3.156 \(\int (a+b \sin (c+d x))^2 \tan ^2(c+d x) \, dx\)
Optimal. Leaf size=94 \[ \frac{a^2 \tan (c+d x)}{d}+a^2 (-x)+\frac{2 a b \cos (c+d x)}{d}+\frac{2 a b \sec (c+d x)}{d}+\frac{3 b^2 \tan (c+d x)}{2 d}-\frac{b^2 \sin ^2(c+d x) \tan (c+d x)}{2 d}-\frac{3 b^2 x}{2} \]
[Out]
-(a^2*x) - (3*b^2*x)/2 + (2*a*b*Cos[c + d*x])/d + (2*a*b*Sec[c + d*x])/d + (a^2*Tan[c + d*x])/d + (3*b^2*Tan[c
+ d*x])/(2*d) - (b^2*Sin[c + d*x]^2*Tan[c + d*x])/(2*d)
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Rubi [A] time = 0.122315, antiderivative size = 94, normalized size of antiderivative = 1.,
number of steps used = 11, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used
= {2722, 3473, 8, 2590, 14, 2591, 288, 321, 203} \[ \frac{a^2 \tan (c+d x)}{d}+a^2 (-x)+\frac{2 a b \cos (c+d x)}{d}+\frac{2 a b \sec (c+d x)}{d}+\frac{3 b^2 \tan (c+d x)}{2 d}-\frac{b^2 \sin ^2(c+d x) \tan (c+d x)}{2 d}-\frac{3 b^2 x}{2} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*Sin[c + d*x])^2*Tan[c + d*x]^2,x]
[Out]
-(a^2*x) - (3*b^2*x)/2 + (2*a*b*Cos[c + d*x])/d + (2*a*b*Sec[c + d*x])/d + (a^2*Tan[c + d*x])/d + (3*b^2*Tan[c
+ d*x])/(2*d) - (b^2*Sin[c + d*x]^2*Tan[c + d*x])/(2*d)
Rule 2722
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_.), x_Symbol] :> Int[Expan
dIntegrand[(g*Tan[e + f*x])^p, (a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2
, 0] && IGtQ[m, 0]
Rule 3473
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Rule 8
Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
Rule 2590
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[f^(-1), Subst[Int[(1 - x^2
)^((m + n - 1)/2)/x^n, x], x, Cos[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n - 1)/2]
Rule 14
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&& !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Rule 2591
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> With[{ff = FreeFactors[Ta
n[e + f*x], x]}, Dist[(b*ff)/f, Subst[Int[(ff*x)^(m + n)/(b^2 + ff^2*x^2)^(m/2 + 1), x], x, (b*Tan[e + f*x])/f
f], x]] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2]
Rule 288
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]
Rule 321
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]
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 (a+b \sin (c+d x))^2 \tan ^2(c+d x) \, dx &=\int \left (a^2 \tan ^2(c+d x)+2 a b \sin (c+d x) \tan ^2(c+d x)+b^2 \sin ^2(c+d x) \tan ^2(c+d x)\right ) \, dx\\ &=a^2 \int \tan ^2(c+d x) \, dx+(2 a b) \int \sin (c+d x) \tan ^2(c+d x) \, dx+b^2 \int \sin ^2(c+d x) \tan ^2(c+d x) \, dx\\ &=\frac{a^2 \tan (c+d x)}{d}-a^2 \int 1 \, dx-\frac{(2 a b) \operatorname{Subst}\left (\int \frac{1-x^2}{x^2} \, dx,x,\cos (c+d x)\right )}{d}+\frac{b^2 \operatorname{Subst}\left (\int \frac{x^4}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-a^2 x+\frac{a^2 \tan (c+d x)}{d}-\frac{b^2 \sin ^2(c+d x) \tan (c+d x)}{2 d}-\frac{(2 a b) \operatorname{Subst}\left (\int \left (-1+\frac{1}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=-a^2 x+\frac{2 a b \cos (c+d x)}{d}+\frac{2 a b \sec (c+d x)}{d}+\frac{a^2 \tan (c+d x)}{d}+\frac{3 b^2 \tan (c+d x)}{2 d}-\frac{b^2 \sin ^2(c+d x) \tan (c+d x)}{2 d}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=-a^2 x-\frac{3 b^2 x}{2}+\frac{2 a b \cos (c+d x)}{d}+\frac{2 a b \sec (c+d x)}{d}+\frac{a^2 \tan (c+d x)}{d}+\frac{3 b^2 \tan (c+d x)}{2 d}-\frac{b^2 \sin ^2(c+d x) \tan (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.457161, size = 77, normalized size = 0.82 \[ \frac{-4 \left (2 a^2+3 b^2\right ) (c+d x)+\left (8 a^2+9 b^2\right ) \tan (c+d x)+b \sec (c+d x) (8 a \cos (2 (c+d x))+24 a+b \sin (3 (c+d x)))}{8 d} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*Sin[c + d*x])^2*Tan[c + d*x]^2,x]
[Out]
(-4*(2*a^2 + 3*b^2)*(c + d*x) + b*Sec[c + d*x]*(24*a + 8*a*Cos[2*(c + d*x)] + b*Sin[3*(c + d*x)]) + (8*a^2 + 9
*b^2)*Tan[c + d*x])/(8*d)
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Maple [A] time = 0.039, size = 116, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( \tan \left ( dx+c \right ) -dx-c \right ) +2\,ab \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{\cos \left ( dx+c \right ) }}+ \left ( 2+ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \right ) \cos \left ( dx+c \right ) \right ) +{b}^{2} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{\cos \left ( dx+c \right ) }}+ \left ( \left ( \sin \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\sin \left ( dx+c \right ) }{2}} \right ) \cos \left ( dx+c \right ) -{\frac{3\,dx}{2}}-{\frac{3\,c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*sin(d*x+c))^2*tan(d*x+c)^2,x)
[Out]
1/d*(a^2*(tan(d*x+c)-d*x-c)+2*a*b*(sin(d*x+c)^4/cos(d*x+c)+(2+sin(d*x+c)^2)*cos(d*x+c))+b^2*(sin(d*x+c)^5/cos(
d*x+c)+(sin(d*x+c)^3+3/2*sin(d*x+c))*cos(d*x+c)-3/2*d*x-3/2*c))
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Maxima [A] time = 2.52019, size = 112, normalized size = 1.19 \begin{align*} -\frac{2 \,{\left (d x + c - \tan \left (d x + c\right )\right )} a^{2} +{\left (3 \, d x + 3 \, c - \frac{\tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 2 \, \tan \left (d x + c\right )\right )} b^{2} - 4 \, a b{\left (\frac{1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sin(d*x+c))^2*tan(d*x+c)^2,x, algorithm="maxima")
[Out]
-1/2*(2*(d*x + c - tan(d*x + c))*a^2 + (3*d*x + 3*c - tan(d*x + c)/(tan(d*x + c)^2 + 1) - 2*tan(d*x + c))*b^2
