∫ tan5 _2 (c+dx)(aB+bB tan(c+dx))______________________

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

Optimal. Leaf size=256 \[ \frac {B (a+b) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )}-\frac {B (a+b) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )}-\frac {B (a-b) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}+\frac {B (a-b) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}-\frac {2 a^{5/2} B \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{3/2} d \left (a^2+b^2\right )}+\frac {2 B \sqrt {\tan (c+d x)}}{b d} \]

[Out]

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

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

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Rubi [A] time = 0.53, antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.361, Rules used = {21, 3566, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ -\frac {2 a^{5/2} B \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{3/2} d \left (a^2+b^2\right )}+\frac {B (a+b) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )}-\frac {B (a+b) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )}-\frac {B (a-b) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}+\frac {B (a-b) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}+\frac {2 B \sqrt {\tan (c+d x)}}{b d} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

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

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

*x]])/(b*d)

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 63

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 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 205

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

&& PosQ[a/b]

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 1168

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 3534

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 3566

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si

mp[(b^2*(a + b*Tan[e + f*x])^(m - 2)*(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 - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(

1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2,

x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]

&& IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || IntegerQ[m]) && !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0]

&& NeQ[a, 0])))

Rule 3634

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 3653

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, -

Rubi steps

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

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Mathematica [C] time = 0.21, size = 156, normalized size = 0.61 \[ \frac {B \left (-2 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )+2 a^2 \sqrt {b} \sqrt {\tan (c+d x)}+\sqrt [4]{-1} b^{3/2} (b-i a) \tan ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+\sqrt [4]{-1} b^{3/2} (b+i a) \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+2 b^{5/2} \sqrt {\tan (c+d x)}\right )}{b^{3/2} d \left (a^2+b^2\right )} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

[c + d*x]])/Sqrt[a]] + (-1)^(1/4)*b^(3/2)*(I*a + b)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*a^2*Sqrt[b]*Sqr

t[Tan[c + d*x]] + 2*b^(5/2)*Sqrt[Tan[c + d*x]]))/(b^(3/2)*(a^2 + b^2)*d)

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fricas [B] time = 15.50, size = 8974, normalized size = 35.05 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(4*sqrt(2)*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*d^5*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 + 2*(a^5*b +

2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*(B^4/((a

^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4)*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a

^2*b^6 + b^8)*d^4))*arctan(-((B^6*a^8 + 2*B^6*a^6*b^2 - 2*B^6*a^2*b^6 - B^6*b^8)*d^4*sqrt(B^4/((a^4 + 2*a^2*b^

2 + b^4)*d^4))*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))

- sqrt(2)*((a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*d^7*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt

((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - (B^2*a^7 + 3*B^2

*a^5*b^2 + 3*B^2*a^3*b^4 + B^2*a*b^6)*d^5*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b

^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(

B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*sqrt(((B^4*a^6 - B^4*a^4*b^2 - B^4*a^

2*b^4 + B^4*b^6)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) + sqrt(2)*((B^3*a^7 - B^3*a^5*b^2 -

B^3*a^3*b^4 + B^3*a*b^6)*d^3*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - (B^5*a^4*b - 2*B^5*a^2*b^3

+ B^5*b^5)*d*cos(d*x + c))*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(B

^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(B^4/(

(a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (B^6*a^4 - 2*B^6*a^2*b^2 + B^6*b^4)*sin(d*x + c))/cos(d*x + c))*(B^4/((a

^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4) - sqrt(2)*((B^3*a^10*b + 3*B^3*a^8*b^3 + 2*B^3*a^6*b^5 - 2*B^3*a^4*b^7 - 3*B

^3*a^2*b^9 - B^3*b^11)*d^7*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((

a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - (B^5*a^9 + 2*B^5*a^7*b^2 - 2*B^5*a^3*b^6 - B^5*a*b^8)*d

^5*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((B^2*

a^4 + 2*B^2*a^2*b^2 + B^2*b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^

2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4))/(

B^10*a^4 - 2*B^10*a^2*b^2 + B^10*b^4)) + 4*sqrt(2)*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*d^5*sqrt((B^2*a^4 + 2

*B^2*a^2*b^2 + B^2*b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 -

2*B^2*a^2*b^2 + B^2*b^4))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4)*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/

((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*arctan(((B^6*a^8 + 2*B^6*a^6*b^2 - 2*B^6*a^2*b^6 - B^6*

b^8)*d^4*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 +

6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + sqrt(2)*((a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*d^7*sqrt(B^4/(

(a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b

^6 + b^8)*d^4)) - (B^2*a^7 + 3*B^2*a^5*b^2 + 3*B^2*a^3*b^4 + B^2*a*b^6)*d^5*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^

4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 + 2*(a^

5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*sqr

t(((B^4*a^6 - B^4*a^4*b^2 - B^4*a^2*b^4 + B^4*b^6)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) -

sqrt(2)*((B^3*a^7 - B^3*a^5*b^2 - B^3*a^3*b^4 + B^3*a*b^6)*d^3*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x

+ c) - (B^5*a^4*b - 2*B^5*a^2*b^3 + B^5*b^5)*d*cos(d*x + c))*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 + 2*(a^5

*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*sqrt

(sin(d*x + c)/cos(d*x + c))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (B^6*a^4 - 2*B^6*a^2*b^2 + B^6*b^4)*si

n(d*x + c))/cos(d*x + c))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4) + sqrt(2)*((B^3*a^10*b + 3*B^3*a^8*b^3 + 2

