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

Optimal. Leaf size=156 \[ \frac {B \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {B \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d}-\frac {2 B}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {B \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {B \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d} \]

[Out]

-1/2*B*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))/d*2^(1/2)-1/2*B*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))/d*2^(1/2)+1/4*B*
ln(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/d*2^(1/2)-1/4*B*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/d*2^(1/2)-
2/3*B/d/tan(d*x+c)^(3/2)

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Rubi [A]  time = 0.10, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {21, 3474, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac {B \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {B \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d}-\frac {2 B}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {B \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {B \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
 n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
 d*x, a + b*x])

Rule 204

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

Rule 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
 && FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 617

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 628

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 1162

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 1165

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 3474

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Tan[c + d*x])^(n + 1)/(b*d*(n + 1)), x] - Dist[
1/b^2, Int[(b*Tan[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1]

Rule 3476

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

Rubi steps

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

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Mathematica [C]  time = 0.04, size = 36, normalized size = 0.23 \[ -\frac {2 B \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};-\tan ^2(c+d x)\right )}{3 d \tan ^{\frac {3}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*B*Hypergeometric2F1[-3/4, 1, 1/4, -Tan[c + d*x]^2])/(3*d*Tan[c + d*x]^(3/2))

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fricas [B]  time = 0.84, size = 615, normalized size = 3.94 \[ \frac {8 \, B \sqrt {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )^{2} + 12 \, {\left (\sqrt {2} d \cos \left (d x + c\right )^{2} - \sqrt {2} d\right )} \left (\frac {B^{4}}{d^{4}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} B d^{3} \left (\frac {B^{4}}{d^{4}}\right )^{\frac {3}{4}} \sqrt {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right )}} - \sqrt {2} d^{3} \left (\frac {B^{4}}{d^{4}}\right )^{\frac {3}{4}} \sqrt {\frac {\sqrt {2} B d \left (\frac {B^{4}}{d^{4}}\right )^{\frac {1}{4}} \sqrt {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) + d^{2} \sqrt {\frac {B^{4}}{d^{4}}} \cos \left (d x + c\right ) + B^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} + B^{4}}{B^{4}}\right ) + 12 \, {\left (\sqrt {2} d \cos \left (d x + c\right )^{2} - \sqrt {2} d\right )} \left (\frac {B^{4}}{d^{4}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} B d^{3} \left (\frac {B^{4}}{d^{4}}\right )^{\frac {3}{4}} \sqrt {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right )}} - \sqrt {2} d^{3} \left (\frac {B^{4}}{d^{4}}\right )^{\frac {3}{4}} \sqrt {-\frac {\sqrt {2} B