- 4*a*b*(1/cos(d*x + c) + cos(d*x + c)))/d
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Fricas [A] time = 1.3702, size = 190, normalized size = 2.02 \begin{align*} -\frac{{\left (2 \, a^{2} + 3 \, b^{2}\right )} d x \cos \left (d x + c\right ) - 4 \, a b \cos \left (d x + c\right )^{2} - 4 \, a b -{\left (b^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} + 2 \, b^{2}\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sin(d*x+c))^2*tan(d*x+c)^2,x, algorithm="fricas")
[Out]
-1/2*((2*a^2 + 3*b^2)*d*x*cos(d*x + c) - 4*a*b*cos(d*x + c)^2 - 4*a*b - (b^2*cos(d*x + c)^2 + 2*a^2 + 2*b^2)*s
in(d*x + c))/(d*cos(d*x + c))
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \sin{\left (c + d x \right )}\right )^{2} \tan ^{2}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sin(d*x+c))**2*tan(d*x+c)**2,x)
[Out]
Integral((a + b*sin(c + d*x))**2*tan(c + d*x)**2, x)
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Giac [B] time = 30.6123, size = 10355, normalized size = 110.16 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sin(d*x+c))^2*tan(d*x+c)^2,x, algorithm="giac")
[Out]
-1/2*(2*a^2*d*x*tan(d*x)^3*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c)^3 + 3*b^2*d*x*tan(d*x)^3*tan(1/2*d*x)^4*tan(1/2*
c)^4*tan(c)^3 + 2*a^2*d*x*tan(d*x)^3*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c) + 3*b^2*d*x*tan(d*x)^3*tan(1/2*d*x)^4*
tan(1/2*c)^4*tan(c) - 2*a^2*d*x*tan(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c)^2 - 3*b^2*d*x*tan(d*x)^2*tan(1/2
*d*x)^4*tan(1/2*c)^4*tan(c)^2 - 8*a^2*d*x*tan(d*x)^3*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c)^3 - 12*b^2*d*x*tan(d*x
)^3*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c)^3 + 2*a^2*d*x*tan(d*x)*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c)^3 + 3*b^2*d*x
*tan(d*x)*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c)^3 - 8*a*b*tan(d*x)^3*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c)^3 + 2*a^2
*tan(d*x)^3*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c)^2 + 3*b^2*tan(d*x)^3*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c)^2 + 2*a
^2*tan(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c)^3 + 3*b^2*tan(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c)^3 - 2
*a^2*d*x*tan(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c)^4 - 3*b^2*d*x*tan(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c)^4 - 8*a^2*d*x
*tan(d*x)^3*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c) - 12*b^2*d*x*tan(d*x)^3*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c) + 2*
a^2*d*x*tan(d*x)*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c) + 3*b^2*d*x*tan(d*x)*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c) -
