∫ (a + b tan4(c + dx))4dx

3.381 \(\int (a+b \tan ^4(c+d x))^4 \, dx\)

Optimal. Leaf size=216 \[ \frac{b^2 \left (6 a^2+4 a b+b^2\right ) \tan ^7(c+d x)}{7 d}-\frac{b^2 \left (6 a^2+4 a b+b^2\right ) \tan ^5(c+d x)}{5 d}+\frac{b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \tan ^3(c+d x)}{3 d}-\frac{b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \tan (c+d x)}{d}+\frac{b^3 (4 a+b) \tan ^{11}(c+d x)}{11 d}-\frac{b^3 (4 a+b) \tan ^9(c+d x)}{9 d}+x (a+b)^4+\frac{b^4 \tan ^{15}(c+d x)}{15 d}-\frac{b^4 \tan ^{13}(c+d x)}{13 d} \]

[Out]

(a + b)^4*x - (b*(2*a + b)*(2*a^2 + 2*a*b + b^2)*Tan[c + d*x])/d + (b*(2*a + b)*(2*a^2 + 2*a*b + b^2)*Tan[c +

d*x]^3)/(3*d) - (b^2*(6*a^2 + 4*a*b + b^2)*Tan[c + d*x]^5)/(5*d) + (b^2*(6*a^2 + 4*a*b + b^2)*Tan[c + d*x]^7)/

(7*d) - (b^3*(4*a + b)*Tan[c + d*x]^9)/(9*d) + (b^3*(4*a + b)*Tan[c + d*x]^11)/(11*d) - (b^4*Tan[c + d*x]^13)/

(13*d) + (b^4*Tan[c + d*x]^15)/(15*d)

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Rubi [A] time = 0.129335, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3661, 1154, 203} \[ \frac{b^2 \left (6 a^2+4 a b+b^2\right ) \tan ^7(c+d x)}{7 d}-\frac{b^2 \left (6 a^2+4 a b+b^2\right ) \tan ^5(c+d x)}{5 d}+\frac{b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \tan ^3(c+d x)}{3 d}-\frac{b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \tan (c+d x)}{d}+\frac{b^3 (4 a+b) \tan ^{11}(c+d x)}{11 d}-\frac{b^3 (4 a+b) \tan ^9(c+d x)}{9 d}+x (a+b)^4+\frac{b^4 \tan ^{15}(c+d x)}{15 d}-\frac{b^4 \tan ^{13}(c+d x)}{13 d} \]

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*Tan[c + d*x]^4)^4,x]

[Out]

(a + b)^4*x - (b*(2*a + b)*(2*a^2 + 2*a*b + b^2)*Tan[c + d*x])/d + (b*(2*a + b)*(2*a^2 + 2*a*b + b^2)*Tan[c +

d*x]^3)/(3*d) - (b^2*(6*a^2 + 4*a*b + b^2)*Tan[c + d*x]^5)/(5*d) + (b^2*(6*a^2 + 4*a*b + b^2)*Tan[c + d*x]^7)/

(7*d) - (b^3*(4*a + b)*Tan[c + d*x]^9)/(9*d) + (b^3*(4*a + b)*Tan[c + d*x]^11)/(11*d) - (b^4*Tan[c + d*x]^13)/

(13*d) + (b^4*Tan[c + d*x]^15)/(15*d)

Rule 3661

Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x]

, x]}, Dist[(c*ff)/f, Subst[Int[(a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ

[{a, b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rule 1154

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a

+ c*x^4)^p, x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rule 203

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

FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \left (a+b \tan ^4(c+d x)\right )^4 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^4\right )^4}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-b (2 a+b) \left (2 a^2+2 a b+b^2\right )+b (2 a+b) \left (2 a^2+2 a b+b^2\right ) x^2-b^2 \left (6 a^2+4 a b+b^2\right ) x^4+b^2 \left (6 a^2+4 a b+b^2\right ) x^6-b^3 (4 a+b) x^8+b^3 (4 a+b) x^{10}-b^4 x^{12}+b^4 x^{14}+\frac{(a+b)^4}{1+x^2}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \tan (c+d x)}{d}+\frac{b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \tan ^3(c+d x)}{3 d}-\frac{b^2 \left (6 a^2+4 a b+b^2\right ) \tan ^5(c+d x)}{5 d}+\frac{b^2 \left (6 a^2+4 a b+b^2\right ) \tan ^7(c+d x)}{7 d}-\frac{b^3 (4 a+b) \tan ^9(c+d x)}{9 d}+\frac{b^3 (4 a+b) \tan ^{11}(c+d x)}{11 d}-\frac{b^4 \tan ^{13}(c+d x)}{13 d}+\frac{b^4 \tan ^{15}(c+d x)}{15 d}+\frac{(a+b)^4 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=(a+b)^4 x-\frac{b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \tan (c+d x)}{d}+\frac{b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \tan ^3(c+d x)}{3 d}-\frac{b^2 \left (6 a^2+4 a b+b^2\right ) \tan ^5(c+d x)}{5 d}+\frac{b^2 \left (6 a^2+4 a b+b^2\right ) \tan ^7(c+d x)}{7 d}-\frac{b^3 (4 a+b) \tan ^9(c+d x)}{9 d}+\frac{b^3 (4 a+b) \tan ^{11}(c+d x)}{11 d}-\frac{b^4 \tan ^{13}(c+d x)}{13 d}+\frac{b^4 \tan ^{15}(c+d x)}{15 d}\\ \end{align*}

Mathematica [A] time = 4.26395, size = 196, normalized size = 0.91 \[ \frac{b \tan (c+d x) \left (6435 b \left (6 a^2+4 a b+b^2\right ) \tan ^6(c+d x)-9009 b \left (6 a^2+4 a b+b^2\right ) \tan ^4(c+d x)+15015 \left (6 a^2 b+4 a^3+4 a b^2+b^3\right ) \tan ^2(c+d x)-45045 \left (6 a^2 b+4 a^3+4 a b^2+b^3\right )+4095 b^2 (4 a+b) \tan ^{10}(c+d x)-5005 b^2 (4 a+b) \tan ^8(c+d x)+3003 b^3 \tan ^{14}(c+d x)-3465 b^3 \tan ^{12}(c+d x)\right )}{45045 d}+\frac{(a+b)^4 \tan ^{-1}(\tan (c+d x))}{d} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*Tan[c + d*x]^4)^4,x]

[Out]

((a + b)^4*ArcTan[Tan[c + d*x]])/d + (b*Tan[c + d*x]*(-45045*(4*a^3 + 6*a^2*b + 4*a*b^2 + b^3) + 15015*(4*a^3