*B^3*a^6*b^5 - 2*B^3*a^4*b^7 - 3*B^3*a^2*b^9 - B^3*b^11)*d^7*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt((B^4

*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - (B^5*a^9 + 2*B^5*a^7*

b^2 - 2*B^5*a^3*b^6 - B^5*a*b^8)*d^5*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 +

4*a^2*b^6 + b^8)*d^4)))*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/(

(a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(B^4/((a^4

+ 2*a^2*b^2 + b^4)*d^4))^(3/4))/(B^10*a^4 - 2*B^10*a^2*b^2 + B^10*b^4)) + 2*B^5*a^2*sqrt(-a/b)*log(-(6*a*b*co

s(d*x + c)*sin(d*x + c) - (a^2 - b^2)*cos(d*x + c)^2 - b^2 - 4*(a*b*cos(d*x + c)^2 - b^2*cos(d*x + c)*sin(d*x

+ c))*sqrt(-a/b)*sqrt(sin(d*x + c)/cos(d*x + c)))/(2*a*b*cos(d*x + c)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^

2 + b^2)) - sqrt(2)*(2*(B^2*a^3*b^2 + B^2*a*b^4)*d^3*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - (B^4*a^2*b + B^

4*b^3)*d)*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b

^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4)*log(((B^4*a^6

- B^4*a^4*b^2 - B^4*a^2*b^4 + B^4*b^6)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) + sqrt(2)*((B^

3*a^7 - B^3*a^5*b^2 - B^3*a^3*b^4 + B^3*a*b^6)*d^3*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - (B^5

*a^4*b - 2*B^5*a^2*b^3 + B^5*b^5)*d*cos(d*x + c))*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 + 2*(a^5*b + 2*a^3*b

^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*sqrt(sin(d*x + c

)/cos(d*x + c))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (B^6*a^4 - 2*B^6*a^2*b^2 + B^6*b^4)*sin(d*x + c))/

cos(d*x + c)) + sqrt(2)*(2*(B^2*a^3*b^2 + B^2*a*b^4)*d^3*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - (B^4*a^2*b

+ B^4*b^3)*d)*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a

^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4)*log(((B^4*

a^6 - B^4*a^4*b^2 - B^4*a^2*b^4 + B^4*b^6)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - sqrt(2)*

((B^3*a^7 - B^3*a^5*b^2 - B^3*a^3*b^4 + B^3*a*b^6)*d^3*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) -

(B^5*a^4*b - 2*B^5*a^2*b^3 + B^5*b^5)*d*cos(d*x + c))*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 + 2*(a^5*b + 2*a

^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*sqrt(sin(d*x

+ c)/cos(d*x + c))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (B^6*a^4 - 2*B^6*a^2*b^2 + B^6*b^4)*sin(d*x +

c))/cos(d*x + c)) + 8*(B^5*a^2 + B^5*b^2)*sqrt(sin(d*x + c)/cos(d*x + c)))/((B^4*a^2*b + B^4*b^3)*d), 1/4*(4*s

qrt(2)*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*d^5*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 + 2*(a^5*b + 2*a^3*b^

3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*(B^4/((a^4 + 2*a^

2*b^2 + b^4)*d^4))^(3/4)*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 +

b^8)*d^4))*arctan(-((B^6*a^8 + 2*B^6*a^6*b^2 - 2*B^6*a^2*b^6 - B^6*b^8)*d^4*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*

d^4))*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - sqrt(2

)*((a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*d^7*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt((B^4*a^4

- 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - (B^2*a^7 + 3*B^2*a^5*b^2

+ 3*B^2*a^3*b^4 + B^2*a*b^6)*d^5*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^

2*b^6 + b^8)*d^4)))*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4

+ 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*sqrt(((B^4*a^6 - B^4*a^4*b^2 - B^4*a^2*b^4 + B

^4*b^6)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) + sqrt(2)*((B^3*a^7 - B^3*a^5*b^2 - B^3*a^3*b

^4 + B^3*a*b^6)*d^3*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - (B^5*a^4*b - 2*B^5*a^2*b^3 + B^5*b^

5)*d*cos(d*x + c))*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4

+ 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(B^4/((a^4 + 2*

a^2*b^2 + b^4)*d^4))^(1/4) + (B^6*a^4 - 2*B^6*a^2*b^2 + B^6*b^4)*sin(d*x + c))/cos(d*x + c))*(B^4/((a^4 + 2*a^

2*b^2 + b^4)*d^4))^(3/4) - sqrt(2)*((B^3*a^10*b + 3*B^3*a^8*b^3 + 2*B^3*a^6*b^5 - 2*B^3*a^4*b^7 - 3*B^3*a^2*b^

9 - B^3*b^11)*d^7*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a

^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - (B^5*a^9 + 2*B^5*a^7*b^2 - 2*B^5*a^3*b^6 - B^5*a*b^8)*d^5*sqrt((

B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((B^2*a^4 + 2*B

^2*a^2*b^2 + B^2*b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2

*B^2*a^2*b^2 + B^2*b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4))/(B^10*a^4

- 2*B^10*a^2*b^2 + B^10*b^4)) + 4*sqrt(2)*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*d^5*sqrt((B^2*a^4 + 2*B^2*a^2*

b^2 + B^2*b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^

2*b^2 + B^2*b^4))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4)*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4