8*a*b*tan(d*x)^3*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c) + 8*a^2*d*x*tan(d*x)^2*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c)^
2 + 12*b^2*d*x*tan(d*x)^2*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c)^2 - 2*a^2*d*x*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c)^
2 - 3*b^2*d*x*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c)^2 + 8*a*b*tan(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c)^2 - 2
*a^2*d*x*tan(d*x)^3*tan(1/2*d*x)^4*tan(c)^3 - 3*b^2*d*x*tan(d*x)^3*tan(1/2*d*x)^4*tan(c)^3 - 8*a^2*d*x*tan(d*x
)^3*tan(1/2*d*x)^3*tan(1/2*c)*tan(c)^3 - 12*b^2*d*x*tan(d*x)^3*tan(1/2*d*x)^3*tan(1/2*c)*tan(c)^3 - 8*a^2*d*x*
tan(d*x)^3*tan(1/2*d*x)*tan(1/2*c)^3*tan(c)^3 - 12*b^2*d*x*tan(d*x)^3*tan(1/2*d*x)*tan(1/2*c)^3*tan(c)^3 - 8*a
^2*d*x*tan(d*x)*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c)^3 - 12*b^2*d*x*tan(d*x)*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c)^
3 + 32*a*b*tan(d*x)^3*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c)^3 - 2*a^2*d*x*tan(d*x)^3*tan(1/2*c)^4*tan(c)^3 - 3*b^
2*d*x*tan(d*x)^3*tan(1/2*c)^4*tan(c)^3 - 8*a*b*tan(d*x)*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c)^3 + 2*a^2*tan(d*x)^
3*tan(1/2*d*x)^4*tan(1/2*c)^4 + 2*b^2*tan(d*x)^3*tan(1/2*d*x)^4*tan(1/2*c)^4 + 2*a^2*tan(d*x)^2*tan(1/2*d*x)^4
*tan(1/2*c)^4*tan(c) - 8*a^2*tan(d*x)^3*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c)^2 - 12*b^2*tan(d*x)^3*tan(1/2*d*x)^
3*tan(1/2*c)^3*tan(c)^2 + 2*a^2*tan(d*x)*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c)^2 - 8*a^2*tan(d*x)^2*tan(1/2*d*x)^
3*tan(1/2*c)^3*tan(c)^3 - 12*b^2*tan(d*x)^2*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c)^3 + 2*a^2*tan(1/2*d*x)^4*tan(1/
2*c)^4*tan(c)^3 + 2*b^2*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c)^3 + 8*a^2*d*x*tan(d*x)^2*tan(1/2*d*x)^3*tan(1/2*c)^
3 + 12*b^2*d*x*tan(d*x)^2*tan(1/2*d*x)^3*tan(1/2*c)^3 - 2*a^2*d*x*tan(1/2*d*x)^4*tan(1/2*c)^4 - 3*b^2*d*x*tan(
1/2*d*x)^4*tan(1/2*c)^4 + 8*a*b*tan(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c)^4 - 2*a^2*d*x*tan(d*x)^3*tan(1/2*d*x)^4*t
an(c) - 3*b^2*d*x*tan(d*x)^3*tan(1/2*d*x)^4*tan(c) - 8*a^2*d*x*tan(d*x)^3*tan(1/2*d*x)^3*tan(1/2*c)*tan(c) - 1
2*b^2*d*x*tan(d*x)^3*tan(1/2*d*x)^3*tan(1/2*c)*tan(c) - 8*a^2*d*x*tan(d*x)^3*tan(1/2*d*x)*tan(1/2*c)^3*tan(c)
- 12*b^2*d*x*tan(d*x)^3*tan(1/2*d*x)*tan(1/2*c)^3*tan(c) - 8*a^2*d*x*tan(d*x)*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(
c) - 12*b^2*d*x*tan(d*x)*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c) + 32*a*b*tan(d*x)^3*tan(1/2*d*x)^3*tan(1/2*c)^3*ta
n(c) - 2*a^2*d*x*tan(d*x)^3*tan(1/2*c)^4*tan(c) - 3*b^2*d*x*tan(d*x)^3*tan(1/2*c)^4*tan(c) - 8*a*b*tan(d*x)*ta
