6 + (B*b^5*(tan(c + d*x)^(1/2)*(512*B^2*a^8*b^8*d^7 - 448*B^2*a^10*b^6*d^7 + 128*B^2*a^12*b^4*d^7 +
64*B^2*a^14*b^2*d^7) - (B*b^5*(512*B*a^8*b^9*d^8 + 640*B*a^10*b^7*d^8 - 256*B*a^12*b^5*d^8 - 384*B*a^14*b^3*d^
8 + (B*b^5*tan(c + d*x)^(1/2)*(512*a^9*b^9*d^9 + 512*a^11*b^7*d^9 - 512*a^13*b^5*d^9 - 512*a^15*b^3*d^9))/(- a
^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/2)))/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/2)))/(- a^3
*b^9*d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/2)))/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/2)))/(- a^3*b
^9*d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/2) - (B*b^5*(tan(c + d*x)^(1/2)*(64*B^4*a^7*b^7*d^5 - 32*B^4*a^9*b^5*
d^5) + (B*b^5*(128*B^3*a^7*b^8*d^6 - 32*B^3*a^11*b^4*d^6 - 32*B^3*a^13*b^2*d^6 + (B*b^5*(tan(c + d*x)^(1/2)*(5
12*B^2*a^8*b^8*d^7 - 448*B^2*a^10*b^6*d^7 + 128*B^2*a^12*b^4*d^7 + 64*B^2*a^14*b^2*d^7) - (B*b^5*(256*B*a^12*b
^5*d^8 - 640*B*a^10*b^7*d^8 - 512*B*a^8*b^9*d^8 + 384*B*a^14*b^3*d^8 + (B*b^5*tan(c + d*x)^(1/2)*(512*a^9*b^9*
d^9 + 512*a^11*b^7*d^9 - 512*a^13*b^5*d^9 - 512*a^15*b^3*d^9))/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(
1/2)))/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/2)))/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/
2)))/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/2)))/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/2)
))*2i)/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/2) + (A*b^7*atan(((A*b^7*(tan(c + d*x)^(1/2)*(64*A^4*a
^14*b^9*d^5 + 32*A^4*a^18*b^5*d^5) - (A*b^7*(32*A^3*a^19*b^5*d^6 - 384*A^3*a^15*b^9*d^6 + 32*A^3*a^21*b^3*d^6
+ (A*b^7*(tan(c + d*x)^(1/2)*(512*A^2*a^15*b^10*d^7 + 448*A^2*a^19*b^6*d^7 - 128*A^2*a^21*b^4*d^7 - 64*A^2*a^2
3*b^2*d^7) + (A*b^7*(512*A*a^16*b^10*d^8 + 512*A*a^18*b^8*d^8 - 384*A*a^20*b^6*d^8 - 256*A*a^22*b^4*d^8 + 128*
A*a^24*b^2*d^8 - (A*b^7*tan(c + d*x)^(1/2)*(512*a^18*b^9*d^9 + 512*a^20*b^7*d^9 - 512*a^22*b^5*d^9 - 512*a^24*
b^3*d^9))/(- a^5*b^11*d^2 - 2*a^7*b^9*d^2 - a^9*b^7*d^2)^(1/2)))/(- a^5*b^11*d^2 - 2*a^7*b^9*d^2 - a^9*b^7*d^2
)^(1/2)))/(- a^5*b^11*d^2 - 2*a^7*b^9*d^2 - a^9*b^7*d^2)^(1/2)))/(- a^5*b^11*d^2 - 2*a^7*b^9*d^2 - a^9*b^7*d^2
)^(1/2))*1i)/(- a^5*b^11*d^2 - 2*a^7*b^9*d^2 - a^9*b^7*d^2)^(1/2) + (A*b^7*(tan(c + d*x)^(1/2)*(64*A^4*a^14*b^