*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*arctan(((B^6*a^8 + 2*B^6*a^6*b^2 - 2*B^6*a^2*b^6 - B^6*b^8)*d^4*

sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4

+ 4*a^2*b^6 + b^8)*d^4)) + sqrt(2)*((a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*d^7*sqrt(B^4/((a^4 + 2*

a^2*b^2 + b^4)*d^4))*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)

*d^4)) - (B^2*a^7 + 3*B^2*a^5*b^2 + 3*B^2*a^3*b^4 + B^2*a*b^6)*d^5*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((

a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 + 2*(a^5*b + 2*a

^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*sqrt(((B^4*a

^6 - B^4*a^4*b^2 - B^4*a^2*b^4 + B^4*b^6)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - sqrt(2)*(

(B^3*a^7 - B^3*a^5*b^2 - B^3*a^3*b^4 + B^3*a*b^6)*d^3*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - (

B^5*a^4*b - 2*B^5*a^2*b^3 + B^5*b^5)*d*cos(d*x + c))*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 + 2*(a^5*b + 2*a^

3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*sqrt(sin(d*x

+ c)/cos(d*x + c))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (B^6*a^4 - 2*B^6*a^2*b^2 + B^6*b^4)*sin(d*x + c

))/cos(d*x + c))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4) + sqrt(2)*((B^3*a^10*b + 3*B^3*a^8*b^3 + 2*B^3*a^6*

b^5 - 2*B^3*a^4*b^7 - 3*B^3*a^2*b^9 - B^3*b^11)*d^7*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt((B^4*a^4 - 2*

B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - (B^5*a^9 + 2*B^5*a^7*b^2 - 2*B

^5*a^3*b^6 - B^5*a*b^8)*d^5*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6

+ b^8)*d^4)))*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*

a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(B^4/((a^4 + 2*a^2*

b^2 + b^4)*d^4))^(3/4))/(B^10*a^4 - 2*B^10*a^2*b^2 + B^10*b^4)) - 8*B^5*a^2*sqrt(a/b)*arctan(b*sqrt(a/b)*sqrt(

sin(d*x + c)/cos(d*x + c))/a) - sqrt(2)*(2*(B^2*a^3*b^2 + B^2*a*b^4)*d^3*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4

)) - (B^4*a^2*b + B^4*b^3)*d)*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt

(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^

(1/4)*log(((B^4*a^6 - B^4*a^4*b^2 - B^4*a^2*b^4 + B^4*b^6)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x

+ c) + sqrt(2)*((B^3*a^7 - B^3*a^5*b^2 - B^3*a^3*b^4 + B^3*a*b^6)*d^3*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))

*cos(d*x + c) - (B^5*a^4*b - 2*B^5*a^2*b^3 + B^5*b^5)*d*cos(d*x + c))*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4

+ 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^

4))*sqrt(sin(d*x + c)/cos(d*x + c))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (B^6*a^4 - 2*B^6*a^2*b^2 + B^6

*b^4)*sin(d*x + c))/cos(d*x + c)) + sqrt(2)*(2*(B^2*a^3*b^2 + B^2*a*b^4)*d^3*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)

*d^4)) - (B^4*a^2*b + B^4*b^3)*d)*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*

sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^

4))^(1/4)*log(((B^4*a^6 - B^4*a^4*b^2 - B^4*a^2*b^4 + B^4*b^6)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos

(d*x + c) - sqrt(2)*((B^3*a^7 - B^3*a^5*b^2 - B^3*a^3*b^4 + B^3*a*b^6)*d^3*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d

^4))*cos(d*x + c) - (B^5*a^4*b - 2*B^5*a^2*b^3 + B^5*b^5)*d*cos(d*x + c))*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*

b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^

2*b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (B^6*a^4 - 2*B^6*a^2*b^2 +

B^6*b^4)*sin(d*x + c))/cos(d*x + c)) + 8*(B^5*a^2 + B^5*b^2)*sqrt(sin(d*x + c)/cos(d*x + c)))/((B^4*a^2*b + B

^4*b^3)*d)]

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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [A] time = 0.33, size = 325, normalized size = 1.27 \[ \frac {2 B \left (\sqrt {\tan }\left (d x +c \right )\right )}{b d}-\frac {2 a^{3} \arctan \left (\frac {\left (\sqrt {\tan }\left (d x +c \right )\right ) b}{\sqrt {a b}}\right ) B}{d b \left (a^{2}+b^{2}\right ) \sqrt {a b}}-\frac {B \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) b}{2 d \left (a^{2}+b^{2}\right )}-\frac {B \sqrt {2}\, \ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right ) b}{4 d \left (a^{2}+b^{2}\right )}-\frac {B \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) b}{2 d \left (a^{2}+b^{2}\right )}-\frac {B \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) a}{2 d \left (a^{2}+b^{2}\right )}-\frac {B \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) a}{2 d \left (a^{2}+b^{2}\right )}-\frac {B \sqrt {2}\, \ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right ) a}{4 d \left (a^{2}+b^{2}\right )} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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