+ 6*a^2*b + 4*a*b^2 + b^3)*Tan[c + d*x]^2 - 9009*b*(6*a^2 + 4*a*b + b^2)*Tan[c + d*x]^4 + 6435*b*(6*a^2 + 4*a*

b + b^2)*Tan[c + d*x]^6 - 5005*b^2*(4*a + b)*Tan[c + d*x]^8 + 4095*b^2*(4*a + b)*Tan[c + d*x]^10 - 3465*b^3*Ta

n[c + d*x]^12 + 3003*b^3*Tan[c + d*x]^14))/(45045*d)

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Maple [B] time = 0.005, size = 412, normalized size = 1.9 \begin{align*}{\frac{{b}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{7}}{7\,d}}-{\frac{{b}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{9}}{9\,d}}+{\frac{{b}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{11}}{11\,d}}-{\frac{{b}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{13}}{13\,d}}+{\frac{{b}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{15}}{15\,d}}-{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{5}{b}^{4}}{5\,d}}+{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{3}{b}^{4}}{3\,d}}-{\frac{{b}^{4}\tan \left ( dx+c \right ) }{d}}+{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{4}}{d}}+{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){b}^{4}}{d}}-{\frac{4\, \left ( \tan \left ( dx+c \right ) \right ) ^{5}a{b}^{3}}{5\,d}}+2\,{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{3}{a}^{2}{b}^{2}}{d}}+{\frac{4\, \left ( \tan \left ( dx+c \right ) \right ) ^{3}a{b}^{3}}{3\,d}}-4\,{\frac{\tan \left ( dx+c \right ){a}^{3}b}{d}}-6\,{\frac{{a}^{2}{b}^{2}\tan \left ( dx+c \right ) }{d}}-4\,{\frac{a{b}^{3}\tan \left ( dx+c \right ) }{d}}+4\,{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{3}b}{d}}+6\,{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{2}{b}^{2}}{d}}+4\,{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ) a{b}^{3}}{d}}+{\frac{4\, \left ( \tan \left ( dx+c \right ) \right ) ^{3}{a}^{3}b}{3\,d}}+{\frac{4\, \left ( \tan \left ( dx+c \right ) \right ) ^{11}a{b}^{3}}{11\,d}}-{\frac{4\,a{b}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{9}}{9\,d}}+{\frac{6\, \left ( \tan \left ( dx+c \right ) \right ) ^{7}{a}^{2}{b}^{2}}{7\,d}}+{\frac{4\, \left ( \tan \left ( dx+c \right ) \right ) ^{7}a{b}^{3}}{7\,d}}-{\frac{6\, \left ( \tan \left ( dx+c \right ) \right ) ^{5}{a}^{2}{b}^{2}}{5\,d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*tan(d*x+c)^4)^4,x)

[Out]

1/7*b^4*tan(d*x+c)^7/d-1/9*b^4*tan(d*x+c)^9/d+1/11*b^4*tan(d*x+c)^11/d-1/13*b^4*tan(d*x+c)^13/d+1/15*b^4*tan(d

*x+c)^15/d-1/5/d*tan(d*x+c)^5*b^4+1/3/d*tan(d*x+c)^3*b^4-1/d*b^4*tan(d*x+c)+1/d*arctan(tan(d*x+c))*a^4+1/d*arc

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

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

4/d*arctan(tan(d*x+c))*a*b^3+4/3/d*tan(d*x+c)^3*a^3*b+4/11/d*tan(d*x+c)^11*a*b^3-4/9/d*a*b^3*tan(d*x+c)^9+6/7/

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

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Maxima [A] time = 1.51872, size = 358, normalized size = 1.66 \begin{align*} a^{4} x + \frac{4 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{3} b}{3 \, d} + \frac{2 \,{\left (15 \, \tan \left (d x + c\right )^{7} - 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 105 \, d x + 105 \, c - 105 \, \tan \left (d x + c\right )\right )} a^{2} b^{2}}{35 \, d} + \frac{4 \,{\left (315 \, \tan \left (d x + c\right )^{11} - 385 \, \tan \left (d x + c\right )^{9} + 495 \, \tan \left (d x + c\right )^{7} - 693 \, \tan \left (d x + c\right )^{5} + 1155 \, \tan \left (d x + c\right )^{3} + 3465 \, d x + 3465 \, c - 3465 \, \tan \left (d x + c\right )\right )} a b^{3}}{3465 \, d} + \frac{{\left (3003 \, \tan \left (d x + c\right )^{15} - 3465 \, \tan \left (d x + c\right )^{13} + 4095 \, \tan \left (d x + c\right )^{11} - 5005 \, \tan \left (d x + c\right )^{9} + 6435 \, \tan \left (d x + c\right )^{7} - 9009 \, \tan \left (d x + c\right )^{5} + 15015 \, \tan \left (d x + c\right )^{3} + 45045 \, d x + 45045 \, c - 45045 \, \tan \left (d x + c\right )\right )} b^{4}}{45045 \, d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+tan(d*x+c)^4*b)^4,x, algorithm="maxima")

[Out]

a^4*x + 4/3*(tan(d*x + c)^3 + 3*d*x + 3*c - 3*tan(d*x + c))*a^3*b/d + 2/35*(15*tan(d*x + c)^7 - 21*tan(d*x + c

)^5 + 35*tan(d*x + c)^3 + 105*d*x + 105*c - 105*tan(d*x + c))*a^2*b^2/d + 4/3465*(315*tan(d*x + c)^11 - 385*ta

n(d*x + c)^9 + 495*tan(d*x + c)^7 - 693*tan(d*x + c)^5 + 1155*tan(d*x + c)^3 + 3465*d*x + 3465*c - 3465*tan(d*

x + c))*a*b^3/d + 1/45045*(3003*tan(d*x + c)^15 - 3465*tan(d*x + c)^13 + 4095*tan(d*x + c)^11 - 5005*tan(d*x +

c)^9 + 6435*tan(d*x + c)^7 - 9009*tan(d*x + c)^5 + 15015*tan(d*x + c)^3 + 45045*d*x + 45045*c - 45045*tan(d*x

+ c))*b^4/d

________________________________________________________________________________________

Fricas [A] time = 1.54464, size = 564, normalized size = 2.61 \begin{align*} \frac{3003 \, b^{4} \tan \left (d x + c\right )^{15} - 3465 \, b^{4} \tan \left (d x + c\right )^{13} + 4095 \,{\left (4 \, a b^{3} + b^{4}\right )} \tan \left (d x + c\right )^{11} - 5005 \,{\left (4 \, a b^{3} + b^{4}\right )} \tan \left (d x + c\right )^{9} + 6435 \,{\left (6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \tan \left (d x + c\right )^{7} - 9009 \,{\left (6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \tan \left (d x + c\right )^{5} + 15015 \,{\left (4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \tan \left (d x + c\right )^{3} + 45045 \,{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} d x - 45045 \,{\left (4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \tan \left (d x + c\right )}{45045 \, d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+tan(d*x+c)^4*b)^4,x, algorithm="fricas")

