3.374 \(\int (a+b \tan ^3(c+d x))^4 \, dx\)

Optimal. Leaf size=255 \[ \frac{b^2 \left (6 a^2-b^2\right ) \tan ^5(c+d x)}{5 d}-\frac{b^2 \left (6 a^2-b^2\right ) \tan ^3(c+d x)}{3 d}+\frac{2 a b \left (a^2-b^2\right ) \tan ^2(c+d x)}{d}+\frac{b^2 \left (6 a^2-b^2\right ) \tan (c+d x)}{d}+\frac{4 a b \left (a^2-b^2\right ) \log (\cos (c+d x))}{d}+x \left (-6 a^2 b^2+a^4+b^4\right )+\frac{a b^3 \tan ^8(c+d x)}{2 d}-\frac{2 a b^3 \tan ^6(c+d x)}{3 d}+\frac{a b^3 \tan ^4(c+d x)}{d}+\frac{b^4 \tan ^{11}(c+d x)}{11 d}-\frac{b^4 \tan ^9(c+d x)}{9 d}+\frac{b^4 \tan ^7(c+d x)}{7 d} \]

[Out]

(a^4 - 6*a^2*b^2 + b^4)*x + (4*a*b*(a^2 - b^2)*Log[Cos[c + d*x]])/d + (b^2*(6*a^2 - b^2)*Tan[c + d*x])/d + (2*
a*b*(a^2 - b^2)*Tan[c + d*x]^2)/d - (b^2*(6*a^2 - b^2)*Tan[c + d*x]^3)/(3*d) + (a*b^3*Tan[c + d*x]^4)/d + (b^2
*(6*a^2 - b^2)*Tan[c + d*x]^5)/(5*d) - (2*a*b^3*Tan[c + d*x]^6)/(3*d) + (b^4*Tan[c + d*x]^7)/(7*d) + (a*b^3*Ta
n[c + d*x]^8)/(2*d) - (b^4*Tan[c + d*x]^9)/(9*d) + (b^4*Tan[c + d*x]^11)/(11*d)

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Rubi [A]  time = 0.147243, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {3661, 1810, 635, 203, 260} \[ \frac{b^2 \left (6 a^2-b^2\right ) \tan ^5(c+d x)}{5 d}-\frac{b^2 \left (6 a^2-b^2\right ) \tan ^3(c+d x)}{3 d}+\frac{2 a b \left (a^2-b^2\right ) \tan ^2(c+d x)}{d}+\frac{b^2 \left (6 a^2-b^2\right ) \tan (c+d x)}{d}+\frac{4 a b \left (a^2-b^2\right ) \log (\cos (c+d x))}{d}+x \left (-6 a^2 b^2+a^4+b^4\right )+\frac{a b^3 \tan ^8(c+d x)}{2 d}-\frac{2 a b^3 \tan ^6(c+d x)}{3 d}+\frac{a b^3 \tan ^4(c+d x)}{d}+\frac{b^4 \tan ^{11}(c+d x)}{11 d}-\frac{b^4 \tan ^9(c+d x)}{9 d}+\frac{b^4 \tan ^7(c+d x)}{7 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Tan[c + d*x]^3)^4,x]

[Out]

(a^4 - 6*a^2*b^2 + b^4)*x + (4*a*b*(a^2 - b^2)*Log[Cos[c + d*x]])/d + (b^2*(6*a^2 - b^2)*Tan[c + d*x])/d + (2*
a*b*(a^2 - b^2)*Tan[c + d*x]^2)/d - (b^2*(6*a^2 - b^2)*Tan[c + d*x]^3)/(3*d) + (a*b^3*Tan[c + d*x]^4)/d + (b^2
*(6*a^2 - b^2)*Tan[c + d*x]^5)/(5*d) - (2*a*b^3*Tan[c + d*x]^6)/(3*d) + (b^4*Tan[c + d*x]^7)/(7*d) + (a*b^3*Ta
n[c + d*x]^8)/(2*d) - (b^4*Tan[c + d*x]^9)/(9*d) + (b^4*Tan[c + d*x]^11)/(11*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 1810

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

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])