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

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maxima [A] time = 0.73, size = 188, normalized size = 0.73 \[ -\frac {\frac {8 \, B a^{3} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{2} b + b^{3}\right )} \sqrt {a b}} + \frac {{\left (2 \, \sqrt {2} {\left (a + b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (a + b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - \sqrt {2} {\left (a - b\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \sqrt {2} {\left (a - b\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )} B}{a^{2} + b^{2}} - \frac {8 \, B \sqrt {\tan \left (d x + c\right )}}{b}}{4 \, d} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 2*sqrt(2)*(a + b)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x +

c)))) - sqrt(2)*(a - b)*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) + sqrt(2)*(a - b)*log(-sqrt(2)*sqr

t(tan(d*x + c)) + tan(d*x + c) + 1))*B/(a^2 + b^2) - 8*B*sqrt(tan(d*x + c))/b)/d

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mupad [B] time = 35.55, size = 18514, normalized size = 72.32 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*((4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)

^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2) + (768*B*a^3*b^3*(a^2 + b^2))/d)*((

4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))

^(1/2))/4 + (64*B^2*a^3*tan(c + d*x)^(1/2)*(2*a^8 + 15*b^8 - 17*a^2*b^6 + 51*a^4*b^4 + 21*a^6*b^2))/(d^2*(a^2

+ b^2)^2))*((4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^5*b*d^2)/(d^4*(a

^2 + b^2)^4))^(1/2))/4 + (32*B^3*a^4*(4*a^8 + b^8 - 77*a^2*b^6 + 47*a^4*b^4 + 33*a^6*b^2))/(d^3*(a^2 + b^2)^3)

)*((4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)

^4))^(1/2))/4 + (16*B^4*a^4*tan(c + d*x)^(1/2)*(a^10 - 2*b^10 - 4*a^2*b^8 - 27*a^4*b^6 + 15*a^6*b^4 + 9*a^8*b^

2))/(b*d^4*(a^2 + b^2)^4))*((4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^

5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (8*B^5*a^7*(a^6 + 10*b^6 + 27*a^2*b^4 + 10*a^4*b^2))/(b*d^5*(a^2 + b^

2)^4))*(((192*B^4*a^6*b^6*d^4 - 16*B^4*a^4*b^8*d^4 - 16*B^4*a^12*d^4 - 608*B^4*a^8*b^4*d^4 + 192*B^4*a^10*b^2*

d^4)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^5*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6

*b^2*d^4))^(1/2))/4 + (log(((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*(-(4*(-B^4*a^4*d^4*(a^

4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2) + (768*B*a^3

*b^3*(a^2 + b^2))/d)*(-(4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a^5*b*d

^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*B^2*a^3*tan(c + d*x)^(1/2)*(2*a^8 + 15*b^8 - 17*a^2*b^6 + 51*a^4*b^4 +

21*a^6*b^2))/(d^2*(a^2 + b^2)^2))*(-(4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 +

16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (32*B^3*a^4*(4*a^8 + b^8 - 77*a^2*b^6 + 47*a^4*b^4 + 33*a^6*

b^2))/(d^3*(a^2 + b^2)^3))*(-(4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a

^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (16*B^4*a^4*tan(c + d*x)^(1/2)*(a^10 - 2*b^10 - 4*a^2*b^8 - 27*a^4*b

^6 + 15*a^6*b^4 + 9*a^8*b^2))/(b*d^4*(a^2 + b^2)^4))*(-(4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*

B^2*a^3*b^3*d^2 + 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (8*B^5*a^7*(a^6 + 10*b^6 + 27*a^2*b^4 + 10

*a^4*b^2))/(b*d^5*(a^2 + b^2)^4))*(-((192*B^4*a^6*b^6*d^4 - 16*B^4*a^4*b^8*d^4 - 16*B^4*a^12*d^4 - 608*B^4*a^8

*b^4*d^4 + 192*B^4*a^10*b^2*d^4)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a^5*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6

*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2))/4 - log((8*B^5*a^7*(a^6 + 10*b^6 + 27*a^2*b^4 + 10*a^4*b^2))/(b*

d^5*(a^2 + b^2)^4) - ((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*((4*(-B^4*a^4*d^4*(a^4 + b^4

- 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2) - (768*B*a^3*b^3*(a

^2 + b^2))/d)*((4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^5*b*d^2)/(d^4

*(a^2 + b^2)^4))^(1/2))/4 + (64*B^2*a^3*tan(c + d*x)^(1/2)*(2*a^8 + 15*b^8 - 17*a^2*b^6 + 51*a^4*b^4 + 21*a^6*

b^2))/(d^2*(a^2 + b^2)^2))*((4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^

5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (32*B^3*a^4*(4*a^8 + b^8 - 77*a^2*b^6 + 47*a^4*b^4 + 33*a^6*b^2))/(d^

3*(a^2 + b^2)^3))*((4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^5*b*d^2)/

(d^4*(a^2 + b^2)^4))^(1/2))/4 + (16*B^4*a^4*tan(c + d*x)^(1/2)*(a^10 - 2*b^10 - 4*a^2*b^8 - 27*a^4*b^6 + 15*a^

6*b^4 + 9*a^8*b^2))/(b*d^4*(a^2 + b^2)^4))*((4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3

*d^2 - 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4)*(((192*B^4*a^6*b^6*d^4 - 16*B^4*a^4*b^8*d^4 - 16*B^4*a

^12*d^4 - 608*B^4*a^8*b^4*d^4 + 192*B^4*a^10*b^2*d^4)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^5*b*d^2)/(16*a^8*d

^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^4))^(1/2) - log((8*B^5*a^7*(a^6 + 10*b^6 + 27

*a^2*b^4 + 10*a^4*b^2))/(b*d^5*(a^2 + b^2)^4) - ((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*(

-(4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4

))^(1/2) - (768*B*a^3*b^3*(a^2 + b^2))/d)*(-(4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3