[Out]

1/45045*(3003*b^4*tan(d*x + c)^15 - 3465*b^4*tan(d*x + c)^13 + 4095*(4*a*b^3 + b^4)*tan(d*x + c)^11 - 5005*(4*

a*b^3 + b^4)*tan(d*x + c)^9 + 6435*(6*a^2*b^2 + 4*a*b^3 + b^4)*tan(d*x + c)^7 - 9009*(6*a^2*b^2 + 4*a*b^3 + b^

4)*tan(d*x + c)^5 + 15015*(4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*tan(d*x + c)^3 + 45045*(a^4 + 4*a^3*b + 6*a^2*

b^2 + 4*a*b^3 + b^4)*d*x - 45045*(4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*tan(d*x + c))/d

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Sympy [A] time = 10.6628, size = 386, normalized size = 1.79 \begin{align*} \begin{cases} a^{4} x + 4 a^{3} b x + \frac{4 a^{3} b \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac{4 a^{3} b \tan{\left (c + d x \right )}}{d} + 6 a^{2} b^{2} x + \frac{6 a^{2} b^{2} \tan ^{7}{\left (c + d x \right )}}{7 d} - \frac{6 a^{2} b^{2} \tan ^{5}{\left (c + d x \right )}}{5 d} + \frac{2 a^{2} b^{2} \tan ^{3}{\left (c + d x \right )}}{d} - \frac{6 a^{2} b^{2} \tan{\left (c + d x \right )}}{d} + 4 a b^{3} x + \frac{4 a b^{3} \tan ^{11}{\left (c + d x \right )}}{11 d} - \frac{4 a b^{3} \tan ^{9}{\left (c + d x \right )}}{9 d} + \frac{4 a b^{3} \tan ^{7}{\left (c + d x \right )}}{7 d} - \frac{4 a b^{3} \tan ^{5}{\left (c + d x \right )}}{5 d} + \frac{4 a b^{3} \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac{4 a b^{3} \tan{\left (c + d x \right )}}{d} + b^{4} x + \frac{b^{4} \tan ^{15}{\left (c + d x \right )}}{15 d} - \frac{b^{4} \tan ^{13}{\left (c + d x \right )}}{13 d} + \frac{b^{4} \tan ^{11}{\left (c + d x \right )}}{11 d} - \frac{b^{4} \tan ^{9}{\left (c + d x \right )}}{9 d} + \frac{b^{4} \tan ^{7}{\left (c + d x \right )}}{7 d} - \frac{b^{4} \tan ^{5}{\left (c + d x \right )}}{5 d} + \frac{b^{4} \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac{b^{4} \tan{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a + b \tan ^{4}{\left (c \right )}\right )^{4} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+tan(d*x+c)**4*b)**4,x)

[Out]

Piecewise((a**4*x + 4*a**3*b*x + 4*a**3*b*tan(c + d*x)**3/(3*d) - 4*a**3*b*tan(c + d*x)/d + 6*a**2*b**2*x + 6*

a**2*b**2*tan(c + d*x)**7/(7*d) - 6*a**2*b**2*tan(c + d*x)**5/(5*d) + 2*a**2*b**2*tan(c + d*x)**3/d - 6*a**2*b

**2*tan(c + d*x)/d + 4*a*b**3*x + 4*a*b**3*tan(c + d*x)**11/(11*d) - 4*a*b**3*tan(c + d*x)**9/(9*d) + 4*a*b**3

*tan(c + d*x)**7/(7*d) - 4*a*b**3*tan(c + d*x)**5/(5*d) + 4*a*b**3*tan(c + d*x)**3/(3*d) - 4*a*b**3*tan(c + d*

x)/d + b**4*x + b**4*tan(c + d*x)**15/(15*d) - b**4*tan(c + d*x)**13/(13*d) + b**4*tan(c + d*x)**11/(11*d) - b

**4*tan(c + d*x)**9/(9*d) + b**4*tan(c + d*x)**7/(7*d) - b**4*tan(c + d*x)**5/(5*d) + b**4*tan(c + d*x)**3/(3*

d) - b**4*tan(c + d*x)/d, Ne(d, 0)), (x*(a + b*tan(c)**4)**4, True))

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Giac [B] time = 156.397, size = 10450, normalized size = 48.38 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+tan(d*x+c)^4*b)^4,x, algorithm="giac")

[Out]

1/45045*(45045*a^4*d*x*tan(d*x)^15*tan(c)^15 + 180180*a^3*b*d*x*tan(d*x)^15*tan(c)^15 + 270270*a^2*b^2*d*x*tan

(d*x)^15*tan(c)^15 + 180180*a*b^3*d*x*tan(d*x)^15*tan(c)^15 + 45045*b^4*d*x*tan(d*x)^15*tan(c)^15 - 675675*a^4

*d*x*tan(d*x)^14*tan(c)^14 - 2702700*a^3*b*d*x*tan(d*x)^14*tan(c)^14 - 4054050*a^2*b^2*d*x*tan(d*x)^14*tan(c)^

14 - 2702700*a*b^3*d*x*tan(d*x)^14*tan(c)^14 - 675675*b^4*d*x*tan(d*x)^14*tan(c)^14 + 180180*a^3*b*tan(d*x)^15

*tan(c)^14 + 270270*a^2*b^2*tan(d*x)^15*tan(c)^14 + 180180*a*b^3*tan(d*x)^15*tan(c)^14 + 45045*b^4*tan(d*x)^15

*tan(c)^14 + 180180*a^3*b*tan(d*x)^14*tan(c)^15 + 270270*a^2*b^2*tan(d*x)^14*tan(c)^15 + 180180*a*b^3*tan(d*x)

^14*tan(c)^15 + 45045*b^4*tan(d*x)^14*tan(c)^15 + 4729725*a^4*d*x*tan(d*x)^13*tan(c)^13 + 18918900*a^3*b*d*x*t

an(d*x)^13*tan(c)^13 + 28378350*a^2*b^2*d*x*tan(d*x)^13*tan(c)^13 + 18918900*a*b^3*d*x*tan(d*x)^13*tan(c)^13 +

4729725*b^4*d*x*tan(d*x)^13*tan(c)^13 - 60060*a^3*b*tan(d*x)^15*tan(c)^12 - 90090*a^2*b^2*tan(d*x)^15*tan(c)^

12 - 60060*a*b^3*tan(d*x)^15*tan(c)^12 - 15015*b^4*tan(d*x)^15*tan(c)^12 - 2702700*a^3*b*tan(d*x)^14*tan(c)^13

