\(\int \frac {\tan ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx\) [398]

Optimal result

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

[Out]

-2*a^(5/2)*(A*b-B*a)*arctan(b^(1/2)*tan(d*x+c)^(1/2)/a^(1/2))/b^(5/2)/(a^2+b^2)/d-1/2*(a*(A-B)+b*(A+B))*arctan
(-1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)/d*2^(1/2)-1/2*(a*(A-B)+b*(A+B))*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))/(a^
2+b^2)/d*2^(1/2)+1/4*(b*(A-B)-a*(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*(b*(A
-B)-a*(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)*tan(d*x+c)^(1/2)/b^2/d+
2/3*B*tan(d*x+c)^(3/2)/b/d

Rubi [A] (verified)

Time = 1.41 (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 = {3688, 3728, 3734, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211} \[ \int \frac {\tan ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\frac {(a (A-B)+b (A+B)) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )}-\frac {(a (A-B)+b (A+B)) \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )}+\frac {(b (A-B)-a (A+B)) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}-\frac {(b (A-B)-a (A+B)) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}-\frac {2 a^{5/2} (A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{5/2} d \left (a^2+b^2\right )}+\frac {2 (A b-a B) \sqrt {\tan (c+d x)}}{b^2 d}+\frac {2 B \tan ^{\frac {3}{2}}(c+d x)}{3 b d} \]

[In]

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

[Out]

((a*(A - B) + b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - ((a*(A - B) + b*(A
+ B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - (2*a^(5/2)*(A*b - a*B)*ArcTan[(Sqrt[b]
*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(b^(5/2)*(a^2 + b^2)*d) + ((b*(A - B) - a*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c +
d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - ((b*(A - B) - a*(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*b - a*B)*Sqrt[Tan[c + d*x]])/(b^2*d) + (2*B*Tan[c + d*x]^
(3/2))/(3*b*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 3688

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*B*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x])^(n + 1)/(d*f
*(m + n))), x] + Dist[1/(d*(m + n)), Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^n*Simp[a^2*A*d*(m +
n) - b*B*(b*c*(m - 1) + a*d*(n + 1)) + d*(m + n)*(2*a*A*b + B*(a^2 - b^2))*Tan[e + f*x] - (b*B*(b*c - a*d)*(m
- 1) - b*(A*b + a*B)*d*(m + n))*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] && GtQ[m, 1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) &&
!(IGtQ[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 3728

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*
tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*(a + b*Tan[e + f*x])^m*((c + d
*Tan[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c +
d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C*(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f
*x] - (C*m*(b*c - a*d) - b*B*d*(m + n + 1))*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] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !Intege
rQ[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 B \tan ^{\frac {3}{2}}(c+d x)}{3 b d}+\frac {2 \int \frac {\sqrt {\tan (c+d x)} \left (-\frac {3 a B}{2}-\frac {3}{2} b B \tan (c+d x)+\frac {3}{2} (A b-a B) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{3 b} \\ & = \frac {2 (A b-a B) \sqrt {\tan (c+d x)}}{b^2 d}+\frac {2 B \tan ^{\frac {3}{2}}(c+d x)}{3 b d}+\frac {4 \int \frac {-\frac {3}{4} a (A b-a B)-\frac {3}{4} A b^2 \tan (c+d x)-\frac {3}{4} \left (a A b-a^2 B+b^2 B\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{3 b^2} \\ & = \frac {2 (A b-a B) \sqrt {\tan (c+d x)}}{b^2 d}+\frac {2 B \tan ^{\frac {3}{2}}(c+d x)}{3 b d}+\frac {4 \int \frac {-\frac {3}{4} b^2 (A b-a B)-\frac {3}{4} b^2 (a A+b B) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{3 b^2 \left (a^2+b^2\right )}-\frac {\left (a^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}{b^2 \left (a^2+b^2\right )} \\ & = \frac {2 (A b-a B) \sqrt {\tan (c+d x)}}{b^2 d}+\frac {2 B \tan ^{\frac {3}{2}}(c+d x)}{3 b d}+\frac {8 \text {Subst}\left (\int \frac {-\frac {3}{4} b^2 (A b-a B)-\frac {3}{4} b^2 (a A+b B) x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{3 b^2 \left (a^2+b^2\right ) d}-\frac {\left (a^3 (A b-a B)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{b^2 \left (a^2+b^2\right ) d} \\ & = \frac {2 (A b-a B) \sqrt {\tan (c+d x)}}{b^2 d}+\frac {2 B \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\left (2 a^3 (A b-a B)\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{b^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 a^{5/2} (A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{5/2} \left (a^2+b^2\right ) d}+\frac {2 (A b-a B) \sqrt {\tan (c+d x)}}{b^2 d}+\frac {2 B \tan ^{\frac {3}{2}}(c+d x)}{3 b d}+\frac {(b (A-B)-a (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 {(b (A-B)-a (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 {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 {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d} \\ & = -\frac {2 a^{5/2} (A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{5/2} \left (a^2+b^2\right ) d}+\frac {(b (A-B)-a (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)) \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 b-a B) \sqrt {\tan (c+d x)}}{b^2 d}+\frac {2 B \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {(a (A-B)+b (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 {(a (A-B)+b (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 {(a (A-B)+b (A+B)) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a (A-B)+b (A+B)) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {2 a^{5/2} (A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{5/2} \left (a^2+b^2\right ) d}+\frac {(b (A-B)-a (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)) \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 b-a B) \sqrt {\tan (c+d x)}}{b^2 d}+\frac {2 B \tan ^{\frac {3}{2}}(c+d x)}{3 b d} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.23 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.58 \[ \int \frac {\tan ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\frac {-3 \sqrt [4]{-1} (a+i b) b^{5/2} (i A+B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+6 a^{5/2} (-A b+a B) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )+3 \sqrt [4]{-1} b^{5/2} (i a+b) (A+i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+2 \sqrt {b} \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (3 A b-3 a B+b B \tan (c+d x))}{3 b^{5/2} \left (a^2+b^2\right ) d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.90