d \left (\frac {B^{4}}{d^{4}}\right )^{\frac {1}{4}} \sqrt {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) - d^{2} \sqrt {\frac {B^{4}}{d^{4}}} \cos \left (d x + c\right ) - B^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} - B^{4}}{B^{4}}\right ) - 3 \, {\left (\sqrt {2} d \cos \left (d x + c\right )^{2} - \sqrt {2} d\right )} \left (\frac {B^{4}}{d^{4}}\right )^{\frac {1}{4}} \log \left (\frac {\sqrt {2} B d \left (\frac {B^{4}}{d^{4}}\right )^{\frac {1}{4}} \sqrt {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) + d^{2} \sqrt {\frac {B^{4}}{d^{4}}} \cos \left (d x + c\right ) + B^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right ) + 3 \, {\left (\sqrt {2} d \cos \left (d x + c\right )^{2} - \sqrt {2} d\right )} \left (\frac {B^{4}}{d^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {\sqrt {2} B d \left (\frac {B^{4}}{d^{4}}\right )^{\frac {1}{4}} \sqrt {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) - d^{2} \sqrt {\frac {B^{4}}{d^{4}}} \cos \left (d x + c\right ) - B^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )}{12 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(8*B*sqrt(sin(d*x + c)/cos(d*x + c))*cos(d*x + c)^2 + 12*(sqrt(2)*d*cos(d*x + c)^2 - sqrt(2)*d)*(B^4/d^4)
^(1/4)*arctan(-(sqrt(2)*B*d^3*(B^4/d^4)^(3/4)*sqrt(sin(d*x + c)/cos(d*x + c)) - sqrt(2)*d^3*(B^4/d^4)^(3/4)*sq
rt((sqrt(2)*B*d*(B^4/d^4)^(1/4)*sqrt(sin(d*x + c)/cos(d*x + c))*cos(d*x + c) + d^2*sqrt(B^4/d^4)*cos(d*x + c)
+ B^2*sin(d*x + c))/cos(d*x + c)) + B^4)/B^4) + 12*(sqrt(2)*d*cos(d*x + c)^2 - sqrt(2)*d)*(B^4/d^4)^(1/4)*arct
an(-(sqrt(2)*B*d^3*(B^4/d^4)^(3/4)*sqrt(sin(d*x + c)/cos(d*x + c)) - sqrt(2)*d^3*(B^4/d^4)^(3/4)*sqrt(-(sqrt(2
)*B*d*(B^4/d^4)^(1/4)*sqrt(sin(d*x + c)/cos(d*x + c))*cos(d*x + c) - d^2*sqrt(B^4/d^4)*cos(d*x + c) - B^2*sin(
d*x + c))/cos(d*x + c)) - B^4)/B^4) - 3*(sqrt(2)*d*cos(d*x + c)^2 - sqrt(2)*d)*(B^4/d^4)^(1/4)*log((sqrt(2)*B*
d*(B^4/d^4)^(1/4)*sqrt(sin(d*x + c)/cos(d*x + c))*cos(d*x + c) + d^2*sqrt(B^4/d^4)*cos(d*x + c) + B^2*sin(d*x
+ c))/cos(d*x + c)) + 3*(sqrt(2)*d*cos(d*x + c)^2 - sqrt(2)*d)*(B^4/d^4)^(1/4)*log(-(sqrt(2)*B*d*(B^4/d^4)^(1/
4)*sqrt(sin(d*x + c)/cos(d*x + c))*cos(d*x + c) - d^2*sqrt(B^4/d^4)*cos(d*x + c) - B^2*sin(d*x + c))/cos(d*x +
 c)))/(d*cos(d*x + c)^2 - d)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B b \tan \left (d x + c\right ) + B a}{{\left (b \tan \left (d x + c\right ) + a\right )} \tan \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.14, size = 118, normalized size = 0.76 \[ -\frac {2 B}{3 d \tan \left (d x +c \right )^{\frac {3}{2}}}-\frac {B \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}}{2 d}-\frac {B \sqrt {2}\, \ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )}{4 d}-\frac {B \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}}{2 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.90, size = 124, normalized size = 0.79 \[ -\frac {6 \, \sqrt {2} B \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 6 \, \sqrt {2} B \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 3 \, \sqrt {2} B \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - 3 \, \sqrt {2} B \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \frac {8 \, B}{\tan \left (d x + c\right )^{\frac {3}{2}}}}{12 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(6*sqrt(2)*B*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 6*sqrt(2)*B*arctan(-1/2*sqrt(2)*(sqr
t(2) - 2*sqrt(tan(d*x + c)))) + 3*sqrt(2)*B*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) - 3*sqrt(2)*B*l
og(-sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) + 8*B/tan(d*x + c)^(3/2))/d