- 4054050*a^2*b^2*tan(d*x)^14*tan(c)^13 - 2702700*a*b^3*tan(d*x)^14*tan(c)^13 - 675675*b^4*tan(d*x)^14*tan(c)

^13 - 2702700*a^3*b*tan(d*x)^13*tan(c)^14 - 4054050*a^2*b^2*tan(d*x)^13*tan(c)^14 - 2702700*a*b^3*tan(d*x)^13*

tan(c)^14 - 675675*b^4*tan(d*x)^13*tan(c)^14 - 60060*a^3*b*tan(d*x)^12*tan(c)^15 - 90090*a^2*b^2*tan(d*x)^12*t

an(c)^15 - 60060*a*b^3*tan(d*x)^12*tan(c)^15 - 15015*b^4*tan(d*x)^12*tan(c)^15 - 20495475*a^4*d*x*tan(d*x)^12*

tan(c)^12 - 81981900*a^3*b*d*x*tan(d*x)^12*tan(c)^12 - 122972850*a^2*b^2*d*x*tan(d*x)^12*tan(c)^12 - 81981900*

a*b^3*d*x*tan(d*x)^12*tan(c)^12 - 20495475*b^4*d*x*tan(d*x)^12*tan(c)^12 + 54054*a^2*b^2*tan(d*x)^15*tan(c)^10

+ 36036*a*b^3*tan(d*x)^15*tan(c)^10 + 9009*b^4*tan(d*x)^15*tan(c)^10 + 720720*a^3*b*tan(d*x)^14*tan(c)^11 + 1

351350*a^2*b^2*tan(d*x)^14*tan(c)^11 + 900900*a*b^3*tan(d*x)^14*tan(c)^11 + 225225*b^4*tan(d*x)^14*tan(c)^11 +

18558540*a^3*b*tan(d*x)^13*tan(c)^12 + 28378350*a^2*b^2*tan(d*x)^13*tan(c)^12 + 18918900*a*b^3*tan(d*x)^13*ta

n(c)^12 + 4729725*b^4*tan(d*x)^13*tan(c)^12 + 18558540*a^3*b*tan(d*x)^12*tan(c)^13 + 28378350*a^2*b^2*tan(d*x)

^12*tan(c)^13 + 18918900*a*b^3*tan(d*x)^12*tan(c)^13 + 4729725*b^4*tan(d*x)^12*tan(c)^13 + 720720*a^3*b*tan(d*

x)^11*tan(c)^14 + 1351350*a^2*b^2*tan(d*x)^11*tan(c)^14 + 900900*a*b^3*tan(d*x)^11*tan(c)^14 + 225225*b^4*tan(

d*x)^11*tan(c)^14 + 54054*a^2*b^2*tan(d*x)^10*tan(c)^15 + 36036*a*b^3*tan(d*x)^10*tan(c)^15 + 9009*b^4*tan(d*x

)^10*tan(c)^15 + 61486425*a^4*d*x*tan(d*x)^11*tan(c)^11 + 245945700*a^3*b*d*x*tan(d*x)^11*tan(c)^11 + 36891855

0*a^2*b^2*d*x*tan(d*x)^11*tan(c)^11 + 245945700*a*b^3*d*x*tan(d*x)^11*tan(c)^11 + 61486425*b^4*d*x*tan(d*x)^11

*tan(c)^11 - 38610*a^2*b^2*tan(d*x)^15*tan(c)^8 - 25740*a*b^3*tan(d*x)^15*tan(c)^8 - 6435*b^4*tan(d*x)^15*tan(

c)^8 - 810810*a^2*b^2*tan(d*x)^14*tan(c)^9 - 540540*a*b^3*tan(d*x)^14*tan(c)^9 - 135135*b^4*tan(d*x)^14*tan(c)

^9 - 3963960*a^3*b*tan(d*x)^13*tan(c)^10 - 9459450*a^2*b^2*tan(d*x)^13*tan(c)^10 - 6306300*a*b^3*tan(d*x)^13*t

an(c)^10 - 1576575*b^4*tan(d*x)^13*tan(c)^10 - 77477400*a^3*b*tan(d*x)^12*tan(c)^11 - 122972850*a^2*b^2*tan(d*

x)^12*tan(c)^11 - 81981900*a*b^3*tan(d*x)^12*tan(c)^11 - 20495475*b^4*tan(d*x)^12*tan(c)^11 - 77477400*a^3*b*t

an(d*x)^11*tan(c)^12 - 122972850*a^2*b^2*tan(d*x)^11*tan(c)^12 - 81981900*a*b^3*tan(d*x)^11*tan(c)^12 - 204954

75*b^4*tan(d*x)^11*tan(c)^12 - 3963960*a^3*b*tan(d*x)^10*tan(c)^13 - 9459450*a^2*b^2*tan(d*x)^10*tan(c)^13 - 6

306300*a*b^3*tan(d*x)^10*tan(c)^13 - 1576575*b^4*tan(d*x)^10*tan(c)^13 - 810810*a^2*b^2*tan(d*x)^9*tan(c)^14 -

540540*a*b^3*tan(d*x)^9*tan(c)^14 - 135135*b^4*tan(d*x)^9*tan(c)^14 - 38610*a^2*b^2*tan(d*x)^8*tan(c)^15 - 25

740*a*b^3*tan(d*x)^8*tan(c)^15 - 6435*b^4*tan(d*x)^8*tan(c)^15 - 135270135*a^4*d*x*tan(d*x)^10*tan(c)^10 - 541

080540*a^3*b*d*x*tan(d*x)^10*tan(c)^10 - 811620810*a^2*b^2*d*x*tan(d*x)^10*tan(c)^10 - 541080540*a*b^3*d*x*tan

(d*x)^10*tan(c)^10 - 135270135*b^4*d*x*tan(d*x)^10*tan(c)^10 + 20020*a*b^3*tan(d*x)^15*tan(c)^6 + 5005*b^4*tan

(d*x)^15*tan(c)^6 + 308880*a^2*b^2*tan(d*x)^14*tan(c)^7 + 386100*a*b^3*tan(d*x)^14*tan(c)^7 + 96525*b^4*tan(d*

x)^14*tan(c)^7 + 4594590*a^2*b^2*tan(d*x)^13*tan(c)^8 + 3783780*a*b^3*tan(d*x)^13*tan(c)^8 + 945945*b^4*tan(d*

x)^13*tan(c)^8 + 13213200*a^3*b*tan(d*x)^12*tan(c)^9 + 38468430*a^2*b^2*tan(d*x)^12*tan(c)^9 + 27327300*a*b^3*

tan(d*x)^12*tan(c)^9 + 6831825*b^4*tan(d*x)^12*tan(c)^9 + 219999780*a^3*b*tan(d*x)^11*tan(c)^10 + 365134770*a^