method	result	size
derivativedivides	\(\frac {\frac {\frac {2 B b \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+2 A b \left (\sqrt {\tan }\left (d x +c \right )\right )-2 B a \left (\sqrt {\tan }\left (d x +c \right )\right )}{b^{2}}-\frac {2 a^{3} \left (A b -B a \right ) \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{b^{2} \left (a^{2}+b^{2}\right ) \sqrt {a b}}+\frac {\frac {\left (-A b +B a \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-a A -B b \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{a^{2}+b^{2}}}{d}\)	\(292\)
default	\(\frac {\frac {\frac {2 B b \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+2 A b \left (\sqrt {\tan }\left (d x +c \right )\right )-2 B a \left (\sqrt {\tan }\left (d x +c \right )\right )}{b^{2}}-\frac {2 a^{3} \left (A b -B a \right ) \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{b^{2} \left (a^{2}+b^{2}\right ) \sqrt {a b}}+\frac {\frac {\left (-A b +B a \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-a A -B b \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{a^{2}+b^{2}}}{d}\)	\(292\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3067 vs. \(2 (281) = 562\).

Time = 21.87 (sec) , antiderivative size = 6160, normalized size of antiderivative = 18.95 \[ \int \frac {\tan ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.79 \[ \int \frac {\tan ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\frac {\frac {24 \, {\left (B a^{4} - A a^{3} b\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{2} b^{2} + b^{4}\right )} \sqrt {a b}} - \frac {3 \, {\left (2 \, \sqrt {2} {\left ({\left (A - B\right )} a + {\left (A + B\right )} b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left ({\left (A - B\right )} a + {\left (A + B\right )} b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - \sqrt {2} {\left ({\left (A + B\right )} a - {\left (A - B\right )} b\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \sqrt {2} {\left ({\left (A + B\right )} a - {\left (A - B\right )} b\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )}}{a^{2} + b^{2}} + \frac {8 \, {\left (B b \tan \left (d x + c\right )^{\frac {3}{2}} - 3 \, {\left (B a - A b\right )} \sqrt {\tan \left (d x + c\right )}\right )}}{b^{2}}}{12 \, d} \]

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 14.94 (sec) , antiderivative size = 16441, normalized size of antiderivative = 50.59 \[ \int \frac {\tan ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