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mupad [B]  time = 11.59, size = 16545, normalized size = 106.06 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*a + B*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*B^4*a^9*b^9*d^5 + 32*B^4*a^13*b^5*d^5) - (-((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^
4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*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^10*d^7 + 448*B^2*a^12*b^6*d^7 - 128*B^2*a^14*b^4*d^7 - 64*B^2*a^16*b^2*d^
7) - (-((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*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^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4
+ 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*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^10*d^8 - 512*B*a^10*b^8*d^
8 + 384*B*a^12*b^6*d^8 + 256*B*a^14*b^4*d^8 - 128*B*a^16*b^2*d^8))*(-((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^
4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - 3
84*B^3*a^9*b^9*d^6 + 32*B^3*a^13*b^5*d^6 + 32*B^3*a^15*b^3*d^6))*(-((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4
+ 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*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*B^4*a^9*b^9*d^5 + 32*B^4*a^13*b^5*d^5) - (-((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4
+ 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*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^10*d^7 + 448*B^2*a^12*b^6*d^7 - 128*B^2*a^14*b^4*d^7 - 64*B^2*a^16*b^2*d^7) - (
-((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*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^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b
^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*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^10*d^8 + 512*B*a^10*b^8*d^8 - 38
4*B*a^12*b^6*d^8 - 256*B*a^14*b^4*d^8 + 128*B*a^16*b^2*d^8))*(-((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16
*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + 384*B^3
*a^9*b^9*d^6 - 32*B^3*a^13*b^5*d^6 - 32*B^3*a^15*b^3*d^6))*(-((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b
^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*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*B^4*a^9*b^9*d^5 + 32*B^4*a^13*b^5*d^5) - (-((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b
^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*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^10*d^7 + 448*B^2*a^12*b^6*d^7 - 128*B^2*a^14*b^4*d^7 - 64*B^2*a^16*b^2*d^7) - (-((64*
B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*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^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4
 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*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^10*d^8 + 512*B*a^10*b^8*d^8 - 384*B*a^
12*b^6*d^8 - 256*B*a^14*b^4*d^8 + 128*B*a^16*b^2*d^8))*(-((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d
^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + 384*B^3*a^9*b
^9*d^6 - 32*B^3*a^13*b^5*d^6 - 32*B^3*a^15*b^3*d^6))*(-((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4
 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*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^9*b^9*d^5 + 32*B^4*a^13*b^5*d^5) - (-((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 +
32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*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^10*d^7 + 448*B^2*a^12*b^6*d^7 - 128*B^2*a^14*b^4*d^7 - 64*B^2*a^16*b^2*d^7) - (-((64*B^4*a^6*b
^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*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^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2