2*b^2*tan(d*x)^11*tan(c)^10 + 245945700*a*b^3*tan(d*x)^11*tan(c)^10 + 61486425*b^4*tan(d*x)^11*tan(c)^10 + 219

999780*a^3*b*tan(d*x)^10*tan(c)^11 + 365134770*a^2*b^2*tan(d*x)^10*tan(c)^11 + 245945700*a*b^3*tan(d*x)^10*tan

(c)^11 + 61486425*b^4*tan(d*x)^10*tan(c)^11 + 13213200*a^3*b*tan(d*x)^9*tan(c)^12 + 38468430*a^2*b^2*tan(d*x)^

9*tan(c)^12 + 27327300*a*b^3*tan(d*x)^9*tan(c)^12 + 6831825*b^4*tan(d*x)^9*tan(c)^12 + 4594590*a^2*b^2*tan(d*x

)^8*tan(c)^13 + 3783780*a*b^3*tan(d*x)^8*tan(c)^13 + 945945*b^4*tan(d*x)^8*tan(c)^13 + 308880*a^2*b^2*tan(d*x)

^7*tan(c)^14 + 386100*a*b^3*tan(d*x)^7*tan(c)^14 + 96525*b^4*tan(d*x)^7*tan(c)^14 + 20020*a*b^3*tan(d*x)^6*tan

(c)^15 + 5005*b^4*tan(d*x)^6*tan(c)^15 + 225450225*a^4*d*x*tan(d*x)^9*tan(c)^9 + 901800900*a^3*b*d*x*tan(d*x)^

9*tan(c)^9 + 1352701350*a^2*b^2*d*x*tan(d*x)^9*tan(c)^9 + 901800900*a*b^3*d*x*tan(d*x)^9*tan(c)^9 + 225450225*

b^4*d*x*tan(d*x)^9*tan(c)^9 - 16380*a*b^3*tan(d*x)^15*tan(c)^4 - 4095*b^4*tan(d*x)^15*tan(c)^4 - 300300*a*b^3*

tan(d*x)^14*tan(c)^5 - 75075*b^4*tan(d*x)^14*tan(c)^5 - 1081080*a^2*b^2*tan(d*x)^13*tan(c)^6 - 2702700*a*b^3*t

an(d*x)^13*tan(c)^6 - 675675*b^4*tan(d*x)^13*tan(c)^6 - 14054040*a^2*b^2*tan(d*x)^12*tan(c)^7 - 16396380*a*b^3

*tan(d*x)^12*tan(c)^7 - 4099095*b^4*tan(d*x)^12*tan(c)^7 - 29729700*a^3*b*tan(d*x)^11*tan(c)^8 - 99729630*a^2*

b^2*tan(d*x)^11*tan(c)^8 - 81981900*a*b^3*tan(d*x)^11*tan(c)^8 - 20495475*b^4*tan(d*x)^11*tan(c)^8 - 449909460

*a^3*b*tan(d*x)^10*tan(c)^9 - 777566790*a^2*b^2*tan(d*x)^10*tan(c)^9 - 541080540*a*b^3*tan(d*x)^10*tan(c)^9 -

135270135*b^4*tan(d*x)^10*tan(c)^9 - 449909460*a^3*b*tan(d*x)^9*tan(c)^10 - 777566790*a^2*b^2*tan(d*x)^9*tan(c

)^10 - 541080540*a*b^3*tan(d*x)^9*tan(c)^10 - 135270135*b^4*tan(d*x)^9*tan(c)^10 - 29729700*a^3*b*tan(d*x)^8*t

an(c)^11 - 99729630*a^2*b^2*tan(d*x)^8*tan(c)^11 - 81981900*a*b^3*tan(d*x)^8*tan(c)^11 - 20495475*b^4*tan(d*x)

^8*tan(c)^11 - 14054040*a^2*b^2*tan(d*x)^7*tan(c)^12 - 16396380*a*b^3*tan(d*x)^7*tan(c)^12 - 4099095*b^4*tan(d

*x)^7*tan(c)^12 - 1081080*a^2*b^2*tan(d*x)^6*tan(c)^13 - 2702700*a*b^3*tan(d*x)^6*tan(c)^13 - 675675*b^4*tan(d

*x)^6*tan(c)^13 - 300300*a*b^3*tan(d*x)^5*tan(c)^14 - 75075*b^4*tan(d*x)^5*tan(c)^14 - 16380*a*b^3*tan(d*x)^4*

tan(c)^15 - 4095*b^4*tan(d*x)^4*tan(c)^15 - 289864575*a^4*d*x*tan(d*x)^8*tan(c)^8 - 1159458300*a^3*b*d*x*tan(d

*x)^8*tan(c)^8 - 1739187450*a^2*b^2*d*x*tan(d*x)^8*tan(c)^8 - 1159458300*a*b^3*d*x*tan(d*x)^8*tan(c)^8 - 28986

4575*b^4*d*x*tan(d*x)^8*tan(c)^8 + 3465*b^4*tan(d*x)^15*tan(c)^2 + 65520*a*b^3*tan(d*x)^14*tan(c)^3 + 61425*b^

4*tan(d*x)^14*tan(c)^3 + 1021020*a*b^3*tan(d*x)^13*tan(c)^4 + 525525*b^4*tan(d*x)^13*tan(c)^4 + 2162160*a^2*b^

2*tan(d*x)^12*tan(c)^5 + 7747740*a*b^3*tan(d*x)^12*tan(c)^5 + 2927925*b^4*tan(d*x)^12*tan(c)^5 + 26486460*a^2*

b^2*tan(d*x)^11*tan(c)^6 + 39279240*a*b^3*tan(d*x)^11*tan(c)^6 + 12297285*b^4*tan(d*x)^11*tan(c)^6 + 47567520*

a^3*b*tan(d*x)^10*tan(c)^7 + 173513340*a^2*b^2*tan(d*x)^10*tan(c)^7 + 162522360*a*b^3*tan(d*x)^10*tan(c)^7 + 4

5090045*b^4*tan(d*x)^10*tan(c)^7 + 683783100*a^3*b*tan(d*x)^9*tan(c)^8 + 1214863650*a^2*b^2*tan(d*x)^9*tan(c)^

8 + 878017140*a*b^3*tan(d*x)^9*tan(c)^8 + 225450225*b^4*tan(d*x)^9*tan(c)^8 + 683783100*a^3*b*tan(d*x)^8*tan(c