atan(((((((32*(4*A*a*b^8*d^4 + 8*A*a^3*b^6*d^4 + 4*A*a^5*b^4*d^4))/(b*d^5) - (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^10*d^4 + 16*a^2*b^8*d^4 - 16*a^4*b^6*d^4 - 16*a^6*b^4*d^4))/(b*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^5*d^2 + 2*A^2*a^5*b^3*d^2 - 14*A^2*a*b^7*d^2 + 16*A
^2*a^7*b*d^2))/(b*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*(A^3*a^2*b^5*d^2 - 15*A^3*a^4*b^3*d^2 + 12*A^3*
a^6*b*d^2))/(b*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) + (32*tan(c + d*x)^(1/2)*(2*A^4*a^6 - A^4*b^6))/(b*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*(4*A*a*b^8*d^4 + 8*A*a^3*b^6*d^4 + 4*A*a^5*b^4*d^4))/(b*d^5) +
(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^10*d^4 + 16*a^2*b^8*d^4 - 16*a^4*b^6*d^4 - 16*a^
6*b^4*d^4))/(b*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^5*d^2 + 2*A^2*a^5*
b^3*d^2 - 14*A^2*a*b^7*d^2 + 16*A^2*a^7*b*d^2))/(b*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*(A^3*a^2*b^5*d
^2 - 15*A^3*a^4*b^3*d^2 + 12*A^3*a^6*b*d^2))/(b*d^5))*(((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) - (32*tan(c + d*x)^(1/2
)*(2*A^4*a^6 - A^4*b^6))/(b*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*(4*A*a*b^8*d^4 + 8*A*a^3*b^6*d
^4 + 4*A*a^5*b^4*d^4))/(b*d^5) - (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^10*d^4 + 16*a^2
*b^8*d^4 - 16*a^4*b^6*d^4 - 16*a^6*b^4*d^4))/(b*d^4))*(((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) + (32*tan(c + d*x)^(1/2
)*(4*A^2*a^3*b^5*d^2 + 2*A^2*a^5*b^3*d^2 - 14*A^2*a*b^7*d^2 + 16*A^2*a^7*b*d^2))/(b*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*(A^3*a^2*b^5*d^2 - 15*A^3*a^4*b^3*d^2 + 12*A^3*a^6*b*d^2))/(b*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) + (32*tan(c + d*x)^(1/2)*(2*A^4*a^6 - A^4*b^6))/(b*d^4))*(((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) + (((((32*
(4*A*a*b^8*d^4 + 8*A*a^3*b^6*d^4 + 4*A*a^5*b^4*d^4))/(b*d^5) + (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^10*d^4 + 16*a^2*b^8*d^4 - 16*a^4*b^6*d^4 - 16*a^6*b^4*d^4))/(b*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^5*d^2 + 2*A^2*a^5*b^3*d^2 - 14*A^2*a*b^7*d^2 + 16*A^2*a^7*b*d^2))
/(b*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*(A^3*a^2*b^5*d^2 - 15*A^3*a^4*b^3*d^2 + 12*A^3*a^6*b*d^2))/(b
*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) - (32*tan(c + d*x)^(1/2)*(2*A^4*a^6 - A^4*b^6))/(b*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^5*a^5 - A^5*a^3*b^2))/(b*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)*2i - atan(((((((
32*(12*B*a^2*b^9*d^4 + 24*B*a^4*b^7*d^4 + 12*B*a^6*b^5*d^4))/(b^3*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^12*d^4 + 16*a^2*b^10*d^4 - 16*a^4*b^8*d^4 - 16*a^6*b^6*d^4))/(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*tan(c + d*x)^(1/2)*(4*B^2*a^3*b^7*d^2 + 2*B^2*a^5*b^5*d^2 - 14*B^2*a*b^9*d^2 - 16*B^