*b^2*d^4))^(1/2) - 8*B^2*a^3*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^1
1*b^7*d^9 - 512*a^13*b^5*d^9 - 512*a^15*b^3*d^9) - 512*B*a^8*b^10*d^8 - 512*B*a^10*b^8*d^8 + 384*B*a^12*b^6*d^
8 + 256*B*a^14*b^4*d^8 - 128*B*a^16*b^2*d^8))*(-((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a
^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - 384*B^3*a^9*b^9*d^6 +
32*B^3*a^13*b^5*d^6 + 32*B^3*a^15*b^3*d^6))*(-((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2
*b^2*d^4))^(1/2) - 8*B^2*a^3*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^10*b^8*d^4))*(-
((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*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*B^4*a^9*b^9*d^5 + 32*B^4*a^13*b^5*d^5
) - (((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3*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^10*d^7 + 448*B^2*a^12*b^6*d^7 -
 128*B^2*a^14*b^4*d^7 - 64*B^2*a^16*b^2*d^7) - (((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a
^2*b^2*d^4))^(1/2) + 8*B^2*a^3*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)*(((6
4*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3*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^10*d^8 - 512*B*a^10*b^8*d^8 + 384*B*a^12*b^6*d^8 + 256*B*a^14*b^4*d^8 - 128*B*a^16*b^2*d^8))*(((
64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3*b*d^2)/(16*(a^4*d^4
 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - 384*B^3*a^9*b^9*d^6 + 32*B^3*a^13*b^5*d^6 + 32*B^3*a^15*b^3*d^6))*(((64*
B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3*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*B^4*a^9*b^9*d^5 + 32*B^4*a^13*b^5*d^5) - (((64*B
^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3*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^10*d^7 + 448*B^2*a^12*b^6*d^7 - 128*B^2*a^
14*b^4*d^7 - 64*B^2*a^16*b^2*d^7) - (((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4)
)^(1/2) + 8*B^2*a^3*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^6*b
^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3*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^10*d^8 + 512*B*a^10*b^8*d^8 - 384*B*a^12*b^6*d^8 - 256*B*a^14*b^4*d^8 + 128*B*a^16*b^2*d^8))*(((64*B^4*a^6*
b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4
+ 2*a^2*b^2*d^4)))^(1/2) + 384*B^3*a^9*b^9*d^6 - 32*B^3*a^13*b^5*d^6 - 32*B^3*a^15*b^3*d^6))*(((64*B^4*a^6*b^2
*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3*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*B^4*a^9*b^9*d^5 + 32*B^4*a^13*b^5*d^5) - (((64*B^4*a^6*b^2*
d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3*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^10*d^7 + 448*B^2*a^12*b^6*d^7 - 128*B^2*a^14*b^4*d^7
- 64*B^2*a^16*b^2*d^7) - (((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8
*B^2*a^3*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^6*b^2*d^4 - B^
4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3*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^10*d^8 +
 512*B*a^10*b^8*d^8 - 384*B*a^12*b^6*d^8 - 256*B*a^14*b^4*d^8 + 128*B*a^16*b^2*d^8))*(((64*B^4*a^6*b^2*d^4 - B
^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2
*d^4)))^(1/2) + 384*B^3*a^9*b^9*d^6 - 32*B^3*a^13*b^5*d^6 - 32*B^3*a^15*b^3*d^6))*(((64*B^4*a^6*b^2*d^4 - B^4*
a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3*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^9*b^9*d^5 + 32*B^4*a^13*b^5*d^5) - (((64*B^4*a^6*b^2*d^4 - B^4*a^4*
(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3*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^10*d^7 + 448*B^2*a^12*b^6*d^7 - 128*B^2*a^14*b^4*d^7 - 64*B^2*a^16*