n(1/2*d*x)^4*tan(1/2*c)^4*tan(c) + 2*a^2*d*x*tan(d*x)^2*tan(1/2*d*x)^4*tan(c)^2 + 3*b^2*d*x*tan(d*x)^2*tan(1/2
*d*x)^4*tan(c)^2 + 8*a^2*d*x*tan(d*x)^2*tan(1/2*d*x)^3*tan(1/2*c)*tan(c)^2 + 12*b^2*d*x*tan(d*x)^2*tan(1/2*d*x
)^3*tan(1/2*c)*tan(c)^2 + 8*a^2*d*x*tan(d*x)^2*tan(1/2*d*x)*tan(1/2*c)^3*tan(c)^2 + 12*b^2*d*x*tan(d*x)^2*tan(
1/2*d*x)*tan(1/2*c)^3*tan(c)^2 + 8*a^2*d*x*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c)^2 + 12*b^2*d*x*tan(1/2*d*x)^3*ta
n(1/2*c)^3*tan(c)^2 - 32*a*b*tan(d*x)^2*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c)^2 + 2*a^2*d*x*tan(d*x)^2*tan(1/2*c)
^4*tan(c)^2 + 3*b^2*d*x*tan(d*x)^2*tan(1/2*c)^4*tan(c)^2 + 8*a*b*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c)^2 - 2*a^2*
d*x*tan(d*x)*tan(1/2*d*x)^4*tan(c)^3 - 3*b^2*d*x*tan(d*x)*tan(1/2*d*x)^4*tan(c)^3 - 8*a*b*tan(d*x)^3*tan(1/2*d
*x)^4*tan(c)^3 - 8*a^2*d*x*tan(d*x)^3*tan(1/2*d*x)*tan(1/2*c)*tan(c)^3 - 12*b^2*d*x*tan(d*x)^3*tan(1/2*d*x)*ta
n(1/2*c)*tan(c)^3 - 8*a^2*d*x*tan(d*x)*tan(1/2*d*x)^3*tan(1/2*c)*tan(c)^3 - 12*b^2*d*x*tan(d*x)*tan(1/2*d*x)^3
*tan(1/2*c)*tan(c)^3 - 32*a*b*tan(d*x)^3*tan(1/2*d*x)^3*tan(1/2*c)*tan(c)^3 - 96*a*b*tan(d*x)^3*tan(1/2*d*x)^2
*tan(1/2*c)^2*tan(c)^3 - 8*a^2*d*x*tan(d*x)*tan(1/2*d*x)*tan(1/2*c)^3*tan(c)^3 - 12*b^2*d*x*tan(d*x)*tan(1/2*d
*x)*tan(1/2*c)^3*tan(c)^3 - 32*a*b*tan(d*x)^3*tan(1/2*d*x)*tan(1/2*c)^3*tan(c)^3 + 32*a*b*tan(d*x)*tan(1/2*d*x
)^3*tan(1/2*c)^3*tan(c)^3 - 2*a^2*d*x*tan(d*x)*tan(1/2*c)^4*tan(c)^3 - 3*b^2*d*x*tan(d*x)*tan(1/2*c)^4*tan(c)^
3 - 8*a*b*tan(d*x)^3*tan(1/2*c)^4*tan(c)^3 - 8*a^2*tan(d*x)^3*tan(1/2*d*x)^3*tan(1/2*c)^3 - 8*b^2*tan(d*x)^3*t
an(1/2*d*x)^3*tan(1/2*c)^3 + 2*a^2*tan(d*x)*tan(1/2*d*x)^4*tan(1/2*c)^4 + 3*b^2*tan(d*x)*tan(1/2*d*x)^4*tan(1/
2*c)^4 - 8*a^2*tan(d*x)^2*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c) + 2*a^2*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c) + 3*b^
2*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c) - 2*a^2*tan(d*x)^3*tan(1/2*d*x)^4*tan(c)^2 - 3*b^2*tan(d*x)^3*tan(1/2*d*x
)^4*tan(c)^2 - 8*a^2*tan(d*x)^3*tan(1/2*d*x)^3*tan(1/2*c)*tan(c)^2 - 12*b^2*tan(d*x)^3*tan(1/2*d*x)^3*tan(1/2*
c)*tan(c)^2 - 8*a^2*tan(d*x)^3*tan(1/2*d*x)*tan(1/2*c)^3*tan(c)^2 - 12*b^2*tan(d*x)^3*tan(1/2*d*x)*tan(1/2*c)^
3*tan(c)^2 - 8*a^2*tan(d*x)*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c)^2 - 2*a^2*tan(d*x)^3*tan(1/2*c)^4*tan(c)^2 - 3*
b^2*tan(d*x)^3*tan(1/2*c)^4*tan(c)^2 - 2*a^2*tan(d*x)^2*tan(1/2*d*x)^4*tan(c)^3 - 3*b^2*tan(d*x)^2*tan(1/2*d*x
)^4*tan(c)^3 - 8*a^2*tan(d*x)^2*tan(1/2*d*x)^3*tan(1/2*c)*tan(c)^3 - 12*b^2*tan(d*x)^2*tan(1/2*d*x)^3*tan(1/2*