2*a^9*b*d^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*(4*B^3*a^9*d^2 + B^3*a^3*b^6*d^2 + 16*B^3*a^5*
b^4*d^2 - 16*B^3*a^7*b^2*d^2 + B^3*a*b^8*d^2))/(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) - (32*tan(c + d*x)^
(1/2)*(2*B^4*a^8 + B^4*b^8))/(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*(12*B*a^2*b^9*d^4 + 24*B
*a^4*b^7*d^4 + 12*B*a^6*b^5*d^4))/(b^3*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^12
*d^4 + 16*a^2*b^10*d^4 - 16*a^4*b^8*d^4 - 16*a^6*b^6*d^4))/(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*t
an(c + d*x)^(1/2)*(4*B^2*a^3*b^7*d^2 + 2*B^2*a^5*b^5*d^2 - 14*B^2*a*b^9*d^2 - 16*B^2*a^9*b*d^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*(4*B^3*a^9*d^2 + B^3*a^3*b^6*d^2 + 16*B^3*a^5*b^4*d^2 - 16*B^3*a^7*b^2*d^
2 + B^3*a*b^8*d^2))/(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) + (32*tan(c + d*x)^(1/2)*(2*B^4*a^8 + B^4*b^8)
)/(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)/(1
6*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*1i)/((((((32*(12*B*a^2*b^9*d^4 + 24*B*a^4*b^7*d^4 + 12*B*a^6*b^5
*d^4))/(b^3*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^12*d^4 + 16*a^2*b^10*d^4 - 16
*a^4*b^8*d^4 - 16*a^6*b^6*d^4))/(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*tan(c + d*x)^(1/2)*(4*B^2*a^
3*b^7*d^2 + 2*B^2*a^5*b^5*d^2 - 14*B^2*a*b^9*d^2 - 16*B^2*a^9*b*d^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*(4*B^3*a^9*d^2 + B^3*a^3*b^6*d^2 + 16*B^3*a^5*b^4*d^2 - 16*B^3*a^7*b^2*d^2 + B^3*a*b^8*d^2))/(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) - (32*tan(c + d*x)^(1/2)*(2*B^4*a^8 + B^4*b^8))/(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*(12*B*a^2*b^9*d^4 + 24*B*a^4*b^7*d^4 + 12*B*a^6*b^5*d^4))/(b^3*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^12*d^4 + 16*a^2*b^10*d^4 - 16*a^4*b^8*d^4 - 16*a^6*b^6*d^4)
)/(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)/(1
6*(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*B^2*a^3*b^7*d^2 + 2*B^2*a^5*b^5*d^2
- 14*B^2*a*b^9*d^2 - 16*B^2*a^9*b*d^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*(4*B^3*a^9*d^2 + B^3
*a^3*b^6*d^2 + 16*B^3*a^5*b^4*d^2 - 16*B^3*a^7*b^2*d^2 + B^3*a*b^8*d^2))/(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) + (32*tan(c + d*x)^(1/2)*(2*B^4*a^8 + B^4*b^8))/(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^6*b - B^5*a^4*b^3))/(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*(12*B*a^2*b^9*d^4 + 24*B
*a^4*b^7*d^4 + 12*B*a^6*b^5*d^4))/(b^3*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^1
2*d^4 + 16*a^2*b^10*d^4 - 16*a^4*b^8*d^4 - 16*a^6*b^6*d^4))/(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