b^2*d^7) - (((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3*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^6*b^2*d^4 - B^4*a^4*(16*a^4*
d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3*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^10*d^8 - 512*B*a^10*b^
8*d^8 + 384*B*a^12*b^6*d^8 + 256*B*a^14*b^4*d^8 - 128*B*a^16*b^2*d^8))*(((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4
*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)
- 384*B^3*a^9*b^9*d^6 + 32*B^3*a^13*b^5*d^6 + 32*B^3*a^15*b^3*d^6))*(((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^
4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + 6
4*B^5*a^10*b^8*d^4))*(((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2
*a^3*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*B^4*a^7*b^11*d^
5 - 32*B^4*a^9*b^9*d^5) - (((64*B^4*a^2*b^6*d^4 - B^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^3*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^6*d^4 - B^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^3*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^6*d^4 - B^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^
3*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^10*d^8 - 640*B*a^10*b^8*d^8 + 256*B*a^12*b^6*d^8 + 384*B*a^14*b^4*d^8) - tan
(c + d*x)^(1/2)*(512*B^2*a^8*b^10*d^7 - 448*B^2*a^10*b^8*d^7 + 128*B^2*a^12*b^6*d^7 + 64*B^2*a^14*b^4*d^7))*((
(64*B^4*a^2*b^6*d^4 - B^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^3*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^11*d^6 + 32*B^3*a^11*b^7*d^6 + 32*B^3*a^13*b^5*d^6))*(((6
4*B^4*a^2*b^6*d^4 - B^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^3*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*B^4*a^7*b^11*d^5 - 32*B^4*a^9*b^9*d^5) - (((64
*B^4*a^2*b^6*d^4 - B^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^3*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^6*d^4 - B^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^3*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
^6*d^4 - B^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^3*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^10*d^8 + 640*B*a^10*b^8*d^8 - 256*B*a^12*b^6*d^8 - 384*B*a^14*b^4*d^8) - tan(c + d*x)^(1/2)*(512*B^2*a^8*b^
10*d^7 - 448*B^2*a^10*b^8*d^7 + 128*B^2*a^12*b^6*d^7 + 64*B^2*a^14*b^4*d^7))*(((64*B^4*a^2*b^6*d^4 - B^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^3*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^11*d^6 - 32*B^3*a^11*b^7*d^6 - 32*B^3*a^13*b^5*d^6))*(((64*B^4*a^2*b^6*d^4 - B^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^3*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*B^4*a^7*b^11*d^5 - 32*B^4*a^9*b^9*d^5) - (((64*B^4*a^2*b^6*d^4 - B^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^3*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^6*d^4 - B^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^3*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^6*d^4 - B^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^3*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^10*d^8 - 640*B*a^10*b^8*d^8
+ 256*B*a^12*b^6*d^8 + 384*B*a^14*b^4*d^8) - tan(c + d*x)^(1/2)*(512*B^2*a^8*b^10*d^7 - 448*B^2*a^10*b^8*d^7 +
 128*B^2*a^12*b^6*d^7 + 64*B^2*a^14*b^4*d^7))*(((64*B^4*a^2*b^6*d^4 - B^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^3*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^11*d^6 +