c)*tan(c)^3 - 8*a^2*tan(d*x)^2*tan(1/2*d*x)*tan(1/2*c)^3*tan(c)^3 - 12*b^2*tan(d*x)^2*tan(1/2*d*x)*tan(1/2*c)^
3*tan(c)^3 - 8*a^2*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c)^3 - 8*b^2*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c)^3 - 2*a^2*t
an(d*x)^2*tan(1/2*c)^4*tan(c)^3 - 3*b^2*tan(d*x)^2*tan(1/2*c)^4*tan(c)^3 + 2*a^2*d*x*tan(d*x)^2*tan(1/2*d*x)^4
+ 3*b^2*d*x*tan(d*x)^2*tan(1/2*d*x)^4 + 8*a^2*d*x*tan(d*x)^2*tan(1/2*d*x)^3*tan(1/2*c) + 12*b^2*d*x*tan(d*x)^
2*tan(1/2*d*x)^3*tan(1/2*c) + 8*a^2*d*x*tan(d*x)^2*tan(1/2*d*x)*tan(1/2*c)^3 + 12*b^2*d*x*tan(d*x)^2*tan(1/2*d
*x)*tan(1/2*c)^3 + 8*a^2*d*x*tan(1/2*d*x)^3*tan(1/2*c)^3 + 12*b^2*d*x*tan(1/2*d*x)^3*tan(1/2*c)^3 - 32*a*b*tan
(d*x)^2*tan(1/2*d*x)^3*tan(1/2*c)^3 + 2*a^2*d*x*tan(d*x)^2*tan(1/2*c)^4 + 3*b^2*d*x*tan(d*x)^2*tan(1/2*c)^4 +
8*a*b*tan(1/2*d*x)^4*tan(1/2*c)^4 - 2*a^2*d*x*tan(d*x)*tan(1/2*d*x)^4*tan(c) - 3*b^2*d*x*tan(d*x)*tan(1/2*d*x)
^4*tan(c) - 8*a*b*tan(d*x)^3*tan(1/2*d*x)^4*tan(c) - 8*a^2*d*x*tan(d*x)^3*tan(1/2*d*x)*tan(1/2*c)*tan(c) - 12*
b^2*d*x*tan(d*x)^3*tan(1/2*d*x)*tan(1/2*c)*tan(c) - 8*a^2*d*x*tan(d*x)*tan(1/2*d*x)^3*tan(1/2*c)*tan(c) - 12*b
^2*d*x*tan(d*x)*tan(1/2*d*x)^3*tan(1/2*c)*tan(c) - 32*a*b*tan(d*x)^3*tan(1/2*d*x)^3*tan(1/2*c)*tan(c) - 96*a*b
*tan(d*x)^3*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(c) - 8*a^2*d*x*tan(d*x)*tan(1/2*d*x)*tan(1/2*c)^3*tan(c) - 12*b^2*
d*x*tan(d*x)*tan(1/2*d*x)*tan(1/2*c)^3*tan(c) - 32*a*b*tan(d*x)^3*tan(1/2*d*x)*tan(1/2*c)^3*tan(c) + 32*a*b*ta
n(d*x)*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c) - 2*a^2*d*x*tan(d*x)*tan(1/2*c)^4*tan(c) - 3*b^2*d*x*tan(d*x)*tan(1/
2*c)^4*tan(c) - 8*a*b*tan(d*x)^3*tan(1/2*c)^4*tan(c) + 2*a^2*d*x*tan(1/2*d*x)^4*tan(c)^2 + 3*b^2*d*x*tan(1/2*d
*x)^4*tan(c)^2 + 8*a*b*tan(d*x)^2*tan(1/2*d*x)^4*tan(c)^2 + 8*a^2*d*x*tan(d*x)^2*tan(1/2*d*x)*tan(1/2*c)*tan(c
)^2 + 12*b^2*d*x*tan(d*x)^2*tan(1/2*d*x)*tan(1/2*c)*tan(c)^2 + 8*a^2*d*x*tan(1/2*d*x)^3*tan(1/2*c)*tan(c)^2 +
12*b^2*d*x*tan(1/2*d*x)^3*tan(1/2*c)*tan(c)^2 + 32*a*b*tan(d*x)^2*tan(1/2*d*x)^3*tan(1/2*c)*tan(c)^2 + 96*a*b*
tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(c)^2 + 8*a^2*d*x*tan(1/2*d*x)*tan(1/2*c)^3*tan(c)^2 + 12*b^2*d*x*ta
n(1/2*d*x)*tan(1/2*c)^3*tan(c)^2 + 32*a*b*tan(d*x)^2*tan(1/2*d*x)*tan(1/2*c)^3*tan(c)^2 - 32*a*b*tan(1/2*d*x)^
3*tan(1/2*c)^3*tan(c)^2 + 2*a^2*d*x*tan(1/2*c)^4*tan(c)^2 + 3*b^2*d*x*tan(1/2*c)^4*tan(c)^2 + 8*a*b*tan(d*x)^2
*tan(1/2*c)^4*tan(c)^2 + 2*a^2*d*x*tan(d*x)^3*tan(c)^3 + 3*b^2*d*x*tan(d*x)^3*tan(c)^3 - 8*a*b*tan(d*x)*tan(1/
2*d*x)^4*tan(c)^3 - 8*a^2*d*x*tan(d*x)*tan(1/2*d*x)*tan(1/2*c)*tan(c)^3 - 12*b^2*d*x*tan(d*x)*tan(1/2*d*x)*tan
(1/2*c)*tan(c)^3 + 32*a*b*tan(d*x)^3*tan(1/2*d*x)*tan(1/2*c)*tan(c)^3 - 32*a*b*tan(d*x)*tan(1/2*d*x)^3*tan(1/2