*tan(c + d*x)^(1/2)*(4*B^2*a^3*b^7*d^2 + 2*B^2*a^5*b^5*d^2 - 14*B^2*a*b^9*d^2 - 16*B^2*a^9*b*d^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*(4*B^3*a^9*d^2 + B^3*a^3*b^6*d^2 + 16*B^3*a^5*b^4*d^2 - 16*B^3*a^7*b^2
*d^2 + B^3*a*b^8*d^2))/(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) - (32*tan(c + d*x)^(1/2)*(2*B^4*a^8 + B^4*
b^8))/(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*(12*B*a^2*b^9*d^4 + 24*B*a^4*b^7*d^4 + 12*B*a^
6*b^5*d^4))/(b^3*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^12*d^4 + 16*a^2*b^10*d^
4 - 16*a^4*b^8*d^4 - 16*a^6*b^6*d^4))/(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*tan(c + d*x)^(1/2)*(4
*B^2*a^3*b^7*d^2 + 2*B^2*a^5*b^5*d^2 - 14*B^2*a*b^9*d^2 - 16*B^2*a^9*b*d^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*(4*B^3*a^9*d^2 + B^3*a^3*b^6*d^2 + 16*B^3*a^5*b^4*d^2 - 16*B^3*a^7*b^2*d^2 + B^3*a*b^8*d^2))
/(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)/(1
6*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(2*B^4*a^8 + B^4*b^8))/(b^3*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)*1i)/((((((32*(12*B*a^2*b^9*d^4 + 24*B*a^4*b^7*d^4 + 12*B*a^6*b^5*d^4))/(b^3*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^12*d^4 + 16*a^2*b^10*d^4 - 16*a^4*b^8*d^4 - 1
6*a^6*b^6*d^4))/(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*tan(c + d*x)^(1/2)*(4*B^2*a^3*b^7*d^2 + 2*B
^2*a^5*b^5*d^2 - 14*B^2*a*b^9*d^2 - 16*B^2*a^9*b*d^2))/(b^3*d^4))*(-((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) - (32*(4*B
^3*a^9*d^2 + B^3*a^3*b^6*d^2 + 16*B^3*a^5*b^4*d^2 - 16*B^3*a^7*b^2*d^2 + B^3*a*b^8*d^2))/(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) - (32*tan(c + d*x)^(1/2)*(2*B^4*a^8 + B^4*b^8))/(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*(12*B*a^2*b^9*d^4 + 24*B*a^4*b^7*d^4 + 12*B*a^6*b^5*d^4))/(b^3*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^12*d^4 + 16*a^2*b^10*d^4 - 16*a^4*b^8*d^4 - 16*a^6*b^6*d^4))/(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*tan(c + d*x)^(1/2)*(4*B^2*a^3*b^7*d^2 + 2*B^2*a^5*b^5*d^2 - 14*B^2*a
*b^9*d^2 - 16*B^2*a^9*b*d^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*(4*B^3*a^9*d^2 + B^3*a^3*b^6*
d^2 + 16*B^3*a^5*b^4*d^2 - 16*B^3*a^7*b^2*d^2 + B^3*a*b^8*d^2))/(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) +
(32*tan(c + d*x)^(1/2)*(2*B^4*a^8 + B^4*b^8))/(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^6*b -
B^5*a^4*b^3))/(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*(4*A*a*b^8*d^4 + 8*A*a^3*b^6*d
^4 + 4*A*a^5*b^4*d^4))/(b*d^5) - (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^10*d^4 + 16*a^