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

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

Mathematica [C]  time = 0.904277, size = 224, normalized size = 0.88 \[ \frac{-1386 b^2 \left (b^2-6 a^2\right ) \tan ^5(c+d x)+2310 b^2 \left (b^2-6 a^2\right ) \tan ^3(c+d x)+13860 a b \left (a^2-b^2\right ) \tan ^2(c+d x)-6930 b^2 \left (b^2-6 a^2\right ) \tan (c+d x)+3465 a b^3 \tan ^8(c+d x)-4620 a b^3 \tan ^6(c+d x)+6930 a b^3 \tan ^4(c+d x)-3465 i \left ((a-i b)^4 \log (-\tan (c+d x)+i)-(a+i b)^4 \log (\tan (c+d x)+i)\right )+630 b^4 \tan ^{11}(c+d x)-770 b^4 \tan ^9(c+d x)+990 b^4 \tan ^7(c+d x)}{6930 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Tan[c + d*x]^3)^4,x]

[Out]

((-3465*I)*((a - I*b)^4*Log[I - Tan[c + d*x]] - (a + I*b)^4*Log[I + Tan[c + d*x]]) - 6930*b^2*(-6*a^2 + b^2)*T
an[c + d*x] + 13860*a*b*(a^2 - b^2)*Tan[c + d*x]^2 + 2310*b^2*(-6*a^2 + b^2)*Tan[c + d*x]^3 + 6930*a*b^3*Tan[c
+ d*x]^4 - 1386*b^2*(-6*a^2 + b^2)*Tan[c + d*x]^5 - 4620*a*b^3*Tan[c + d*x]^6 + 990*b^4*Tan[c + d*x]^7 + 3465
*a*b^3*Tan[c + d*x]^8 - 770*b^4*Tan[c + d*x]^9 + 630*b^4*Tan[c + d*x]^11)/(6930*d)

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Maple [A]  time = 0.008, size = 321, normalized size = 1.3 \begin{align*}{\frac{{b}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{11}}{11\,d}}-{\frac{{b}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{9}}{9\,d}}+{\frac{a{b}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{8}}{2\,d}}+{\frac{{b}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{7}}{7\,d}}-{\frac{2\,a{b}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{6}}{3\,d}}+{\frac{6\, \left ( \tan \left ( dx+c \right ) \right ) ^{5}{a}^{2}{b}^{2}}{5\,d}}-{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{5}{b}^{4}}{5\,d}}+{\frac{a{b}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{d}}-2\,{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{3}{a}^{2}{b}^{2}}{d}}+{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{3}{b}^{4}}{3\,d}}+2\,{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{2}{a}^{3}b}{d}}-2\,{\frac{a{b}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{d}}+6\,{\frac{{a}^{2}{b}^{2}\tan \left ( dx+c \right ) }{d}}-{\frac{{b}^{4}\tan \left ( dx+c \right ) }{d}}-2\,{\frac{\ln \left ( \left ( \tan \left ( dx+c \right ) \right ) ^{2}+1 \right ){a}^{3}b}{d}}+2\,{\frac{\ln \left ( \left ( \tan \left ( dx+c \right ) \right ) ^{2}+1 \right ) a{b}^{3}}{d}}+{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{4}}{d}}-6\,{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{2}{b}^{2}}{d}}+{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){b}^{4}}{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*tan(d*x+c)^3)^4,x)

[Out]

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

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Maxima [A]  time = 1.51901, size = 351, normalized size = 1.38 \begin{align*} a^{4} x + \frac{2 \,{\left (3 \, \tan \left (d x + c\right )^{5} - 5 \, \tan \left (d x + c\right )^{3} - 15 \, d x - 15 \, c + 15 \, \tan \left (d x + c\right )\right )} a^{2} b^{2}}{5 \, d} + \frac{{\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 )} b^{4}}{3465 \, d} + \frac{a b^{3}{\left (\frac{48 \, \sin \left (d x + c\right )^{6} - 108 \, \sin \left (d x + c\right )^{4} + 88 \, \sin \left (d x + c\right )^{2} - 25}{\sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{2} + 1} - 12 \, \log \left (\sin \left (d x + c\right )^{2} - 1\right )\right )}}{6 \, d} - \frac{2 \, a^{3} b{\left (\frac{1}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right )^{2} - 1\right )\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(d*x+c)^3)^4,x, algorithm="maxima")