*c)*tan(c)^3 - 96*a*b*tan(d*x)*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(c)^3 - 32*a*b*tan(d*x)*tan(1/2*d*x)*tan(1/2*c)^
3*tan(c)^3 - 8*a*b*tan(d*x)*tan(1/2*c)^4*tan(c)^3 - 2*a^2*tan(d*x)^3*tan(1/2*d*x)^4 - 2*b^2*tan(d*x)^3*tan(1/2
*d*x)^4 - 8*a^2*tan(d*x)^3*tan(1/2*d*x)^3*tan(1/2*c) - 8*b^2*tan(d*x)^3*tan(1/2*d*x)^3*tan(1/2*c) - 8*a^2*tan(
d*x)^3*tan(1/2*d*x)*tan(1/2*c)^3 - 8*b^2*tan(d*x)^3*tan(1/2*d*x)*tan(1/2*c)^3 - 8*a^2*tan(d*x)*tan(1/2*d*x)^3*
tan(1/2*c)^3 - 12*b^2*tan(d*x)*tan(1/2*d*x)^3*tan(1/2*c)^3 - 2*a^2*tan(d*x)^3*tan(1/2*c)^4 - 2*b^2*tan(d*x)^3*
tan(1/2*c)^4 - 2*a^2*tan(d*x)^2*tan(1/2*d*x)^4*tan(c) - 8*a^2*tan(d*x)^2*tan(1/2*d*x)^3*tan(1/2*c)*tan(c) - 8*
a^2*tan(d*x)^2*tan(1/2*d*x)*tan(1/2*c)^3*tan(c) - 8*a^2*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c) - 12*b^2*tan(1/2*d*
x)^3*tan(1/2*c)^3*tan(c) - 2*a^2*tan(d*x)^2*tan(1/2*c)^4*tan(c) - 2*a^2*tan(d*x)*tan(1/2*d*x)^4*tan(c)^2 - 8*a
^2*tan(d*x)^3*tan(1/2*d*x)*tan(1/2*c)*tan(c)^2 - 12*b^2*tan(d*x)^3*tan(1/2*d*x)*tan(1/2*c)*tan(c)^2 - 8*a^2*ta
n(d*x)*tan(1/2*d*x)^3*tan(1/2*c)*tan(c)^2 - 8*a^2*tan(d*x)*tan(1/2*d*x)*tan(1/2*c)^3*tan(c)^2 - 2*a^2*tan(d*x)
*tan(1/2*c)^4*tan(c)^2 - 2*a^2*tan(1/2*d*x)^4*tan(c)^3 - 2*b^2*tan(1/2*d*x)^4*tan(c)^3 - 8*a^2*tan(d*x)^2*tan(
1/2*d*x)*tan(1/2*c)*tan(c)^3 - 12*b^2*tan(d*x)^2*tan(1/2*d*x)*tan(1/2*c)*tan(c)^3 - 8*a^2*tan(1/2*d*x)^3*tan(1
/2*c)*tan(c)^3 - 8*b^2*tan(1/2*d*x)^3*tan(1/2*c)*tan(c)^3 - 8*a^2*tan(1/2*d*x)*tan(1/2*c)^3*tan(c)^3 - 8*b^2*t
an(1/2*d*x)*tan(1/2*c)^3*tan(c)^3 - 2*a^2*tan(1/2*c)^4*tan(c)^3 - 2*b^2*tan(1/2*c)^4*tan(c)^3 + 2*a^2*d*x*tan(
1/2*d*x)^4 + 3*b^2*d*x*tan(1/2*d*x)^4 + 8*a*b*tan(d*x)^2*tan(1/2*d*x)^4 + 8*a^2*d*x*tan(d*x)^2*tan(1/2*d*x)*ta
n(1/2*c) + 12*b^2*d*x*tan(d*x)^2*tan(1/2*d*x)*tan(1/2*c) + 8*a^2*d*x*tan(1/2*d*x)^3*tan(1/2*c) + 12*b^2*d*x*ta
n(1/2*d*x)^3*tan(1/2*c) + 32*a*b*tan(d*x)^2*tan(1/2*d*x)^3*tan(1/2*c) + 96*a*b*tan(d*x)^2*tan(1/2*d*x)^2*tan(1
/2*c)^2 + 8*a^2*d*x*tan(1/2*d*x)*tan(1/2*c)^3 + 12*b^2*d*x*tan(1/2*d*x)*tan(1/2*c)^3 + 32*a*b*tan(d*x)^2*tan(1
/2*d*x)*tan(1/2*c)^3 - 32*a*b*tan(1/2*d*x)^3*tan(1/2*c)^3 + 2*a^2*d*x*tan(1/2*c)^4 + 3*b^2*d*x*tan(1/2*c)^4 +
8*a*b*tan(d*x)^2*tan(1/2*c)^4 + 2*a^2*d*x*tan(d*x)^3*tan(c) + 3*b^2*d*x*tan(d*x)^3*tan(c) - 8*a*b*tan(d*x)*tan
(1/2*d*x)^4*tan(c) - 8*a^2*d*x*tan(d*x)*tan(1/2*d*x)*tan(1/2*c)*tan(c) - 12*b^2*d*x*tan(d*x)*tan(1/2*d*x)*tan(
1/2*c)*tan(c) + 32*a*b*tan(d*x)^3*tan(1/2*d*x)*tan(1/2*c)*tan(c) - 32*a*b*tan(d*x)*tan(1/2*d*x)^3*tan(1/2*c)*t
an(c) - 96*a*b*tan(d*x)*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(c) - 32*a*b*tan(d*x)*tan(1/2*d*x)*tan(1/2*c)^3*tan(c)