)^9 + 1214863650*a^2*b^2*tan(d*x)^8*tan(c)^9 + 878017140*a*b^3*tan(d*x)^8*tan(c)^9 + 225450225*b^4*tan(d*x)^8*

tan(c)^9 + 47567520*a^3*b*tan(d*x)^7*tan(c)^10 + 173513340*a^2*b^2*tan(d*x)^7*tan(c)^10 + 162522360*a*b^3*tan(

d*x)^7*tan(c)^10 + 45090045*b^4*tan(d*x)^7*tan(c)^10 + 26486460*a^2*b^2*tan(d*x)^6*tan(c)^11 + 39279240*a*b^3*

tan(d*x)^6*tan(c)^11 + 12297285*b^4*tan(d*x)^6*tan(c)^11 + 2162160*a^2*b^2*tan(d*x)^5*tan(c)^12 + 7747740*a*b^

3*tan(d*x)^5*tan(c)^12 + 2927925*b^4*tan(d*x)^5*tan(c)^12 + 1021020*a*b^3*tan(d*x)^4*tan(c)^13 + 525525*b^4*ta

n(d*x)^4*tan(c)^13 + 65520*a*b^3*tan(d*x)^3*tan(c)^14 + 61425*b^4*tan(d*x)^3*tan(c)^14 + 3465*b^4*tan(d*x)^2*t

an(c)^15 + 289864575*a^4*d*x*tan(d*x)^7*tan(c)^7 + 1159458300*a^3*b*d*x*tan(d*x)^7*tan(c)^7 + 1739187450*a^2*b

^2*d*x*tan(d*x)^7*tan(c)^7 + 1159458300*a*b^3*d*x*tan(d*x)^7*tan(c)^7 + 289864575*b^4*d*x*tan(d*x)^7*tan(c)^7

- 3003*b^4*tan(d*x)^15 - 51975*b^4*tan(d*x)^14*tan(c) - 98280*a*b^3*tan(d*x)^13*tan(c)^2 - 429975*b^4*tan(d*x)

^13*tan(c)^2 - 1481480*a*b^3*tan(d*x)^12*tan(c)^3 - 2277275*b^4*tan(d*x)^12*tan(c)^3 - 2702700*a^2*b^2*tan(d*x

)^11*tan(c)^4 - 10810800*a*b^3*tan(d*x)^11*tan(c)^4 - 8783775*b^4*tan(d*x)^11*tan(c)^4 - 32540508*a^2*b^2*tan(

d*x)^10*tan(c)^5 - 52324272*a*b^3*tan(d*x)^10*tan(c)^5 - 27054027*b^4*tan(d*x)^10*tan(c)^5 - 55495440*a^3*b*ta

n(d*x)^9*tan(c)^6 - 208107900*a^2*b^2*tan(d*x)^9*tan(c)^6 - 204804600*a*b^3*tan(d*x)^9*tan(c)^6 - 75150075*b^4

*tan(d*x)^9*tan(c)^6 - 784864080*a^3*b*tan(d*x)^8*tan(c)^7 - 1408106700*a^2*b^2*tan(d*x)^8*tan(c)^7 - 10345935

60*a*b^3*tan(d*x)^8*tan(c)^7 - 289864575*b^4*tan(d*x)^8*tan(c)^7 - 784864080*a^3*b*tan(d*x)^7*tan(c)^8 - 14081

06700*a^2*b^2*tan(d*x)^7*tan(c)^8 - 1034593560*a*b^3*tan(d*x)^7*tan(c)^8 - 289864575*b^4*tan(d*x)^7*tan(c)^8 -

55495440*a^3*b*tan(d*x)^6*tan(c)^9 - 208107900*a^2*b^2*tan(d*x)^6*tan(c)^9 - 204804600*a*b^3*tan(d*x)^6*tan(c

)^9 - 75150075*b^4*tan(d*x)^6*tan(c)^9 - 32540508*a^2*b^2*tan(d*x)^5*tan(c)^10 - 52324272*a*b^3*tan(d*x)^5*tan

(c)^10 - 27054027*b^4*tan(d*x)^5*tan(c)^10 - 2702700*a^2*b^2*tan(d*x)^4*tan(c)^11 - 10810800*a*b^3*tan(d*x)^4*

tan(c)^11 - 8783775*b^4*tan(d*x)^4*tan(c)^11 - 1481480*a*b^3*tan(d*x)^3*tan(c)^12 - 2277275*b^4*tan(d*x)^3*tan

(c)^12 - 98280*a*b^3*tan(d*x)^2*tan(c)^13 - 429975*b^4*tan(d*x)^2*tan(c)^13 - 51975*b^4*tan(d*x)*tan(c)^14 - 3

003*b^4*tan(c)^15 - 225450225*a^4*d*x*tan(d*x)^6*tan(c)^6 - 901800900*a^3*b*d*x*tan(d*x)^6*tan(c)^6 - 13527013

50*a^2*b^2*d*x*tan(d*x)^6*tan(c)^6 - 901800900*a*b^3*d*x*tan(d*x)^6*tan(c)^6 - 225450225*b^4*d*x*tan(d*x)^6*ta

n(c)^6 + 3465*b^4*tan(d*x)^13 + 65520*a*b^3*tan(d*x)^12*tan(c) + 61425*b^4*tan(d*x)^12*tan(c) + 1021020*a*b^3*

tan(d*x)^11*tan(c)^2 + 525525*b^4*tan(d*x)^11*tan(c)^2 + 2162160*a^2*b^2*tan(d*x)^10*tan(c)^3 + 7747740*a*b^3*

tan(d*x)^10*tan(c)^3 + 2927925*b^4*tan(d*x)^10*tan(c)^3 + 26486460*a^2*b^2*tan(d*x)^9*tan(c)^4 + 39279240*a*b^

3*tan(d*x)^9*tan(c)^4 + 12297285*b^4*tan(d*x)^9*tan(c)^4 + 47567520*a^3*b*tan(d*x)^8*tan(c)^5 + 173513340*a^2*

b^2*tan(d*x)^8*tan(c)^5 + 162522360*a*b^3*tan(d*x)^8*tan(c)^5 + 45090045*b^4*tan(d*x)^8*tan(c)^5 + 683783100*a

^3*b*tan(d*x)^7*tan(c)^6 + 1214863650*a^2*b^2*tan(d*x)^7*tan(c)^6 + 878017140*a*b^3*tan(d*x)^7*tan(c)^6 + 2254

50225*b^4*tan(d*x)^7*tan(c)^6 + 683783100*a^3*b*tan(d*x)^6*tan(c)^7 + 1214863650*a^2*b^2*tan(d*x)^6*tan(c)^7 +

878017140*a*b^3*tan(d*x)^6*tan(c)^7 + 225450225*b^4*tan(d*x)^6*tan(c)^7 + 47567520*a^3*b*tan(d*x)^5*tan(c)^8

+ 173513340*a^2*b^2*tan(d*x)^5*tan(c)^8 + 162522360*a*b^3*tan(d*x)^5*tan(c)^8 + 45090045*b^4*tan(d*x)^5*tan(c)