32*B^3*a^11*b^7*d^6 + 32*B^3*a^13*b^5*d^6))*(((64*B^4*a^2*b^6*d^4 - B^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^3*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^7*b^11*d^5 - 32*B^4*a^9*b^9*d^5) - (((64*B^4*a^2*b^6*d^4 - B^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^3*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^6*d^4 - B^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^3*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^6*d^4 - B^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^3*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^10*d^8 + 640*B*a^10*b^8*d^8 - 256*B*a^12*b^6*d^8 - 384*B*a^14*
b^4*d^8) - tan(c + d*x)^(1/2)*(512*B^2*a^8*b^10*d^7 - 448*B^2*a^10*b^8*d^7 + 128*B^2*a^12*b^6*d^7 + 64*B^2*a^1
4*b^4*d^7))*(((64*B^4*a^2*b^6*d^4 - B^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^3*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^11*d^6 - 32*B^3*a^11*b^7*d^6 - 32*B^3*a^13*
b^5*d^6))*(((64*B^4*a^2*b^6*d^4 - B^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^3*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^6*d^4 - B^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^3*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*B^4*a^7*b^11*d^5 - 32*B^4*a^9*b^9*d^5) - (-((64*B^4*a^2*b^6*d^4 - B^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^3*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^6*d^4 - B^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^3*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^6*d^4 - B^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^3*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^10*d^8 - 640*B*a^10*b^8*d^8 + 256*B*a^12*b^
6*d^8 + 384*B*a^14*b^4*d^8) - tan(c + d*x)^(1/2)*(512*B^2*a^8*b^10*d^7 - 448*B^2*a^10*b^8*d^7 + 128*B^2*a^12*b
^6*d^7 + 64*B^2*a^14*b^4*d^7))*(-((64*B^4*a^2*b^6*d^4 - B^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^3*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^11*d^6 + 32*B^3*a^11*b^
7*d^6 + 32*B^3*a^13*b^5*d^6))*(-((64*B^4*a^2*b^6*d^4 - B^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^3*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*B^4*a^7*b^1
1*d^5 - 32*B^4*a^9*b^9*d^5) - (-((64*B^4*a^2*b^6*d^4 - B^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^3*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^6*d^4 - B^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^3*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^6*d^4 - B^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^3*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^10*d^8 + 640*B*a^10*b^8*d^8 - 256*B*a^12*b^6*d^8 - 384*B*a^14*b^4*d^8
) - tan(c + d*x)^(1/2)*(512*B^2*a^8*b^10*d^7 - 448*B^2*a^10*b^8*d^7 + 128*B^2*a^12*b^6*d^7 + 64*B^2*a^14*b^4*d
^7))*(-((64*B^4*a^2*b^6*d^4 - B^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^3*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^11*d^6 - 32*B^3*a^11*b^7*d^6 - 32*B^3*a^13*b^5*d^
6))*(-((64*B^4*a^2*b^6*d^4 - B^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^3*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*B^4*a^7*b^11*d^5 - 32*B^4*a^9*b^9*d^5
) - (-((64*B^4*a^2*b^6*d^4 - B^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^3*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^6*d^4 - B^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^3*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^6*d^4 - B^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^3*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^10*d^8 - 640*B*a^10*b^8*d^8 + 256*B*a^12*b^6*d^8 + 384*B*a^14*b^4*d^8) - tan(c + d*x)^(1/2)*(51
2*B^2*a^8*b^10*d^7 - 448*B^2*a^10*b^8*d^7 + 128*B^2*a^12*b^6*d^7 + 64*B^2*a^14*b^4*d^7))*(-((64*B^4*a^2*b^6*d^
4 - B^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^3*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^11*d^6 + 32*B^3*a^11*b^7*d^6 + 32*B^3*a^13*b^5*d^6))*(-((64*B^4*a^2*b^6*d^4