2*b^8*d^4 - 16*a^4*b^6*d^4 - 16*a^6*b^4*d^4))/(b*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^5*d^2 + 2*A^2*a^5*b^3*d^2 - 14*A^2*a*b^7*d^2 + 16*A^2*a^7*b*d^2))/(b*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*(A^3*a^2*b^5*d^2 - 15*A^3*a^4*b^3*d^2 + 12*A^3*a^6*b*d^2))/(b*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) + (32*tan(c + d*x)^(1/2)*(2*A^4*a^6 - A^4*b^6))/(b*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*(4*A*a*b^8*d^4 + 8*A*a^3*b^6*d^4 + 4*A*a^5*b^4*d^4))/(b*d^5) + (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^10*d^4 + 16*a^2*b^8*d^4 - 16*a^4*b^6*d^4 - 16*a^6*b^4*d^4))/(b*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^5*d^2 + 2*A^2*a^5*b^3*d^2 - 14*A^2*a*b^7*d^2 + 16*A^2*a
^7*b*d^2))/(b*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*(A^3*a^2*b^5*d^2 - 15*A^3*a^4*b^3*d^2 + 12*A^3*a^6
*b*d^2))/(b*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) - (32*tan(c + d*x)^(1/2)*(2*A^4*a^6 - A^4*b^6))/(b*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*(4*A*a*b^8*d^4 + 8*A*a^3*b^6*d^4 + 4*A*a^5*b^4*d^4))/(b*d^5) - (
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^10*d^4 + 16*a^2*b^8*d^4 - 16*a^4*b^6*d^4 - 16*a^
6*b^4*d^4))/(b*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^5*d^2 + 2*A^2*a^5
*b^3*d^2 - 14*A^2*a*b^7*d^2 + 16*A^2*a^7*b*d^2))/(b*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*(A^3*a^2*b^5
*d^2 - 15*A^3*a^4*b^3*d^2 + 12*A^3*a^6*b*d^2))/(b*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) + (32*tan(c + d*x)^(
1/2)*(2*A^4*a^6 - A^4*b^6))/(b*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*(4*A*a*b^8*d^4 + 8*A*a^3*b^6*
d^4 + 4*A*a^5*b^4*d^4))/(b*d^5) + (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^10*d^4 + 16*a
^2*b^8*d^4 - 16*a^4*b^6*d^4 - 16*a^6*b^4*d^4))/(b*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^5*d^2 + 2*A^2*a^5*b^3*d^2 - 14*A^2*a*b^7*d^2 + 16*A^2*a^7*b*d^2))/(b*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*(A^3*a^2*b^5*d^2 - 15*A^3*a^4*b^3*d^2 + 12*A^3*a^6*b*d^2))/(b*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) - (32*tan(c + d*x)^(1/2)*(2*A^4*a^6 - A^4*b^6))/(b*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) - (6
4*(A^5*a^5 - A^5*a^3*b^2))/(b*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)*2i - (B*a^7*atan(((B*a^7*((32*tan(c + d
*x)^(1/2)*(2*B^4*a^8 + B^4*b^8))/(b^3*d^4) + (B*a^7*((32*(4*B^3*a^9*d^2 + B^3*a^3*b^6*d^2 + 16*B^3*a^5*b^4*d^2
- 16*B^3*a^7*b^2*d^2 + B^3*a*b^8*d^2))/(b^3*d^5) + (B*a^7*((32*tan(c + d*x)^(1/2)*(4*B^2*a^3*b^7*d^2 + 2*B^2*
a^5*b^5*d^2 - 14*B^2*a*b^9*d^2 - 16*B^2*a^9*b*d^2))/(b^3*d^4) - (B*a^7*((32*(12*B*a^2*b^9*d^4 + 24*B*a^4*b^7*d
^4 + 12*B*a^6*b^5*d^4))/(b^3*d^5) - (32*B*a^7*tan(c + d*x)^(1/2)*(16*b^12*d^4 + 16*a^2*b^10*d^4 - 16*a^4*b^8*d
^4 - 16*a^6*b^6*d^4))/(b^3*d^4*(- a^7*b^9*d^2 - 2*a^9*b^7*d^2 - a^11*b^5*d^2)^(1/2))))/(- a^7*b^9*d^2 - 2*a^9*
b^7*d^2 - a^11*b^5*d^2)^(1/2)))/(- a^7*b^9*d^2 - 2*a^9*b^7*d^2 - a^11*b^5*d^2)^(1/2)))/(- a^7*b^9*d^2 - 2*a^9*