^8 + 26486460*a^2*b^2*tan(d*x)^4*tan(c)^9 + 39279240*a*b^3*tan(d*x)^4*tan(c)^9 + 12297285*b^4*tan(d*x)^4*tan(c

)^9 + 2162160*a^2*b^2*tan(d*x)^3*tan(c)^10 + 7747740*a*b^3*tan(d*x)^3*tan(c)^10 + 2927925*b^4*tan(d*x)^3*tan(c

)^10 + 1021020*a*b^3*tan(d*x)^2*tan(c)^11 + 525525*b^4*tan(d*x)^2*tan(c)^11 + 65520*a*b^3*tan(d*x)*tan(c)^12 +

61425*b^4*tan(d*x)*tan(c)^12 + 3465*b^4*tan(c)^13 + 135270135*a^4*d*x*tan(d*x)^5*tan(c)^5 + 541080540*a^3*b*d

*x*tan(d*x)^5*tan(c)^5 + 811620810*a^2*b^2*d*x*tan(d*x)^5*tan(c)^5 + 541080540*a*b^3*d*x*tan(d*x)^5*tan(c)^5 +

135270135*b^4*d*x*tan(d*x)^5*tan(c)^5 - 16380*a*b^3*tan(d*x)^11 - 4095*b^4*tan(d*x)^11 - 300300*a*b^3*tan(d*x

)^10*tan(c) - 75075*b^4*tan(d*x)^10*tan(c) - 1081080*a^2*b^2*tan(d*x)^9*tan(c)^2 - 2702700*a*b^3*tan(d*x)^9*ta

n(c)^2 - 675675*b^4*tan(d*x)^9*tan(c)^2 - 14054040*a^2*b^2*tan(d*x)^8*tan(c)^3 - 16396380*a*b^3*tan(d*x)^8*tan

(c)^3 - 4099095*b^4*tan(d*x)^8*tan(c)^3 - 29729700*a^3*b*tan(d*x)^7*tan(c)^4 - 99729630*a^2*b^2*tan(d*x)^7*tan

(c)^4 - 81981900*a*b^3*tan(d*x)^7*tan(c)^4 - 20495475*b^4*tan(d*x)^7*tan(c)^4 - 449909460*a^3*b*tan(d*x)^6*tan

(c)^5 - 777566790*a^2*b^2*tan(d*x)^6*tan(c)^5 - 541080540*a*b^3*tan(d*x)^6*tan(c)^5 - 135270135*b^4*tan(d*x)^6

*tan(c)^5 - 449909460*a^3*b*tan(d*x)^5*tan(c)^6 - 777566790*a^2*b^2*tan(d*x)^5*tan(c)^6 - 541080540*a*b^3*tan(

d*x)^5*tan(c)^6 - 135270135*b^4*tan(d*x)^5*tan(c)^6 - 29729700*a^3*b*tan(d*x)^4*tan(c)^7 - 99729630*a^2*b^2*ta

n(d*x)^4*tan(c)^7 - 81981900*a*b^3*tan(d*x)^4*tan(c)^7 - 20495475*b^4*tan(d*x)^4*tan(c)^7 - 14054040*a^2*b^2*t

an(d*x)^3*tan(c)^8 - 16396380*a*b^3*tan(d*x)^3*tan(c)^8 - 4099095*b^4*tan(d*x)^3*tan(c)^8 - 1081080*a^2*b^2*ta

n(d*x)^2*tan(c)^9 - 2702700*a*b^3*tan(d*x)^2*tan(c)^9 - 675675*b^4*tan(d*x)^2*tan(c)^9 - 300300*a*b^3*tan(d*x)

*tan(c)^10 - 75075*b^4*tan(d*x)*tan(c)^10 - 16380*a*b^3*tan(c)^11 - 4095*b^4*tan(c)^11 - 61486425*a^4*d*x*tan(

d*x)^4*tan(c)^4 - 245945700*a^3*b*d*x*tan(d*x)^4*tan(c)^4 - 368918550*a^2*b^2*d*x*tan(d*x)^4*tan(c)^4 - 245945

700*a*b^3*d*x*tan(d*x)^4*tan(c)^4 - 61486425*b^4*d*x*tan(d*x)^4*tan(c)^4 + 20020*a*b^3*tan(d*x)^9 + 5005*b^4*t

an(d*x)^9 + 308880*a^2*b^2*tan(d*x)^8*tan(c) + 386100*a*b^3*tan(d*x)^8*tan(c) + 96525*b^4*tan(d*x)^8*tan(c) +

4594590*a^2*b^2*tan(d*x)^7*tan(c)^2 + 3783780*a*b^3*tan(d*x)^7*tan(c)^2 + 945945*b^4*tan(d*x)^7*tan(c)^2 + 132

13200*a^3*b*tan(d*x)^6*tan(c)^3 + 38468430*a^2*b^2*tan(d*x)^6*tan(c)^3 + 27327300*a*b^3*tan(d*x)^6*tan(c)^3 +

6831825*b^4*tan(d*x)^6*tan(c)^3 + 219999780*a^3*b*tan(d*x)^5*tan(c)^4 + 365134770*a^2*b^2*tan(d*x)^5*tan(c)^4

+ 245945700*a*b^3*tan(d*x)^5*tan(c)^4 + 61486425*b^4*tan(d*x)^5*tan(c)^4 + 219999780*a^3*b*tan(d*x)^4*tan(c)^5

+ 365134770*a^2*b^2*tan(d*x)^4*tan(c)^5 + 245945700*a*b^3*tan(d*x)^4*tan(c)^5 + 61486425*b^4*tan(d*x)^4*tan(c

)^5 + 13213200*a^3*b*tan(d*x)^3*tan(c)^6 + 38468430*a^2*b^2*tan(d*x)^3*tan(c)^6 + 27327300*a*b^3*tan(d*x)^3*ta

n(c)^6 + 6831825*b^4*tan(d*x)^3*tan(c)^6 + 4594590*a^2*b^2*tan(d*x)^2*tan(c)^7 + 3783780*a*b^3*tan(d*x)^2*tan(

c)^7 + 945945*b^4*tan(d*x)^2*tan(c)^7 + 308880*a^2*b^2*tan(d*x)*tan(c)^8 + 386100*a*b^3*tan(d*x)*tan(c)^8 + 96

525*b^4*tan(d*x)*tan(c)^8 + 20020*a*b^3*tan(c)^9 + 5005*b^4*tan(c)^9 + 20495475*a^4*d*x*tan(d*x)^3*tan(c)^3 +