*d^2 + 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*B^2*a^3*tan(c + d*x)^(1/2)*(2*a^8 + 15*b^8 - 17*a

^2*b^6 + 51*a^4*b^4 + 21*a^6*b^2))/(d^2*(a^2 + b^2)^2))*(-(4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) -

16*B^2*a^3*b^3*d^2 + 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (32*B^3*a^4*(4*a^8 + b^8 - 77*a^2*b^6 +

47*a^4*b^4 + 33*a^6*b^2))/(d^3*(a^2 + b^2)^3))*(-(4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a

^3*b^3*d^2 + 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (16*B^4*a^4*tan(c + d*x)^(1/2)*(a^10 - 2*b^10 -

4*a^2*b^8 - 27*a^4*b^6 + 15*a^6*b^4 + 9*a^8*b^2))/(b*d^4*(a^2 + b^2)^4))*(-(4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^

2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4)*(-((192*B^4*a^6*b^6*d^

4 - 16*B^4*a^4*b^8*d^4 - 16*B^4*a^12*d^4 - 608*B^4*a^8*b^4*d^4 + 192*B^4*a^10*b^2*d^4)^(1/2) - 16*B^2*a^3*b^3*

d^2 + 16*B^2*a^5*b*d^2)/(16*a^8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^4))^(1/2) +

(log(((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*((4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)

^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2) - (128*B*a*b^3*(7*a^4 + b^4 + 8*a^2

*b^2))/d)*((4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a*b^5*d^2)/(d^4*(a^

2 + b^2)^4))^(1/2))/4 + (64*B^2*a*tan(c + d*x)^(1/2)*(18*a^10 - 15*b^10 + 17*a^2*b^8 - a^4*b^6 + 97*a^6*b^4 +

84*a^8*b^2))/(d^2*(a^2 + b^2)^2))*((4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16

*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (32*B^3*a^4*(127*a^2*b^6 - 112*b^8 - 9*a^8 + 173*a^4*b^4 + 21*

a^6*b^2))/(d^3*(a^2 + b^2)^3))*((4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^

2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (16*B^4*b*tan(c + d*x)^(1/2)*(9*a^12 + 2*b^12 + 4*a^2*b^10 + 2*a^

4*b^8 - 49*a^6*b^6 + 7*a^8*b^4 + 33*a^10*b^2))/(d^4*(a^2 + b^2)^4))*((4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^

2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (16*B^5*a^3*b^7*(3*a^2 + 7*b

^2))/(d^5*(a^2 + b^2)^4))*(((192*B^4*a^2*b^10*d^4 - 16*B^4*b^12*d^4 - 608*B^4*a^4*b^8*d^4 + 192*B^4*a^6*b^6*d^

4 - 16*B^4*a^8*b^4*d^4)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a*b^5*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*

a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2))/4 + (log(((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*(-(

4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))

^(1/2) - (128*B*a*b^3*(7*a^4 + b^4 + 8*a^2*b^2))/d)*(-(4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B

^2*a^3*b^3*d^2 + 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*B^2*a*tan(c + d*x)^(1/2)*(18*a^10 - 15*

b^10 + 17*a^2*b^8 - a^4*b^6 + 97*a^6*b^4 + 84*a^8*b^2))/(d^2*(a^2 + b^2)^2))*(-(4*(-B^4*b^4*d^4*(a^4 + b^4 - 6

*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (32*B^3*a^4*(127*a

^2*b^6 - 112*b^8 - 9*a^8 + 173*a^4*b^4 + 21*a^6*b^2))/(d^3*(a^2 + b^2)^3))*(-(4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a

^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (16*B^4*b*tan(c + d*

x)^(1/2)*(9*a^12 + 2*b^12 + 4*a^2*b^10 + 2*a^4*b^8 - 49*a^6*b^6 + 7*a^8*b^4 + 33*a^10*b^2))/(d^4*(a^2 + b^2)^4

))*(-(4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^

2)^4))^(1/2))/4 - (16*B^5*a^3*b^7*(3*a^2 + 7*b^2))/(d^5*(a^2 + b^2)^4))*(-((192*B^4*a^2*b^10*d^4 - 16*B^4*b^12

*d^4 - 608*B^4*a^4*b^8*d^4 + 192*B^4*a^6*b^6*d^4 - 16*B^4*a^8*b^4*d^4)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a*b

^5*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2))/4 - log(- ((((((((128*b^3*

tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*((4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b

^3*d^2 - 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2) + (128*B*a*b^3*(7*a^4 + b^4 + 8*a^2*b^2))/d)*((4*(-B^4*b

^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4

+ (64*B^2*a*tan(c + d*x)^(1/2)*(18*a^10 - 15*b^10 + 17*a^2*b^8 - a^4*b^6 + 97*a^6*b^4 + 84*a^8*b^2))/(d^2*(a^

2 + b^2)^2))*((4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a*b^5*d^2)/(d^4*

(a^2 + b^2)^4))^(1/2))/4 - (32*B^3*a^4*(127*a^2*b^6 - 112*b^8 - 9*a^8 + 173*a^4*b^4 + 21*a^6*b^2))/(d^3*(a^2 +

b^2)^3))*((4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a*b^5*d^2)/(d^4*(a^

2 + b^2)^4))^(1/2))/4 - (16*B^4*b*tan(c + d*x)^(1/2)*(9*a^12 + 2*b^12 + 4*a^2*b^10 + 2*a^4*b^8 - 49*a^6*b^6 +