[Out]

a^4*x + 2/5*(3*tan(d*x + c)^5 - 5*tan(d*x + c)^3 - 15*d*x - 15*c + 15*tan(d*x + c))*a^2*b^2/d + 1/3465*(315*ta
n(d*x + c)^11 - 385*tan(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))*b^4/d + 1/6*a*b^3*((48*sin(d*x + c)^6 - 108*sin(d*x + c)^4 + 88*sin(d*x + c)^2 -
25)/(sin(d*x + c)^8 - 4*sin(d*x + c)^6 + 6*sin(d*x + c)^4 - 4*sin(d*x + c)^2 + 1) - 12*log(sin(d*x + c)^2 - 1
))/d - 2*a^3*b*(1/(sin(d*x + c)^2 - 1) - log(sin(d*x + c)^2 - 1))/d

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Fricas [A]  time = 1.77588, size = 559, normalized size = 2.19 \begin{align*} \frac{630 \, b^{4} \tan \left (d x + c\right )^{11} - 770 \, b^{4} \tan \left (d x + c\right )^{9} + 3465 \, a b^{3} \tan \left (d x + c\right )^{8} + 990 \, b^{4} \tan \left (d x + c\right )^{7} - 4620 \, a b^{3} \tan \left (d x + c\right )^{6} + 6930 \, a b^{3} \tan \left (d x + c\right )^{4} + 1386 \,{\left (6 \, a^{2} b^{2} - b^{4}\right )} \tan \left (d x + c\right )^{5} - 2310 \,{\left (6 \, a^{2} b^{2} - b^{4}\right )} \tan \left (d x + c\right )^{3} + 6930 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} d x + 13860 \,{\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )^{2} + 13860 \,{\left (a^{3} b - a b^{3}\right )} \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right ) + 6930 \,{\left (6 \, a^{2} b^{2} - b^{4}\right )} \tan \left (d x + c\right )}{6930 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(d*x+c)^3)^4,x, algorithm="fricas")

[Out]

1/6930*(630*b^4*tan(d*x + c)^11 - 770*b^4*tan(d*x + c)^9 + 3465*a*b^3*tan(d*x + c)^8 + 990*b^4*tan(d*x + c)^7
- 4620*a*b^3*tan(d*x + c)^6 + 6930*a*b^3*tan(d*x + c)^4 + 1386*(6*a^2*b^2 - b^4)*tan(d*x + c)^5 - 2310*(6*a^2*
b^2 - b^4)*tan(d*x + c)^3 + 6930*(a^4 - 6*a^2*b^2 + b^4)*d*x + 13860*(a^3*b - a*b^3)*tan(d*x + c)^2 + 13860*(a
^3*b - a*b^3)*log(1/(tan(d*x + c)^2 + 1)) + 6930*(6*a^2*b^2 - b^4)*tan(d*x + c))/d

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Sympy [A]  time = 4.69613, size = 301, normalized size = 1.18 \begin{align*} \begin{cases} a^{4} x - \frac{2 a^{3} b \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac{2 a^{3} b \tan ^{2}{\left (c + d x \right )}}{d} - 6 a^{2} b^{2} x + \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} + \frac{2 a b^{3} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac{a b^{3} \tan ^{8}{\left (c + d x \right )}}{2 d} - \frac{2 a b^{3} \tan ^{6}{\left (c + d x \right )}}{3 d} + \frac{a b^{3} \tan ^{4}{\left (c + d x \right )}}{d} - \frac{2 a b^{3} \tan ^{2}{\left (c + d x \right )}}{d} + b^{4} x + \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 ^{3}{\left (c \right )}\right )^{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(d*x+c)**3)**4,x)

[Out]

Piecewise((a**4*x - 2*a**3*b*log(tan(c + d*x)**2 + 1)/d + 2*a**3*b*tan(c + d*x)**2/d - 6*a**2*b**2*x + 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 + 2*a*b**3*log(tan(c +
d*x)**2 + 1)/d + a*b**3*tan(c + d*x)**8/(2*d) - 2*a*b**3*tan(c + d*x)**6/(3*d) + a*b**3*tan(c + d*x)**4/d - 2
*a*b**3*tan(c + d*x)**2/d + b**4*x + 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)**3)**4, True))

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Giac [B]  time = 75.9038, size = 8031, normalized size = 31.49 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(d*x+c)^3)^4,x, algorithm="giac")