9*d^5 + 32*A^4*a^18*b^5*d^5) - (A*b^7*(384*A^3*a^15*b^9*d^6 - 32*A^3*a^19*b^5*d^6 - 32*A^3*a^21*b^3*d^6 + (A*b
^7*(tan(c + d*x)^(1/2)*(512*A^2*a^15*b^10*d^7 + 448*A^2*a^19*b^6*d^7 - 128*A^2*a^21*b^4*d^7 - 64*A^2*a^23*b^2*
d^7) - (A*b^7*(512*A*a^16*b^10*d^8 + 512*A*a^18*b^8*d^8 - 384*A*a^20*b^6*d^8 - 256*A*a^22*b^4*d^8 + 128*A*a^24
*b^2*d^8 + (A*b^7*tan(c + d*x)^(1/2)*(512*a^18*b^9*d^9 + 512*a^20*b^7*d^9 - 512*a^22*b^5*d^9 - 512*a^24*b^3*d^
9))/(- a^5*b^11*d^2 - 2*a^7*b^9*d^2 - a^9*b^7*d^2)^(1/2)))/(- a^5*b^11*d^2 - 2*a^7*b^9*d^2 - a^9*b^7*d^2)^(1/2
)))/(- a^5*b^11*d^2 - 2*a^7*b^9*d^2 - a^9*b^7*d^2)^(1/2)))/(- a^5*b^11*d^2 - 2*a^7*b^9*d^2 - a^9*b^7*d^2)^(1/2
))*1i)/(- a^5*b^11*d^2 - 2*a^7*b^9*d^2 - a^9*b^7*d^2)^(1/2))/(64*A^5*a^14*b^8*d^4 - (A*b^7*(tan(c + d*x)^(1/2)
*(64*A^4*a^14*b^9*d^5 + 32*A^4*a^18*b^5*d^5) - (A*b^7*(32*A^3*a^19*b^5*d^6 - 384*A^3*a^15*b^9*d^6 + 32*A^3*a^2
1*b^3*d^6 + (A*b^7*(tan(c + d*x)^(1/2)*(512*A^2*a^15*b^10*d^7 + 448*A^2*a^19*b^6*d^7 - 128*A^2*a^21*b^4*d^7 -
64*A^2*a^23*b^2*d^7) + (A*b^7*(512*A*a^16*b^10*d^8 + 512*A*a^18*b^8*d^8 - 384*A*a^20*b^6*d^8 - 256*A*a^22*b^4*
d^8 + 128*A*a^24*b^2*d^8 - (A*b^7*tan(c + d*x)^(1/2)*(512*a^18*b^9*d^9 + 512*a^20*b^7*d^9 - 512*a^22*b^5*d^9 -
512*a^24*b^3*d^9))/(- a^5*b^11*d^2 - 2*a^7*b^9*d^2 - a^9*b^7*d^2)^(1/2)))/(- a^5*b^11*d^2 - 2*a^7*b^9*d^2 - a
^9*b^7*d^2)^(1/2)))/(- a^5*b^11*d^2 - 2*a^7*b^9*d^2 - a^9*b^7*d^2)^(1/2)))/(- a^5*b^11*d^2 - 2*a^7*b^9*d^2 - a
^9*b^7*d^2)^(1/2)))/(- a^5*b^11*d^2 - 2*a^7*b^9*d^2 - a^9*b^7*d^2)^(1/2) + (A*b^7*(tan(c + d*x)^(1/2)*(64*A^4*
a^14*b^9*d^5 + 32*A^4*a^18*b^5*d^5) - (A*b^7*(384*A^3*a^15*b^9*d^6 - 32*A^3*a^19*b^5*d^6 - 32*A^3*a^21*b^3*d^6
+ (A*b^7*(tan(c + d*x)^(1/2)*(512*A^2*a^15*b^10*d^7 + 448*A^2*a^19*b^6*d^7 - 128*A^2*a^21*b^4*d^7 - 64*A^2*a^
23*b^2*d^7) - (A*b^7*(512*A*a^16*b^10*d^8 + 512*A*a^18*b^8*d^8 - 384*A*a^20*b^6*d^8 - 256*A*a^22*b^4*d^8 + 128
*A*a^24*b^2*d^8 + (A*b^7*tan(c + d*x)^(1/2)*(512*a^18*b^9*d^9 + 512*a^20*b^7*d^9 - 512*a^22*b^5*d^9 - 512*a^24
*b^3*d^9))/(- a^5*b^11*d^2 - 2*a^7*b^9*d^2 - a^9*b^7*d^2)^(1/2)))/(- a^5*b^11*d^2 - 2*a^7*b^9*d^2 - a^9*b^7*d^
2)^(1/2)))/(- a^5*b^11*d^2 - 2*a^7*b^9*d^2 - a^9*b^7*d^2)^(1/2)))/(- a^5*b^11*d^2 - 2*a^7*b^9*d^2 - a^9*b^7*d^
2)^(1/2)))/(- a^5*b^11*d^2 - 2*a^7*b^9*d^2 - a^9*b^7*d^2)^(1/2)))*2i)/(- a^5*b^11*d^2 - 2*a^7*b^9*d^2 - a^9*b^
7*d^2)^(1/2) - (2*B)/(a*d*tan(c + d*x)^(1/2))