 - B^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^3*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^7*b^11*d^5 - 32*B^4*a^9*b^9*d^5) - (-((64*B^4*a^2*b^6*d^4 -
B^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^3*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^6*d^4 - B^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^3*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^6*d^4 - B^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^3*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^10*d^8 + 640*B
*a^10*b^8*d^8 - 256*B*a^12*b^6*d^8 - 384*B*a^14*b^4*d^8) - tan(c + d*x)^(1/2)*(512*B^2*a^8*b^10*d^7 - 448*B^2*
a^10*b^8*d^7 + 128*B^2*a^12*b^6*d^7 + 64*B^2*a^14*b^4*d^7))*(-((64*B^4*a^2*b^6*d^4 - B^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^3*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^11*d^6 - 32*B^3*a^11*b^7*d^6 - 32*B^3*a^13*b^5*d^6))*(-((64*B^4*a^2*b^6*d^4 - B^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^3*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^6*d^4 - B^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^3*d^2)/(16*(a^4*d^4 + b^
4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*2i - ((2*B)/3 - (2*B*b*tan(c + d*x))/a)/(d*tan(c + d*x)^(3/2)) - (B*b^7*atan(((
B*b^7*(tan(c + d*x)^(1/2)*(64*B^4*a^7*b^11*d^5 - 32*B^4*a^9*b^9*d^5) + (B*b^7*(32*B^3*a^11*b^7*d^6 - 128*B^3*a
^7*b^11*d^6 + 32*B^3*a^13*b^5*d^6 + (B*b^7*(tan(c + d*x)^(1/2)*(512*B^2*a^8*b^10*d^7 - 448*B^2*a^10*b^8*d^7 +
128*B^2*a^12*b^6*d^7 + 64*B^2*a^14*b^4*d^7) - (B*b^7*(512*B*a^8*b^10*d^8 + 640*B*a^10*b^8*d^8 - 256*B*a^12*b^6
*d^8 - 384*B*a^14*b^4*d^8 + (B*b^7*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^11*d^2 - 2*a^5*b^9*d^2 - a^7*b^7*d^2)^(1/2)))/(- a^3*b^11*d^2 - 2*a^5*b^9*d^2 - a
^7*b^7*d^2)^(1/2)))/(- a^3*b^11*d^2 - 2*a^5*b^9*d^2 - a^7*b^7*d^2)^(1/2)))/(- a^3*b^11*d^2 - 2*a^5*b^9*d^2 - a
^7*b^7*d^2)^(1/2))*1i)/(- a^3*b^11*d^2 - 2*a^5*b^9*d^2 - a^7*b^7*d^2)^(1/2) + (B*b^7*(tan(c + d*x)^(1/2)*(64*B
^4*a^7*b^11*d^5 - 32*B^4*a^9*b^9*d^5) + (B*b^7*(128*B^3*a^7*b^11*d^6 - 32*B^3*a^11*b^7*d^6 - 32*B^3*a^13*b^5*d
^6 + (B*b^7*(tan(c + d*x)^(1/2)*(512*B^2*a^8*b^10*d^7 - 448*B^2*a^10*b^8*d^7 + 128*B^2*a^12*b^6*d^7 + 64*B^2*a
^14*b^4*d^7) - (B*b^7*(256*B*a^12*b^6*d^8 - 640*B*a^10*b^8*d^8 - 512*B*a^8*b^10*d^8 + 384*B*a^14*b^4*d^8 + (B*
b^7*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^11
*d^2 - 2*a^5*b^9*d^2 - a^7*b^7*d^2)^(1/2)))/(- a^3*b^11*d^2 - 2*a^5*b^9*d^2 - a^7*b^7*d^2)^(1/2)))/(- a^3*b^11
*d^2 - 2*a^5*b^9*d^2 - a^7*b^7*d^2)^(1/2)))/(- a^3*b^11*d^2 - 2*a^5*b^9*d^2 - a^7*b^7*d^2)^(1/2))*1i)/(- a^3*b
^11*d^2 - 2*a^5*b^9*d^2 - a^7*b^7*d^2)^(1/2))/((B*b^7*(tan(c + d*x)^(1/2)*(64*B^4*a^7*b^11*d^5 - 32*B^4*a^9*b^
9*d^5) + (B*b^7*(32*B^3*a^11*b^7*d^6 - 128*B^3*a^7*b^11*d^6 + 32*B^3*a^13*b^5*d^6 + (B*b^7*(tan(c + d*x)^(1/2)
*(512*B^2*a^8*b^10*d^7 - 448*B^2*a^10*b^8*d^7 + 128*B^2*a^12*b^6*d^7 + 64*B^2*a^14*b^4*d^7) - (B*b^7*(512*B*a^
8*b^10*d^8 + 640*B*a^10*b^8*d^8 - 256*B*a^12*b^6*d^8 - 384*B*a^14*b^4*d^8 + (B*b^7*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^11*d^2 - 2*a^5*b^9*d^2 - a^7*b^7*
d^2)^(1/2)))/(- a^3*b^11*d^2 - 2*a^5*b^9*d^2 - a^7*b^7*d^2)^(1/2)))/(- a^3*b^11*d^2 - 2*a^5*b^9*d^2 - a^7*b^7*
d^2)^(1/2)))/(- a^3*b^11*d^2 - 2*a^5*b^9*d^2 - a^7*b^7*d^2)^(1/2)))/(- a^3*b^11*d^2 - 2*a^5*b^9*d^2 - a^7*b^7*
d^2)^(1/2) - (B*b^7*(tan(c + d*x)^(1/2)*(64*B^4*a^7*b^11*d^5 - 32*B^4*a^9*b^9*d^5) + (B*b^7*(128*B^3*a^7*b^11*
d^6 - 32*B^3*a^11*b^7*d^6 - 32*B^3*a^13*b^5*d^6 + (B*b^7*(tan(c + d*x)^(1/2)*(512*B^2*a^8*b^10*d^7 - 448*B^2*a
^10*b^8*d^7 + 128*B^2*a^12*b^6*d^7 + 64*B^2*a^14*b^4*d^7) - (B*b^7*(256*B*a^12*b^6*d^8 - 640*B*a^10*b^8*d^8 -