[Out]

1/6930*(6930*a^4*d*x*tan(d*x)^11*tan(c)^11 - 41580*a^2*b^2*d*x*tan(d*x)^11*tan(c)^11 + 6930*b^4*d*x*tan(d*x)^1
1*tan(c)^11 + 13860*a^3*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^
2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^11*tan(c)^11 - 13860*a*b^3*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*
tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^11*tan(c)
^11 - 76230*a^4*d*x*tan(d*x)^10*tan(c)^10 + 457380*a^2*b^2*d*x*tan(d*x)^10*tan(c)^10 - 76230*b^4*d*x*tan(d*x)^
10*tan(c)^10 + 13860*a^3*b*tan(d*x)^11*tan(c)^11 - 28875*a*b^3*tan(d*x)^11*tan(c)^11 - 152460*a^3*b*log(4*(tan
(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) +
1))*tan(d*x)^10*tan(c)^10 + 152460*a*b^3*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + ta
n(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^10*tan(c)^10 - 41580*a^2*b^2*tan(d*x)^11*tan
(c)^10 + 6930*b^4*tan(d*x)^11*tan(c)^10 - 41580*a^2*b^2*tan(d*x)^10*tan(c)^11 + 6930*b^4*tan(d*x)^10*tan(c)^11
+ 381150*a^4*d*x*tan(d*x)^9*tan(c)^9 - 2286900*a^2*b^2*d*x*tan(d*x)^9*tan(c)^9 + 381150*b^4*d*x*tan(d*x)^9*ta
n(c)^9 + 13860*a^3*b*tan(d*x)^11*tan(c)^9 - 13860*a*b^3*tan(d*x)^11*tan(c)^9 - 124740*a^3*b*tan(d*x)^10*tan(c)
^10 + 289905*a*b^3*tan(d*x)^10*tan(c)^10 + 13860*a^3*b*tan(d*x)^9*tan(c)^11 - 13860*a*b^3*tan(d*x)^9*tan(c)^11
+ 13860*a^2*b^2*tan(d*x)^11*tan(c)^8 - 2310*b^4*tan(d*x)^11*tan(c)^8 + 762300*a^3*b*log(4*(tan(c)^2 + 1)/(tan
(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^9
*tan(c)^9 - 762300*a*b^3*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2
+ tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^9*tan(c)^9 + 457380*a^2*b^2*tan(d*x)^10*tan(c)^9 - 76230*b^4*
tan(d*x)^10*tan(c)^9 + 457380*a^2*b^2*tan(d*x)^9*tan(c)^10 - 76230*b^4*tan(d*x)^9*tan(c)^10 + 13860*a^2*b^2*ta
n(d*x)^8*tan(c)^11 - 2310*b^4*tan(d*x)^8*tan(c)^11 + 6930*a*b^3*tan(d*x)^11*tan(c)^7 - 1143450*a^4*d*x*tan(d*x
)^8*tan(c)^8 + 6860700*a^2*b^2*d*x*tan(d*x)^8*tan(c)^8 - 1143450*b^4*d*x*tan(d*x)^8*tan(c)^8 - 124740*a^3*b*ta
n(d*x)^10*tan(c)^8 + 152460*a*b^3*tan(d*x)^10*tan(c)^8 + 512820*a^3*b*tan(d*x)^9*tan(c)^9 - 1297065*a*b^3*tan(
d*x)^9*tan(c)^9 - 124740*a^3*b*tan(d*x)^8*tan(c)^10 + 152460*a*b^3*tan(d*x)^8*tan(c)^10 + 6930*a*b^3*tan(d*x)^
7*tan(c)^11 - 8316*a^2*b^2*tan(d*x)^11*tan(c)^6 + 1386*b^4*tan(d*x)^11*tan(c)^6 - 152460*a^2*b^2*tan(d*x)^10*t
an(c)^7 + 25410*b^4*tan(d*x)^10*tan(c)^7 - 2286900*a^3*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x