b^7*d^2 - a^11*b^5*d^2)^(1/2))*1i)/(- a^7*b^9*d^2 - 2*a^9*b^7*d^2 - a^11*b^5*d^2)^(1/2) + (B*a^7*((32*tan(c +
d*x)^(1/2)*(2*B^4*a^8 + B^4*b^8))/(b^3*d^4) - (B*a^7*((32*(4*B^3*a^9*d^2 + B^3*a^3*b^6*d^2 + 16*B^3*a^5*b^4*d^
2 - 16*B^3*a^7*b^2*d^2 + B^3*a*b^8*d^2))/(b^3*d^5) - (B*a^7*((32*tan(c + d*x)^(1/2)*(4*B^2*a^3*b^7*d^2 + 2*B^2
*a^5*b^5*d^2 - 14*B^2*a*b^9*d^2 - 16*B^2*a^9*b*d^2))/(b^3*d^4) + (B*a^7*((32*(12*B*a^2*b^9*d^4 + 24*B*a^4*b^7*
d^4 + 12*B*a^6*b^5*d^4))/(b^3*d^5) + (32*B*a^7*tan(c + d*x)^(1/2)*(16*b^12*d^4 + 16*a^2*b^10*d^4 - 16*a^4*b^8*
d^4 - 16*a^6*b^6*d^4))/(b^3*d^4*(- a^7*b^9*d^2 - 2*a^9*b^7*d^2 - a^11*b^5*d^2)^(1/2))))/(- a^7*b^9*d^2 - 2*a^9
*b^7*d^2 - a^11*b^5*d^2)^(1/2)))/(- a^7*b^9*d^2 - 2*a^9*b^7*d^2 - a^11*b^5*d^2)^(1/2)))/(- a^7*b^9*d^2 - 2*a^9
*b^7*d^2 - a^11*b^5*d^2)^(1/2))*1i)/(- a^7*b^9*d^2 - 2*a^9*b^7*d^2 - a^11*b^5*d^2)^(1/2))/((64*(B^5*a^6*b - B^
5*a^4*b^3))/(b^3*d^5) + (B*a^7*((32*tan(c + d*x)^(1/2)*(2*B^4*a^8 + B^4*b^8))/(b^3*d^4) + (B*a^7*((32*(4*B^3*a
^9*d^2 + B^3*a^3*b^6*d^2 + 16*B^3*a^5*b^4*d^2 - 16*B^3*a^7*b^2*d^2 + B^3*a*b^8*d^2))/(b^3*d^5) + (B*a^7*((32*t
an(c + d*x)^(1/2)*(4*B^2*a^3*b^7*d^2 + 2*B^2*a^5*b^5*d^2 - 14*B^2*a*b^9*d^2 - 16*B^2*a^9*b*d^2))/(b^3*d^4) - (
B*a^7*((32*(12*B*a^2*b^9*d^4 + 24*B*a^4*b^7*d^4 + 12*B*a^6*b^5*d^4))/(b^3*d^5) - (32*B*a^7*tan(c + d*x)^(1/2)*
(16*b^12*d^4 + 16*a^2*b^10*d^4 - 16*a^4*b^8*d^4 - 16*a^6*b^6*d^4))/(b^3*d^4*(- a^7*b^9*d^2 - 2*a^9*b^7*d^2 - a
^11*b^5*d^2)^(1/2))))/(- a^7*b^9*d^2 - 2*a^9*b^7*d^2 - a^11*b^5*d^2)^(1/2)))/(- a^7*b^9*d^2 - 2*a^9*b^7*d^2 -
a^11*b^5*d^2)^(1/2)))/(- a^7*b^9*d^2 - 2*a^9*b^7*d^2 - a^11*b^5*d^2)^(1/2)))/(- a^7*b^9*d^2 - 2*a^9*b^7*d^2 -
a^11*b^5*d^2)^(1/2) - (B*a^7*((32*tan(c + d*x)^(1/2)*(2*B^4*a^8 + B^4*b^8))/(b^3*d^4) - (B*a^7*((32*(4*B^3*a^9
*d^2 + B^3*a^3*b^6*d^2 + 16*B^3*a^5*b^4*d^2 - 16*B^3*a^7*b^2*d^2 + B^3*a*b^8*d^2))/(b^3*d^5) - (B*a^7*((32*tan
(c + d*x)^(1/2)*(4*B^2*a^3*b^7*d^2 + 2*B^2*a^5*b^5*d^2 - 14*B^2*a*b^9*d^2 - 16*B^2*a^9*b*d^2))/(b^3*d^4) + (B*
a^7*((32*(12*B*a^2*b^9*d^4 + 24*B*a^4*b^7*d^4 + 12*B*a^6*b^5*d^4))/(b^3*d^5) + (32*B*a^7*tan(c + d*x)^(1/2)*(1
6*b^12*d^4 + 16*a^2*b^10*d^4 - 16*a^4*b^8*d^4 - 16*a^6*b^6*d^4))/(b^3*d^4*(- a^7*b^9*d^2 - 2*a^9*b^7*d^2 - a^1
1*b^5*d^2)^(1/2))))/(- a^7*b^9*d^2 - 2*a^9*b^7*d^2 - a^11*b^5*d^2)^(1/2)))/(- a^7*b^9*d^2 - 2*a^9*b^7*d^2 - a^
11*b^5*d^2)^(1/2)))/(- a^7*b^9*d^2 - 2*a^9*b^7*d^2 - a^11*b^5*d^2)^(1/2)))/(- a^7*b^9*d^2 - 2*a^9*b^7*d^2 - a^
11*b^5*d^2)^(1/2)))*2i)/(- a^7*b^9*d^2 - 2*a^9*b^7*d^2 - a^11*b^5*d^2)^(1/2) - (A*a^5*atan(((A*a^5*((A*a^5*((3
2*(A^3*a^2*b^5*d^2 - 15*A^3*a^4*b^3*d^2 + 12*A^3*a^6*b*d^2))/(b*d^5) + (A*a^5*((32*tan(c + d*x)^(1/2)*(4*A^2*a
^3*b^5*d^2 + 2*A^2*a^5*b^3*d^2 - 14*A^2*a*b^7*d^2 + 16*A^2*a^7*b*d^2))/(b*d^4) - (A*a^5*((32*(4*A*a*b^8*d^4 +