7*a^8*b^4 + 33*a^10*b^2))/(d^4*(a^2 + b^2)^4))*((4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3

*b^3*d^2 - 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (16*B^5*a^3*b^7*(3*a^2 + 7*b^2))/(d^5*(a^2 + b^2)

^4))*(((192*B^4*a^2*b^10*d^4 - 16*B^4*b^12*d^4 - 608*B^4*a^4*b^8*d^4 + 192*B^4*a^6*b^6*d^4 - 16*B^4*a^8*b^4*d^

4)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a*b^5*d^2)/(16*a^8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 +

64*a^6*b^2*d^4))^(1/2) - log(- ((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*(-(4*(-B^4*b^4*d^

4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2) + (128*

B*a*b^3*(7*a^4 + b^4 + 8*a^2*b^2))/d)*(-(4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2

+ 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*B^2*a*tan(c + d*x)^(1/2)*(18*a^10 - 15*b^10 + 17*a^2*

b^8 - a^4*b^6 + 97*a^6*b^4 + 84*a^8*b^2))/(d^2*(a^2 + b^2)^2))*(-(4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(

1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (32*B^3*a^4*(127*a^2*b^6 - 112*b

^8 - 9*a^8 + 173*a^4*b^4 + 21*a^6*b^2))/(d^3*(a^2 + b^2)^3))*(-(4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/

2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (16*B^4*b*tan(c + d*x)^(1/2)*(9*a^

12 + 2*b^12 + 4*a^2*b^10 + 2*a^4*b^8 - 49*a^6*b^6 + 7*a^8*b^4 + 33*a^10*b^2))/(d^4*(a^2 + b^2)^4))*(-(4*(-B^4*

b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/

4 - (16*B^5*a^3*b^7*(3*a^2 + 7*b^2))/(d^5*(a^2 + b^2)^4))*(-((192*B^4*a^2*b^10*d^4 - 16*B^4*b^12*d^4 - 608*B^4

*a^4*b^8*d^4 + 192*B^4*a^6*b^6*d^4 - 16*B^4*a^8*b^4*d^4)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a*b^5*d^2)/(16*a^

8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^4))^(1/2) - (atan(((((16*tan(c + d*x)^(1/2

)*(2*B^4*b^13 + 9*B^4*a^12*b + 4*B^4*a^2*b^11 + 2*B^4*a^4*b^9 - 49*B^4*a^6*b^7 + 7*B^4*a^8*b^5 + 33*B^4*a^10*b

^3))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4) + (((16*(30*B^3*a^6*b^8*d^2 - 224*B^3

*a^4*b^10*d^2 - 18*B^3*a^14*d^2 + 600*B^3*a^8*b^6*d^2 + 388*B^3*a^10*b^4*d^2 + 24*B^3*a^12*b^2*d^2))/(a^8*d^5

+ b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) - (((16*tan(c + d*x)^(1/2)*(72*B^2*a^15*d^2 - 52*B^

2*a^3*b^12*d^2 + 72*B^2*a^5*b^10*d^2 + 448*B^2*a^7*b^8*d^2 + 1108*B^2*a^9*b^6*d^2 + 1132*B^2*a^11*b^4*d^2 + 48

0*B^2*a^13*b^2*d^2 - 60*B^2*a*b^14*d^2))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4) +

(((16*(8*B*a*b^15*d^4 + 96*B*a^3*b^13*d^4 + 360*B*a^5*b^11*d^4 + 640*B*a^7*b^9*d^4 + 600*B*a^9*b^7*d^4 + 288*

B*a^11*b^5*d^4 + 56*B*a^13*b^3*d^4))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) - (4*

tan(c + d*x)^(1/2)*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2

+ 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2)*(32*b^17*d^4 + 160*a^2*b^15*d^4 + 288*a^4*b^13*d^4 + 160*a^6*b^11*d^4 -

160*a^8*b^9*d^4 - 288*a^10*b^7*d^4 - 160*a^12*b^5*d^4 - 32*a^14*b^3*d^4))/((a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4

+ 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(

-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a

^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(9*B^2*a

^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)

)^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(9*B^2*a^9 + 49*B^2

*a^5*b^4 + 42*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4

*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 +

42*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2)*1i)/(4*(b^11*d

^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)) + (((16*tan(c + d*x)^(1/2)*(2*B^4*b^13 + 9*

B^4*a^12*b + 4*B^4*a^2*b^11 + 2*B^4*a^4*b^9 - 49*B^4*a^6*b^7 + 7*B^4*a^8*b^5 + 33*B^4*a^10*b^3))/(a^8*d^4 + b^

8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4) - (((16*(30*B^3*a^6*b^8*d^2 - 224*B^3*a^4*b^10*d^2 - 18

*B^3*a^14*d^2 + 600*B^3*a^8*b^6*d^2 + 388*B^3*a^10*b^4*d^2 + 24*B^3*a^12*b^2*d^2))/(a^8*d^5 + b^8*d^5 + 4*a^2*

b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) + (((16*tan(c + d*x)^(1/2)*(72*B^2*a^15*d^2 - 52*B^2*a^3*b^12*d^2 + 7

2*B^2*a^5*b^10*d^2 + 448*B^2*a^7*b^8*d^2 + 1108*B^2*a^9*b^6*d^2 + 1132*B^2*a^11*b^4*d^2 + 480*B^2*a^13*b^2*d^2