)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^8*tan(c)^8 + 2286900*a*b^3*lo
g(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*
tan(c) + 1))*tan(d*x)^8*tan(c)^8 - 2286900*a^2*b^2*tan(d*x)^9*tan(c)^8 + 381150*b^4*tan(d*x)^9*tan(c)^8 - 2286
900*a^2*b^2*tan(d*x)^8*tan(c)^9 + 381150*b^4*tan(d*x)^8*tan(c)^9 - 152460*a^2*b^2*tan(d*x)^7*tan(c)^10 + 25410
*b^4*tan(d*x)^7*tan(c)^10 - 8316*a^2*b^2*tan(d*x)^6*tan(c)^11 + 1386*b^4*tan(d*x)^6*tan(c)^11 - 4620*a*b^3*tan
(d*x)^11*tan(c)^5 - 76230*a*b^3*tan(d*x)^10*tan(c)^6 + 2286900*a^4*d*x*tan(d*x)^7*tan(c)^7 - 13721400*a^2*b^2*
d*x*tan(d*x)^7*tan(c)^7 + 2286900*b^4*d*x*tan(d*x)^7*tan(c)^7 + 498960*a^3*b*tan(d*x)^9*tan(c)^7 - 762300*a*b^
3*tan(d*x)^9*tan(c)^7 - 1288980*a^3*b*tan(d*x)^8*tan(c)^8 + 3382995*a*b^3*tan(d*x)^8*tan(c)^8 + 498960*a^3*b*t
an(d*x)^7*tan(c)^9 - 762300*a*b^3*tan(d*x)^7*tan(c)^9 - 76230*a*b^3*tan(d*x)^6*tan(c)^10 - 4620*a*b^3*tan(d*x)
^5*tan(c)^11 - 990*b^4*tan(d*x)^11*tan(c)^4 + 49896*a^2*b^2*tan(d*x)^10*tan(c)^5 - 15246*b^4*tan(d*x)^10*tan(c
)^5 + 637560*a^2*b^2*tan(d*x)^9*tan(c)^6 - 127050*b^4*tan(d*x)^9*tan(c)^6 + 4573800*a^3*b*log(4*(tan(c)^2 + 1)
/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d
*x)^7*tan(c)^7 - 4573800*a*b^3*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*ta
n(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^7*tan(c)^7 + 6652800*a^2*b^2*tan(d*x)^8*tan(c)^7 - 1143
450*b^4*tan(d*x)^8*tan(c)^7 + 6652800*a^2*b^2*tan(d*x)^7*tan(c)^8 - 1143450*b^4*tan(d*x)^7*tan(c)^8 + 637560*a
^2*b^2*tan(d*x)^6*tan(c)^9 - 127050*b^4*tan(d*x)^6*tan(c)^9 + 49896*a^2*b^2*tan(d*x)^5*tan(c)^10 - 15246*b^4*t
an(d*x)^5*tan(c)^10 - 990*b^4*tan(d*x)^4*tan(c)^11 + 3465*a*b^3*tan(d*x)^11*tan(c)^3 + 50820*a*b^3*tan(d*x)^10
*tan(c)^4 + 381150*a*b^3*tan(d*x)^9*tan(c)^5 - 3201660*a^4*d*x*tan(d*x)^6*tan(c)^6 + 19209960*a^2*b^2*d*x*tan(
d*x)^6*tan(c)^6 - 3201660*b^4*d*x*tan(d*x)^6*tan(c)^6 - 1164240*a^3*b*tan(d*x)^8*tan(c)^6 + 2286900*a*b^3*tan(
d*x)^8*tan(c)^6 + 2245320*a^3*b*tan(d*x)^7*tan(c)^7 - 5622540*a*b^3*tan(d*x)^7*tan(c)^7 - 1164240*a^3*b*tan(d*
x)^6*tan(c)^8 + 2286900*a*b^3*tan(d*x)^6*tan(c)^8 + 381150*a*b^3*tan(d*x)^5*tan(c)^9 + 50820*a*b^3*tan(d*x)^4*
tan(c)^10 + 3465*a*b^3*tan(d*x)^3*tan(c)^11 + 770*b^4*tan(d*x)^11*tan(c)^2 + 10890*b^4*tan(d*x)^10*tan(c)^3 -
124740*a^2*b^2*tan(d*x)^9*tan(c)^4 + 76230*b^4*tan(d*x)^9*tan(c)^4 - 1399860*a^2*b^2*tan(d*x)^8*tan(c)^5 + 381