81981900*a^3*b*d*x*tan(d*x)^3*tan(c)^3 + 122972850*a^2*b^2*d*x*tan(d*x)^3*tan(c)^3 + 81981900*a*b^3*d*x*tan(d*

x)^3*tan(c)^3 + 20495475*b^4*d*x*tan(d*x)^3*tan(c)^3 - 38610*a^2*b^2*tan(d*x)^7 - 25740*a*b^3*tan(d*x)^7 - 643

5*b^4*tan(d*x)^7 - 810810*a^2*b^2*tan(d*x)^6*tan(c) - 540540*a*b^3*tan(d*x)^6*tan(c) - 135135*b^4*tan(d*x)^6*t

an(c) - 3963960*a^3*b*tan(d*x)^5*tan(c)^2 - 9459450*a^2*b^2*tan(d*x)^5*tan(c)^2 - 6306300*a*b^3*tan(d*x)^5*tan

(c)^2 - 1576575*b^4*tan(d*x)^5*tan(c)^2 - 77477400*a^3*b*tan(d*x)^4*tan(c)^3 - 122972850*a^2*b^2*tan(d*x)^4*ta

n(c)^3 - 81981900*a*b^3*tan(d*x)^4*tan(c)^3 - 20495475*b^4*tan(d*x)^4*tan(c)^3 - 77477400*a^3*b*tan(d*x)^3*tan

(c)^4 - 122972850*a^2*b^2*tan(d*x)^3*tan(c)^4 - 81981900*a*b^3*tan(d*x)^3*tan(c)^4 - 20495475*b^4*tan(d*x)^3*t

an(c)^4 - 3963960*a^3*b*tan(d*x)^2*tan(c)^5 - 9459450*a^2*b^2*tan(d*x)^2*tan(c)^5 - 6306300*a*b^3*tan(d*x)^2*t

an(c)^5 - 1576575*b^4*tan(d*x)^2*tan(c)^5 - 810810*a^2*b^2*tan(d*x)*tan(c)^6 - 540540*a*b^3*tan(d*x)*tan(c)^6

- 135135*b^4*tan(d*x)*tan(c)^6 - 38610*a^2*b^2*tan(c)^7 - 25740*a*b^3*tan(c)^7 - 6435*b^4*tan(c)^7 - 4729725*a

^4*d*x*tan(d*x)^2*tan(c)^2 - 18918900*a^3*b*d*x*tan(d*x)^2*tan(c)^2 - 28378350*a^2*b^2*d*x*tan(d*x)^2*tan(c)^2

- 18918900*a*b^3*d*x*tan(d*x)^2*tan(c)^2 - 4729725*b^4*d*x*tan(d*x)^2*tan(c)^2 + 54054*a^2*b^2*tan(d*x)^5 + 3

6036*a*b^3*tan(d*x)^5 + 9009*b^4*tan(d*x)^5 + 720720*a^3*b*tan(d*x)^4*tan(c) + 1351350*a^2*b^2*tan(d*x)^4*tan(

c) + 900900*a*b^3*tan(d*x)^4*tan(c) + 225225*b^4*tan(d*x)^4*tan(c) + 18558540*a^3*b*tan(d*x)^3*tan(c)^2 + 2837

8350*a^2*b^2*tan(d*x)^3*tan(c)^2 + 18918900*a*b^3*tan(d*x)^3*tan(c)^2 + 4729725*b^4*tan(d*x)^3*tan(c)^2 + 1855

8540*a^3*b*tan(d*x)^2*tan(c)^3 + 28378350*a^2*b^2*tan(d*x)^2*tan(c)^3 + 18918900*a*b^3*tan(d*x)^2*tan(c)^3 + 4

729725*b^4*tan(d*x)^2*tan(c)^3 + 720720*a^3*b*tan(d*x)*tan(c)^4 + 1351350*a^2*b^2*tan(d*x)*tan(c)^4 + 900900*a

*b^3*tan(d*x)*tan(c)^4 + 225225*b^4*tan(d*x)*tan(c)^4 + 54054*a^2*b^2*tan(c)^5 + 36036*a*b^3*tan(c)^5 + 9009*b

^4*tan(c)^5 + 675675*a^4*d*x*tan(d*x)*tan(c) + 2702700*a^3*b*d*x*tan(d*x)*tan(c) + 4054050*a^2*b^2*d*x*tan(d*x

)*tan(c) + 2702700*a*b^3*d*x*tan(d*x)*tan(c) + 675675*b^4*d*x*tan(d*x)*tan(c) - 60060*a^3*b*tan(d*x)^3 - 90090

*a^2*b^2*tan(d*x)^3 - 60060*a*b^3*tan(d*x)^3 - 15015*b^4*tan(d*x)^3 - 2702700*a^3*b*tan(d*x)^2*tan(c) - 405405

0*a^2*b^2*tan(d*x)^2*tan(c) - 2702700*a*b^3*tan(d*x)^2*tan(c) - 675675*b^4*tan(d*x)^2*tan(c) - 2702700*a^3*b*t

an(d*x)*tan(c)^2 - 4054050*a^2*b^2*tan(d*x)*tan(c)^2 - 2702700*a*b^3*tan(d*x)*tan(c)^2 - 675675*b^4*tan(d*x)*t

an(c)^2 - 60060*a^3*b*tan(c)^3 - 90090*a^2*b^2*tan(c)^3 - 60060*a*b^3*tan(c)^3 - 15015*b^4*tan(c)^3 - 45045*a^

4*d*x - 180180*a^3*b*d*x - 270270*a^2*b^2*d*x - 180180*a*b^3*d*x - 45045*b^4*d*x + 180180*a^3*b*tan(d*x) + 270

270*a^2*b^2*tan(d*x) + 180180*a*b^3*tan(d*x) + 45045*b^4*tan(d*x) + 180180*a^3*b*tan(c) + 270270*a^2*b^2*tan(c

) + 180180*a*b^3*tan(c) + 45045*b^4*tan(c))/(d*tan(d*x)^15*tan(c)^15 - 15*d*tan(d*x)^14*tan(c)^14 + 105*d*tan(

d*x)^13*tan(c)^13 - 455*d*tan(d*x)^12*tan(c)^12 + 1365*d*tan(d*x)^11*tan(c)^11 - 3003*d*tan(d*x)^10*tan(c)^10

+ 5005*d*tan(d*x)^9*tan(c)^9 - 6435*d*tan(d*x)^8*tan(c)^8 + 6435*d*tan(d*x)^7*tan(c)^7 - 5005*d*tan(d*x)^6*tan

(c)^6 + 3003*d*tan(d*x)^5*tan(c)^5 - 1365*d*tan(d*x)^4*tan(c)^4 + 455*d*tan(d*x)^3*tan(c)^3 - 105*d*tan(d*x)^2

*tan(c)^2 + 15*d*tan(d*x)*tan(c) - d)

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