- 60*B^2*a*b^14*d^2))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4) - (((16*(8*B*a*b^15

*d^4 + 96*B*a^3*b^13*d^4 + 360*B*a^5*b^11*d^4 + 640*B*a^7*b^9*d^4 + 600*B*a^9*b^7*d^4 + 288*B*a^11*b^5*d^4 + 5

6*B*a^13*b^3*d^4))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) + (4*tan(c + d*x)^(1/2)

*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 +

a^8*b^3*d^2))^(1/2)*(32*b^17*d^4 + 160*a^2*b^15*d^4 + 288*a^4*b^13*d^4 + 160*a^6*b^11*d^4 - 160*a^8*b^9*d^4 -

288*a^10*b^7*d^4 - 160*a^12*b^5*d^4 - 32*a^14*b^3*d^4))/((a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 +

4*a^6*b^2*d^4)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(9*B^2*a^9 + 49

*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2)

)/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^

4 + 42*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*

d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*

a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2

*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b

^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2)*1i)/(4*(b^11*d^2 + 4*a^2*b^9*d^2

+ 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))/((32*(7*B^5*a^3*b^9 + 3*B^5*a^5*b^7))/(a^8*d^5 + b^8*d^5 + 4

*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) - (((16*tan(c + d*x)^(1/2)*(2*B^4*b^13 + 9*B^4*a^12*b + 4*B^4*a^

2*b^11 + 2*B^4*a^4*b^9 - 49*B^4*a^6*b^7 + 7*B^4*a^8*b^5 + 33*B^4*a^10*b^3))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4

+ 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4) + (((16*(30*B^3*a^6*b^8*d^2 - 224*B^3*a^4*b^10*d^2 - 18*B^3*a^14*d^2 + 600*B

^3*a^8*b^6*d^2 + 388*B^3*a^10*b^4*d^2 + 24*B^3*a^12*b^2*d^2))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d

^5 + 4*a^6*b^2*d^5) - (((16*tan(c + d*x)^(1/2)*(72*B^2*a^15*d^2 - 52*B^2*a^3*b^12*d^2 + 72*B^2*a^5*b^10*d^2 +

448*B^2*a^7*b^8*d^2 + 1108*B^2*a^9*b^6*d^2 + 1132*B^2*a^11*b^4*d^2 + 480*B^2*a^13*b^2*d^2 - 60*B^2*a*b^14*d^2)

)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4) + (((16*(8*B*a*b^15*d^4 + 96*B*a^3*b^13*

d^4 + 360*B*a^5*b^11*d^4 + 640*B*a^7*b^9*d^4 + 600*B*a^9*b^7*d^4 + 288*B*a^11*b^5*d^4 + 56*B*a^13*b^3*d^4))/(a

^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) - (4*tan(c + d*x)^(1/2)*(-4*(9*B^2*a^9 + 49*

B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2)*

(32*b^17*d^4 + 160*a^2*b^15*d^4 + 288*a^4*b^13*d^4 + 160*a^6*b^11*d^4 - 160*a^8*b^9*d^4 - 288*a^10*b^7*d^4 - 1

60*a^12*b^5*d^4 - 32*a^14*b^3*d^4))/((a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4)*(b^11

*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2

*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^

2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(

b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 +

6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^11*d^2 +

4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*

d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d

^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6

*b^5*d^2 + a^8*b^3*d^2)) + (((16*tan(c + d*x)^(1/2)*(2*B^4*b^13 + 9*B^4*a^12*b + 4*B^4*a^2*b^11 + 2*B^4*a^4*b^

9 - 49*B^4*a^6*b^7 + 7*B^4*a^8*b^5 + 33*B^4*a^10*b^3))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*

a^6*b^2*d^4) - (((16*(30*B^3*a^6*b^8*d^2 - 224*B^3*a^4*b^10*d^2 - 18*B^3*a^14*d^2 + 600*B^3*a^8*b^6*d^2 + 388*

B^3*a^10*b^4*d^2 + 24*B^3*a^12*b^2*d^2))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) +

(((16*tan(c + d*x)^(1/2)*(72*B^2*a^15*d^2 - 52*B^2*a^3*b^12*d^2 + 72*B^2*a^5*b^10*d^2 + 448*B^2*a^7*b^8*d^2 +

1108*B^2*a^9*b^6*d^2 + 1132*B^2*a^11*b^4*d^2 + 480*B^2*a^13*b^2*d^2 - 60*B^2*a*b^14*d^2))/(a^8*d^4 + b^8*d^4

+ 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4) - (((16*(8*B*a*b^15*d^4 + 96*B*a^3*b^13*d^4 + 360*B*a^5*b^11*

d^4 + 640*B*a^7*b^9*d^4 + 600*B*a^9*b^7*d^4 + 288*B*a^11*b^5*d^4 + 56*B*a^13*b^3*d^4))/(a^8*d^5 + b^8*d^5 + 4*

a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) + (4*tan(c + d*x)^(1/2)*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*

a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2)*(32*b^17*d^4 + 160*a^

2*b^15*d^4 + 288*a^4*b^13*d^4 + 160*a^6*b^11*d^4 - 160*a^8*b^9*d^4 - 288*a^10*b^7*d^4 - 160*a^12*b^5*d^4 - 32*

a^14*b^3*d^4))/((a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4)*(b^11*d^2 + 4*a^2*b^9*d^2