512*B*a^8*b^10*d^8 + 384*B*a^14*b^4*d^8 + (B*b^7*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^11*d^2 - 2*a^5*b^9*d^2 - a^7*b^7*d^2)^(1/2)))/(- a^3*b^11*d^2 - 2*a
^5*b^9*d^2 - a^7*b^7*d^2)^(1/2)))/(- a^3*b^11*d^2 - 2*a^5*b^9*d^2 - a^7*b^7*d^2)^(1/2)))/(- a^3*b^11*d^2 - 2*a
^5*b^9*d^2 - a^7*b^7*d^2)^(1/2)))/(- a^3*b^11*d^2 - 2*a^5*b^9*d^2 - a^7*b^7*d^2)^(1/2)))*2i)/(- a^3*b^11*d^2 -
 2*a^5*b^9*d^2 - a^7*b^7*d^2)^(1/2) + (B*b^7*atan(((B*b^7*(tan(c + d*x)^(1/2)*(64*B^4*a^9*b^9*d^5 + 32*B^4*a^1
3*b^5*d^5) - (B*b^7*(32*B^3*a^13*b^5*d^6 - 384*B^3*a^9*b^9*d^6 + 32*B^3*a^15*b^3*d^6 + (B*b^7*(tan(c + d*x)^(1
/2)*(512*B^2*a^8*b^10*d^7 + 448*B^2*a^12*b^6*d^7 - 128*B^2*a^14*b^4*d^7 - 64*B^2*a^16*b^2*d^7) + (B*b^7*(512*B
*a^8*b^10*d^8 + 512*B*a^10*b^8*d^8 - 384*B*a^12*b^6*d^8 - 256*B*a^14*b^4*d^8 + 128*B*a^16*b^2*d^8 - (B*b^7*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^11*d^2 -
2*a^5*b^9*d^2 - a^7*b^7*d^2)^(1/2)))/(- a^3*b^11*d^2 - 2*a^5*b^9*d^2 - a^7*b^7*d^2)^(1/2)))/(- a^3*b^11*d^2 -
2*a^5*b^9*d^2 - a^7*b^7*d^2)^(1/2)))/(- a^3*b^11*d^2 - 2*a^5*b^9*d^2 - a^7*b^7*d^2)^(1/2))*1i)/(- a^3*b^11*d^2
 - 2*a^5*b^9*d^2 - a^7*b^7*d^2)^(1/2) + (B*b^7*(tan(c + d*x)^(1/2)*(64*B^4*a^9*b^9*d^5 + 32*B^4*a^13*b^5*d^5)
- (B*b^7*(384*B^3*a^9*b^9*d^6 - 32*B^3*a^13*b^5*d^6 - 32*B^3*a^15*b^3*d^6 + (B*b^7*(tan(c + d*x)^(1/2)*(512*B^
2*a^8*b^10*d^7 + 448*B^2*a^12*b^6*d^7 - 128*B^2*a^14*b^4*d^7 - 64*B^2*a^16*b^2*d^7) - (B*b^7*(512*B*a^8*b^10*d
^8 + 512*B*a^10*b^8*d^8 - 384*B*a^12*b^6*d^8 - 256*B*a^14*b^4*d^8 + 128*B*a^16*b^2*d^8 + (B*b^7*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^11*d^2 - 2*a^5*b^9*d
^2 - a^7*b^7*d^2)^(1/2)))/(- a^3*b^11*d^2 - 2*a^5*b^9*d^2 - a^7*b^7*d^2)^(1/2)))/(- a^3*b^11*d^2 - 2*a^5*b^9*d
^2 - a^7*b^7*d^2)^(1/2)))/(- a^3*b^11*d^2 - 2*a^5*b^9*d^2 - a^7*b^7*d^2)^(1/2))*1i)/(- a^3*b^11*d^2 - 2*a^5*b^
9*d^2 - a^7*b^7*d^2)^(1/2))/(64*B^5*a^10*b^8*d^4 - (B*b^7*(tan(c + d*x)^(1/2)*(64*B^4*a^9*b^9*d^5 + 32*B^4*a^1
3*b^5*d^5) - (B*b^7*(32*B^3*a^13*b^5*d^6 - 384*B^3*a^9*b^9*d^6 + 32*B^3*a^15*b^3*d^6 + (B*b^7*(tan(c + d*x)^(1
/2)*(512*B^2*a^8*b^10*d^7 + 448*B^2*a^12*b^6*d^7 - 128*B^2*a^14*b^4*d^7 - 64*B^2*a^16*b^2*d^7) + (B*b^7*(512*B
*a^8*b^10*d^8 + 512*B*a^10*b^8*d^8 - 384*B*a^12*b^6*d^8 - 256*B*a^14*b^4*d^8 + 128*B*a^16*b^2*d^8 - (B*b^7*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^11*d^2 -
2*a^5*b^9*d^2 - a^7*b^7*d^2)^(1/2)))/(- a^3*b^11*d^2 - 2*a^5*b^9*d^2 - a^7*b^7*d^2)^(1/2)))/(- a^3*b^11*d^2 -
2*a^5*b^9*d^2 - a^7*b^7*d^2)^(1/2)))/(- a^3*b^11*d^2 - 2*a^5*b^9*d^2 - a^7*b^7*d^2)^(1/2)))/(- a^3*b^11*d^2 -
2*a^5*b^9*d^2 - a^7*b^7*d^2)^(1/2) + (B*b^7*(tan(c + d*x)^(1/2)*(64*B^4*a^9*b^9*d^5 + 32*B^4*a^13*b^5*d^5) - (
B*b^7*(384*B^3*a^9*b^9*d^6 - 32*B^3*a^13*b^5*d^6 - 32*B^3*a^15*b^3*d^6 + (B*b^7*(tan(c + d*x)^(1/2)*(512*B^2*a
^8*b^10*d^7 + 448*B^2*a^12*b^6*d^7 - 128*B^2*a^14*b^4*d^7 - 64*B^2*a^16*b^2*d^7) - (B*b^7*(512*B*a^8*b^10*d^8
+ 512*B*a^10*b^8*d^8 - 384*B*a^12*b^6*d^8 - 256*B*a^14*b^4*d^8 + 128*B*a^16*b^2*d^8 + (B*b^7*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^11*d^2 - 2*a^5*b^9*d^2
- a^7*b^7*d^2)^(1/2)))/(- a^3*b^11*d^2 - 2*a^5*b^9*d^2 - a^7*b^7*d^2)^(1/2)))/(- a^3*b^11*d^2 - 2*a^5*b^9*d^2
- a^7*b^7*d^2)^(1/2)))/(- a^3*b^11*d^2 - 2*a^5*b^9*d^2 - a^7*b^7*d^2)^(1/2)))/(- a^3*b^11*d^2 - 2*a^5*b^9*d^2
- a^7*b^7*d^2)^(1/2)))*2i)/(- a^3*b^11*d^2 - 2*a^5*b^9*d^2 - a^7*b^7*d^2)^(1/2) - (2*B*b)/(a*d*tan(c + d*x)^(1
/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ B \int \frac {1}{\tan ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*Integral(tan(c + d*x)**(-5/2), x)

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