8*A*a^3*b^6*d^4 + 4*A*a^5*b^4*d^4))/(b*d^5) + (32*A*a^5*tan(c + d*x)^(1/2)*(16*b^10*d^4 + 16*a^2*b^8*d^4 - 16*
a^4*b^6*d^4 - 16*a^6*b^4*d^4))/(b*d^4*(- a^5*b^7*d^2 - 2*a^7*b^5*d^2 - a^9*b^3*d^2)^(1/2))))/(- a^5*b^7*d^2 -
2*a^7*b^5*d^2 - a^9*b^3*d^2)^(1/2)))/(- a^5*b^7*d^2 - 2*a^7*b^5*d^2 - a^9*b^3*d^2)^(1/2)))/(- a^5*b^7*d^2 - 2*
a^7*b^5*d^2 - a^9*b^3*d^2)^(1/2) + (32*tan(c + d*x)^(1/2)*(2*A^4*a^6 - A^4*b^6))/(b*d^4))*1i)/(- a^5*b^7*d^2 -
2*a^7*b^5*d^2 - a^9*b^3*d^2)^(1/2) - (A*a^5*((A*a^5*((32*(A^3*a^2*b^5*d^2 - 15*A^3*a^4*b^3*d^2 + 12*A^3*a^6*b
*d^2))/(b*d^5) - (A*a^5*((32*tan(c + d*x)^(1/2)*(4*A^2*a^3*b^5*d^2 + 2*A^2*a^5*b^3*d^2 - 14*A^2*a*b^7*d^2 + 16
*A^2*a^7*b*d^2))/(b*d^4) + (A*a^5*((32*(4*A*a*b^8*d^4 + 8*A*a^3*b^6*d^4 + 4*A*a^5*b^4*d^4))/(b*d^5) - (32*A*a^
5*tan(c + d*x)^(1/2)*(16*b^10*d^4 + 16*a^2*b^8*d^4 - 16*a^4*b^6*d^4 - 16*a^6*b^4*d^4))/(b*d^4*(- a^5*b^7*d^2 -
2*a^7*b^5*d^2 - a^9*b^3*d^2)^(1/2))))/(- a^5*b^7*d^2 - 2*a^7*b^5*d^2 - a^9*b^3*d^2)^(1/2)))/(- a^5*b^7*d^2 -
2*a^7*b^5*d^2 - a^9*b^3*d^2)^(1/2)))/(- a^5*b^7*d^2 - 2*a^7*b^5*d^2 - a^9*b^3*d^2)^(1/2) - (32*tan(c + d*x)^(1
/2)*(2*A^4*a^6 - A^4*b^6))/(b*d^4))*1i)/(- a^5*b^7*d^2 - 2*a^7*b^5*d^2 - a^9*b^3*d^2)^(1/2))/((64*(A^5*a^5 - A
^5*a^3*b^2))/(b*d^5) + (A*a^5*((A*a^5*((32*(A^3*a^2*b^5*d^2 - 15*A^3*a^4*b^3*d^2 + 12*A^3*a^6*b*d^2))/(b*d^5)
+ (A*a^5*((32*tan(c + d*x)^(1/2)*(4*A^2*a^3*b^5*d^2 + 2*A^2*a^5*b^3*d^2 - 14*A^2*a*b^7*d^2 + 16*A^2*a^7*b*d^2)
)/(b*d^4) - (A*a^5*((32*(4*A*a*b^8*d^4 + 8*A*a^3*b^6*d^4 + 4*A*a^5*b^4*d^4))/(b*d^5) + (32*A*a^5*tan(c + d*x)^
(1/2)*(16*b^10*d^4 + 16*a^2*b^8*d^4 - 16*a^4*b^6*d^4 - 16*a^6*b^4*d^4))/(b*d^4*(- a^5*b^7*d^2 - 2*a^7*b^5*d^2
- a^9*b^3*d^2)^(1/2))))/(- a^5*b^7*d^2 - 2*a^7*b^5*d^2 - a^9*b^3*d^2)^(1/2)))/(- a^5*b^7*d^2 - 2*a^7*b^5*d^2 -
a^9*b^3*d^2)^(1/2)))/(- a^5*b^7*d^2 - 2*a^7*b^5*d^2 - a^9*b^3*d^2)^(1/2) + (32*tan(c + d*x)^(1/2)*(2*A^4*a^6
- A^4*b^6))/(b*d^4)))/(- a^5*b^7*d^2 - 2*a^7*b^5*d^2 - a^9*b^3*d^2)^(1/2) + (A*a^5*((A*a^5*((32*(A^3*a^2*b^5*d
^2 - 15*A^3*a^4*b^3*d^2 + 12*A^3*a^6*b*d^2))/(b*d^5) - (A*a^5*((32*tan(c + d*x)^(1/2)*(4*A^2*a^3*b^5*d^2 + 2*A
^2*a^5*b^3*d^2 - 14*A^2*a*b^7*d^2 + 16*A^2*a^7*b*d^2))/(b*d^4) + (A*a^5*((32*(4*A*a*b^8*d^4 + 8*A*a^3*b^6*d^4
+ 4*A*a^5*b^4*d^4))/(b*d^5) - (32*A*a^5*tan(c + d*x)^(1/2)*(16*b^10*d^4 + 16*a^2*b^8*d^4 - 16*a^4*b^6*d^4 - 16
*a^6*b^4*d^4))/(b*d^4*(- a^5*b^7*d^2 - 2*a^7*b^5*d^2 - a^9*b^3*d^2)^(1/2))))/(- a^5*b^7*d^2 - 2*a^7*b^5*d^2 -
a^9*b^3*d^2)^(1/2)))/(- a^5*b^7*d^2 - 2*a^7*b^5*d^2 - a^9*b^3*d^2)^(1/2)))/(- a^5*b^7*d^2 - 2*a^7*b^5*d^2 - a^
9*b^3*d^2)^(1/2) - (32*tan(c + d*x)^(1/2)*(2*A^4*a^6 - A^4*b^6))/(b*d^4)))/(- a^5*b^7*d^2 - 2*a^7*b^5*d^2 - a^
9*b^3*d^2)^(1/2)))*2i)/(- a^5*b^7*d^2 - 2*a^7*b^5*d^2 - a^9*b^3*d^2)^(1/2) + (2*A*tan(c + d*x)^(1/2))/(b*d) +
(2*B*tan(c + d*x)^(3/2))/(3*b*d) - (2*B*a*tan(c + d*x)^(1/2))/(b^2*d)