+ 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^11*d^2 +

4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7

*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*

d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^

6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4

*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 +

a^8*b^3*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 +

4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^

2))))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*

d^2 + a^8*b^3*d^2))^(1/2)*1i)/(2*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)) - (

atan(((((((8*(4*B^3*a^4*b^11*d^2 - 304*B^3*a^6*b^9*d^2 - 120*B^3*a^8*b^7*d^2 + 320*B^3*a^10*b^5*d^2 + 148*B^3*

a^12*b^3*d^2 + 16*B^3*a^14*b*d^2))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5) + (((

((8*(96*B*a^3*b^14*d^4 + 480*B*a^5*b^12*d^4 + 960*B*a^7*b^10*d^4 + 960*B*a^9*b^8*d^4 + 480*B*a^11*b^6*d^4 + 96

*B*a^13*b^4*d^4))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5) - (4*tan(c + d*x)^(1/2

)*(-4*(B^2*a^9 + 25*B^2*a^5*b^4 + 10*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 +

a^8*b^3*d^2))^(1/2)*(32*b^18*d^4 + 160*a^2*b^16*d^4 + 288*a^4*b^14*d^4 + 160*a^6*b^12*d^4 - 160*a^8*b^10*d^4 -

288*a^10*b^8*d^4 - 160*a^12*b^6*d^4 - 32*a^14*b^4*d^4))/((b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4

+ 4*a^6*b^3*d^4)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(B^2*a^9 + 25

*B^2*a^5*b^4 + 10*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2)

)/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)) + (16*tan(c + d*x)^(1/2)*(60*B^

2*a^3*b^13*d^2 + 52*B^2*a^5*b^11*d^2 + 128*B^2*a^7*b^9*d^2 + 424*B^2*a^9*b^7*d^2 + 380*B^2*a^11*b^5*d^2 + 100*

B^2*a^13*b^3*d^2 + 8*B^2*a^15*b*d^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*(

-4*(B^2*a^9 + 25*B^2*a^5*b^4 + 10*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8

*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(B^2*a^9 +

25*B^2*a^5*b^4 + 10*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1

/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)) + (16*tan(c + d*x)^(1/2)*(B^

4*a^14 - 2*B^4*a^4*b^10 - 4*B^4*a^6*b^8 - 27*B^4*a^8*b^6 + 15*B^4*a^10*b^4 + 9*B^4*a^12*b^2))/(b^9*d^4 + a^8*b

*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*(-4*(B^2*a^9 + 25*B^2*a^5*b^4 + 10*B^2*a^7*b^2)*(b^11*d

^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2)*1i)/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*

a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)) - (((((8*(4*B^3*a^4*b^11*d^2 - 304*B^3*a^6*b^9*d^2 - 120*B^3*a^8*b

^7*d^2 + 320*B^3*a^10*b^5*d^2 + 148*B^3*a^12*b^3*d^2 + 16*B^3*a^14*b*d^2))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^

5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5) + (((((8*(96*B*a^3*b^14*d^4 + 480*B*a^5*b^12*d^4 + 960*B*a^7*b^10*d^4 + 960

*B*a^9*b^8*d^4 + 480*B*a^11*b^6*d^4 + 96*B*a^13*b^4*d^4))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5

+ 4*a^6*b^3*d^5) + (4*tan(c + d*x)^(1/2)*(-4*(B^2*a^9 + 25*B^2*a^5*b^4 + 10*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^

9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2)*(32*b^18*d^4 + 160*a^2*b^16*d^4 + 288*a^4*b^14*d^4

+ 160*a^6*b^12*d^4 - 160*a^8*b^10*d^4 - 288*a^10*b^8*d^4 - 160*a^12*b^6*d^4 - 32*a^14*b^4*d^4))/((b^9*d^4 + a

^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^

5*d^2 + a^8*b^3*d^2)))*(-4*(B^2*a^9 + 25*B^2*a^5*b^4 + 10*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d

^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b

^3*d^2)) - (16*tan(c + d*x)^(1/2)*(60*B^2*a^3*b^13*d^2 + 52*B^2*a^5*b^11*d^2 + 128*B^2*a^7*b^9*d^2 + 424*B^2*a

^9*b^7*d^2 + 380*B^2*a^11*b^5*d^2 + 100*B^2*a^13*b^3*d^2 + 8*B^2*a^15*b*d^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7

*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*(-4*(B^2*a^9 + 25*B^2*a^5*b^4 + 10*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d

^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6

*b^5*d^2 + a^8*b^3*d^2)))*(-4*(B^2*a^9 + 25*B^2*a^5*b^4 + 10*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^

7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^

8*b^3*d^2)) - (16*tan(c + d*x)^(1/2)*(B^4*a^14 - 2*B^4*a^4*b^10 - 4*B^4*a^6*b^8 - 27*B^4*a^8*b^6 + 15*B^4*a^10

*b^4 + 9*B^4*a^12*b^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*(-4*(B^2*a^9 +

25*B^2*a^5*b^4 + 10*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/

2)*1i)/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))/((16*(B^5*a^13 + 10*B^5*a

^7*b^6 + 27*B^5*a^9*b^4 + 10*B^5*a^11*b^2))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d

^5) + (((((8*(4*B^3*a^4*b^11*d^2 - 304*B^3*a^6*b^9*d^2 - 120*B^3*a^8*b^7*d^2 + 320*B^3*a^10*b^5*d^2 + 148*B^3*