150*b^4*tan(d*x)^8*tan(c)^5 - 6403320*a^3*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) +
tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^6*tan(c)^6 + 6403320*a*b^3*log(4*(tan(c)^2
+ 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*
tan(d*x)^6*tan(c)^6 - 12307680*a^2*b^2*tan(d*x)^7*tan(c)^6 + 2286900*b^4*tan(d*x)^7*tan(c)^6 - 12307680*a^2*b^
2*tan(d*x)^6*tan(c)^7 + 2286900*b^4*tan(d*x)^6*tan(c)^7 - 1399860*a^2*b^2*tan(d*x)^5*tan(c)^8 + 381150*b^4*tan
(d*x)^5*tan(c)^8 - 124740*a^2*b^2*tan(d*x)^4*tan(c)^9 + 76230*b^4*tan(d*x)^4*tan(c)^9 + 10890*b^4*tan(d*x)^3*t
an(c)^10 + 770*b^4*tan(d*x)^2*tan(c)^11 - 10395*a*b^3*tan(d*x)^10*tan(c)^2 - 129360*a*b^3*tan(d*x)^9*tan(c)^3
- 810810*a*b^3*tan(d*x)^8*tan(c)^4 + 3201660*a^4*d*x*tan(d*x)^5*tan(c)^5 - 19209960*a^2*b^2*d*x*tan(d*x)^5*tan
(c)^5 + 3201660*b^4*d*x*tan(d*x)^5*tan(c)^5 + 1746360*a^3*b*tan(d*x)^7*tan(c)^5 - 3991680*a*b^3*tan(d*x)^7*tan
(c)^5 - 2910600*a^3*b*tan(d*x)^6*tan(c)^6 + 6740580*a*b^3*tan(d*x)^6*tan(c)^6 + 1746360*a^3*b*tan(d*x)^5*tan(c
)^7 - 3991680*a*b^3*tan(d*x)^5*tan(c)^7 - 810810*a*b^3*tan(d*x)^4*tan(c)^8 - 129360*a*b^3*tan(d*x)^3*tan(c)^9
- 10395*a*b^3*tan(d*x)^2*tan(c)^10 - 630*b^4*tan(d*x)^11 - 8470*b^4*tan(d*x)^10*tan(c) - 54450*b^4*tan(d*x)^9*
tan(c)^2 + 166320*a^2*b^2*tan(d*x)^8*tan(c)^3 - 228690*b^4*tan(d*x)^8*tan(c)^3 + 1801800*a^2*b^2*tan(d*x)^7*ta
n(c)^4 - 762300*b^4*tan(d*x)^7*tan(c)^4 + 6403320*a^3*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)
^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^5*tan(c)^5 - 6403320*a*b^3*log
(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*t
an(c) + 1))*tan(d*x)^5*tan(c)^5 + 15051960*a^2*b^2*tan(d*x)^6*tan(c)^5 - 3201660*b^4*tan(d*x)^6*tan(c)^5 + 150
51960*a^2*b^2*tan(d*x)^5*tan(c)^6 - 3201660*b^4*tan(d*x)^5*tan(c)^6 + 1801800*a^2*b^2*tan(d*x)^4*tan(c)^7 - 76
2300*b^4*tan(d*x)^4*tan(c)^7 + 166320*a^2*b^2*tan(d*x)^3*tan(c)^8 - 228690*b^4*tan(d*x)^3*tan(c)^8 - 54450*b^4
*tan(d*x)^2*tan(c)^9 - 8470*b^4*tan(d*x)*tan(c)^10 - 630*b^4*tan(c)^11 + 10395*a*b^3*tan(d*x)^9*tan(c) + 12936
0*a*b^3*tan(d*x)^8*tan(c)^2 + 810810*a*b^3*tan(d*x)^7*tan(c)^3 - 2286900*a^4*d*x*tan(d*x)^4*tan(c)^4 + 1372140
0*a^2*b^2*d*x*tan(d*x)^4*tan(c)^4 - 2286900*b^4*d*x*tan(d*x)^4*tan(c)^4 - 1746360*a^3*b*tan(d*x)^6*tan(c)^4 +
3991680*a*b^3*tan(d*x)^6*tan(c)^4 + 2910600*a^3*b*tan(d*x)^5*tan(c)^5 - 6740580*a*b^3*tan(d*x)^5*tan(c)^5 - 17
46360*a^3*b*tan(d*x)^4*tan(c)^6 + 3991680*a*b^3*tan(d*x)^4*tan(c)^6 + 810810*a*b^3*tan(d*x)^3*tan(c)^7 + 12936
0*a*b^3*tan(d*x)^2*tan(c)^8 + 10395*a*b^3*tan(d*x)*tan(c)^9 + 770*b^4*tan(d*x)^9 + 10890*b^4*tan(d*x)^8*tan(c)
- 124740*a^2*b^2*tan(d*x)^7*tan(c)^2 + 76230*b^4*tan(d*x)^7*tan(c)^2 - 1399860*a^2*b^2*tan(d*x)^6*tan(c)^3 +
381150*b^4*tan(d*x)^6*tan(c)^3 - 4573800*a^3*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c)
+ tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^4*tan(c)^4 + 4573800*a*b^3*log(4*(tan(c
)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1
))*tan(d*x)^4*tan(c)^4 - 12307680*a^2*b^2*tan(d*x)^5*tan(c)^4 + 2286900*b^4*tan(d*x)^5*tan(c)^4 - 12307680*a^2
*b^2*tan(d*x)^4*tan(c)^5 + 2286900*b^4*tan(d*x)^4*tan(c)^5 - 1399860*a^2*b^2*tan(d*x)^3*tan(c)^6 + 381150*b^4*
tan(d*x)^3*tan(c)^6 - 124740*a^2*b^2*tan(d*x)^2*tan(c)^7 + 76230*b^4*tan(d*x)^2*tan(c)^7 + 10890*b^4*tan(d*x)*
tan(c)^8 + 770*b^4*tan(c)^9 - 3465*a*b^3*tan(d*x)^8 - 50820*a*b^3*tan(d*x)^7*tan(c) - 381150*a*b^3*tan(d*x)^6*
tan(c)^2 + 1143450*a^4*d*x*tan(d*x)^3*tan(c)^3 - 6860700*a^2*b^2*d*x*tan(d*x)^3*tan(c)^3 + 1143450*b^4*d*x*tan
(d*x)^3*tan(c)^3 + 1164240*a^3*b*tan(d*x)^5*tan(c)^3 - 2286900*a*b^3*tan(d*x)^5*tan(c)^3 - 2245320*a^3*b*tan(d
*x)^4*tan(c)^4 + 5622540*a*b^3*tan(d*x)^4*tan(c)^4 + 1164240*a^3*b*tan(d*x)^3*tan(c)^5 - 2286900*a*b^3*tan(d*x
)^3*tan(c)^5 - 381150*a*b^3*tan(d*x)^2*tan(c)^6 - 50820*a*b^3*tan(d*x)*tan(c)^7 - 3465*a*b^3*tan(c)^8 - 990*b^
4*tan(d*x)^7 + 49896*a^2*b^2*tan(d*x)^6*tan(c) - 15246*b^4*tan(d*x)^6*tan(c) + 637560*a^2*b^2*tan(d*x)^5*tan(c
)^2 - 127050*b^4*tan(d*x)^5*tan(c)^2 + 2286900*a^3*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*
tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^3*tan(c)^3 - 2286900*a*b^3*log(4*
(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(
c) + 1))*tan(d*x)^3*tan(c)^3 + 6652800*a^2*b^2*tan(d*x)^4*tan(c)^3 - 1143450*b^4*tan(d*x)^4*tan(c)^3 + 6652800
*a^2*b^2*tan(d*x)^3*tan(c)^4 - 1143450*b^4*tan(d*x)^3*tan(c)^4 + 637560*a^2*b^2*tan(d*x)^2*tan(c)^5 - 127050*b
^4*tan(d*x)^2*tan(c)^5 + 49896*a^2*b^2*tan(d*x)*tan(c)^6 - 15246*b^4*tan(d*x)*tan(c)^6 - 990*b^4*tan(c)^7 + 46
20*a*b^3*tan(d*x)^6 + 76230*a*b^3*tan(d*x)^5*tan(c) - 381150*a^4*d*x*tan(d*x)^2*tan(c)^2 + 2286900*a^2*b^2*d*x
*tan(d*x)^2*tan(c)^2 - 381150*b^4*d*x*tan(d*x)^2*tan(c)^2 - 498960*a^3*b*tan(d*x)^4*tan(c)^2 + 762300*a*b^3*ta
n(d*x)^4*tan(c)^2 + 1288980*a^3*b*tan(d*x)^3*tan(c)^3 - 3382995*a*b^3*tan(d*x)^3*tan(c)^3 - 498960*a^3*b*tan(d
*x)^2*tan(c)^4 + 762300*a*b^3*tan(d*x)^2*tan(c)^4 + 76230*a*b^3*tan(d*x)*tan(c)^5 + 4620*a*b^3*tan(c)^6 - 8316
*a^2*b^2*tan(d*x)^5 + 1386*b^4*tan(d*x)^5 - 152460*a^2*b^2*tan(d*x)^4*tan(c) + 25410*b^4*tan(d*x)^4*tan(c) - 7
62300*a^3*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2
- 2*tan(d*x)*tan(c) + 1))*tan(d*x)^2*tan(c)^2 + 762300*a*b^3*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*ta
n(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^2*tan(c)^2 - 2286900*a^2
*b^2*tan(d*x)^3*tan(c)^2 + 381150*b^4*tan(d*x)^3*tan(c)^2 - 2286900*a^2*b^2*tan(d*x)^2*tan(c)^3 + 381150*b^4*t
an(d*x)^2*tan(c)^3 - 152460*a^2*b^2*tan(d*x)*tan(c)^4 + 25410*b^4*tan(d*x)*tan(c)^4 - 8316*a^2*b^2*tan(c)^5 +
1386*b^4*tan(c)^5 - 6930*a*b^3*tan(d*x)^4 + 76230*a^4*d*x*tan(d*x)*tan(c) - 457380*a^2*b^2*d*x*tan(d*x)*tan(c)
+ 76230*b^4*d*x*tan(d*x)*tan(c) + 124740*a^3*b*tan(d*x)^3*tan(c) - 152460*a*b^3*tan(d*x)^3*tan(c) - 512820*a^
3*b*tan(d*x)^2*tan(c)^2 + 1297065*a*b^3*tan(d*x)^2*tan(c)^2 + 124740*a^3*b*tan(d*x)*tan(c)^3 - 152460*a*b^3*ta
n(d*x)*tan(c)^3 - 6930*a*b^3*tan(c)^4 + 13860*a^2*b^2*tan(d*x)^3 - 2310*b^4*tan(d*x)^3 + 152460*a^3*b*log(4*(t
an(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c)
+ 1))*tan(d*x)*tan(c) - 152460*a*b^3*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*
x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)*tan(c) + 457380*a^2*b^2*tan(d*x)^2*tan(c) - 7623
0*b^4*tan(d*x)^2*tan(c) + 457380*a^2*b^2*tan(d*x)*tan(c)^2 - 76230*b^4*tan(d*x)*tan(c)^2 + 13860*a^2*b^2*tan(c
)^3 - 2310*b^4*tan(c)^3 - 6930*a^4*d*x + 41580*a^2*b^2*d*x - 6930*b^4*d*x - 13860*a^3*b*tan(d*x)^2 + 13860*a*b
^3*tan(d*x)^2 + 124740*a^3*b*tan(d*x)*tan(c) - 289905*a*b^3*tan(d*x)*tan(c) - 13860*a^3*b*tan(c)^2 + 13860*a*b
^3*tan(c)^2 - 13860*a^3*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^
2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)) + 13860*a*b^3*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^
3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)) - 41580*a^2*b^2*tan(d*x) + 6930*b^4*tan(
d*x) - 41580*a^2*b^2*tan(c) + 6930*b^4*tan(c) - 13860*a^3*b + 28875*a*b^3)/(d*tan(d*x)^11*tan(c)^11 - 11*d*tan
(d*x)^10*tan(c)^10 + 55*d*tan(d*x)^9*tan(c)^9 - 165*d*tan(d*x)^8*tan(c)^8 + 330*d*tan(d*x)^7*tan(c)^7 - 462*d*
tan(d*x)^6*tan(c)^6 + 462*d*tan(d*x)^5*tan(c)^5 - 330*d*tan(d*x)^4*tan(c)^4 + 165*d*tan(d*x)^3*tan(c)^3 - 55*d
*tan(d*x)^2*tan(c)^2 + 11*d*tan(d*x)*tan(c) - d)