- 8*a*b*tan(d*x)*tan(1/2*c)^4*tan(c) - 2*a^2*d*x*tan(d*x)^2*tan(c)^2 - 3*b^2*d*x*tan(d*x)^2*tan(c)^2 + 8*a*b*t
an(1/2*d*x)^4*tan(c)^2 + 8*a^2*d*x*tan(1/2*d*x)*tan(1/2*c)*tan(c)^2 + 12*b^2*d*x*tan(1/2*d*x)*tan(1/2*c)*tan(c
)^2 - 32*a*b*tan(d*x)^2*tan(1/2*d*x)*tan(1/2*c)*tan(c)^2 + 32*a*b*tan(1/2*d*x)^3*tan(1/2*c)*tan(c)^2 + 96*a*b*
tan(1/2*d*x)^2*tan(1/2*c)^2*tan(c)^2 + 32*a*b*tan(1/2*d*x)*tan(1/2*c)^3*tan(c)^2 + 8*a*b*tan(1/2*c)^4*tan(c)^2
+ 2*a^2*d*x*tan(d*x)*tan(c)^3 + 3*b^2*d*x*tan(d*x)*tan(c)^3 - 8*a*b*tan(d*x)^3*tan(c)^3 + 32*a*b*tan(d*x)*tan
(1/2*d*x)*tan(1/2*c)*tan(c)^3 - 2*a^2*tan(d*x)*tan(1/2*d*x)^4 - 3*b^2*tan(d*x)*tan(1/2*d*x)^4 - 8*a^2*tan(d*x)
^3*tan(1/2*d*x)*tan(1/2*c) - 8*b^2*tan(d*x)^3*tan(1/2*d*x)*tan(1/2*c) - 8*a^2*tan(d*x)*tan(1/2*d*x)^3*tan(1/2*
c) - 12*b^2*tan(d*x)*tan(1/2*d*x)^3*tan(1/2*c) - 8*a^2*tan(d*x)*tan(1/2*d*x)*tan(1/2*c)^3 - 12*b^2*tan(d*x)*ta
n(1/2*d*x)*tan(1/2*c)^3 - 2*a^2*tan(d*x)*tan(1/2*c)^4 - 3*b^2*tan(d*x)*tan(1/2*c)^4 - 2*a^2*tan(1/2*d*x)^4*tan
(c) - 3*b^2*tan(1/2*d*x)^4*tan(c) - 8*a^2*tan(d*x)^2*tan(1/2*d*x)*tan(1/2*c)*tan(c) - 8*a^2*tan(1/2*d*x)^3*tan
(1/2*c)*tan(c) - 12*b^2*tan(1/2*d*x)^3*tan(1/2*c)*tan(c) - 8*a^2*tan(1/2*d*x)*tan(1/2*c)^3*tan(c) - 12*b^2*tan
(1/2*d*x)*tan(1/2*c)^3*tan(c) - 2*a^2*tan(1/2*c)^4*tan(c) - 3*b^2*tan(1/2*c)^4*tan(c) + 2*a^2*tan(d*x)^3*tan(c
)^2 + 3*b^2*tan(d*x)^3*tan(c)^2 - 8*a^2*tan(d*x)*tan(1/2*d*x)*tan(1/2*c)*tan(c)^2 + 2*a^2*tan(d*x)^2*tan(c)^3
+ 3*b^2*tan(d*x)^2*tan(c)^3 - 8*a^2*tan(1/2*d*x)*tan(1/2*c)*tan(c)^3 - 8*b^2*tan(1/2*d*x)*tan(1/2*c)*tan(c)^3
- 2*a^2*d*x*tan(d*x)^2 - 3*b^2*d*x*tan(d*x)^2 + 8*a*b*tan(1/2*d*x)^4 + 8*a^2*d*x*tan(1/2*d*x)*tan(1/2*c) + 12*
b^2*d*x*tan(1/2*d*x)*tan(1/2*c) - 32*a*b*tan(d*x)^2*tan(1/2*d*x)*tan(1/2*c) + 32*a*b*tan(1/2*d*x)^3*tan(1/2*c)
+ 96*a*b*tan(1/2*d*x)^2*tan(1/2*c)^2 + 32*a*b*tan(1/2*d*x)*tan(1/2*c)^3 + 8*a*b*tan(1/2*c)^4 + 2*a^2*d*x*tan(
d*x)*tan(c) + 3*b^2*d*x*tan(d*x)*tan(c) - 8*a*b*tan(d*x)^3*tan(c) + 32*a*b*tan(d*x)*tan(1/2*d*x)*tan(1/2*c)*ta
n(c) - 2*a^2*d*x*tan(c)^2 - 3*b^2*d*x*tan(c)^2 + 8*a*b*tan(d*x)^2*tan(c)^2 - 32*a*b*tan(1/2*d*x)*tan(1/2*c)*ta
n(c)^2 - 8*a*b*tan(d*x)*tan(c)^3 + 2*a^2*tan(d*x)^3 + 2*b^2*tan(d*x)^3 - 8*a^2*tan(d*x)*tan(1/2*d*x)*tan(1/2*c
) - 12*b^2*tan(d*x)*tan(1/2*d*x)*tan(1/2*c) + 2*a^2*tan(d*x)^2*tan(c) - 8*a^2*tan(1/2*d*x)*tan(1/2*c)*tan(c) -
12*b^2*tan(1/2*d*x)*tan(1/2*c)*tan(c) + 2*a^2*tan(d*x)*tan(c)^2 + 2*a^2*tan(c)^3 + 2*b^2*tan(c)^3 - 2*a^2*d*x
- 3*b^2*d*x + 8*a*b*tan(d*x)^2 - 32*a*b*tan(1/2*d*x)*tan(1/2*c) - 8*a*b*tan(d*x)*tan(c) + 8*a*b*tan(c)^2 + 2*
a^2*tan(d*x) + 3*b^2*tan(d*x) + 2*a^2*tan(c) + 3*b^2*tan(c) + 8*a*b)/(d*tan(d*x)^3*tan(1/2*d*x)^4*tan(1/2*c)^4
*tan(c)^3 + d*tan(d*x)^3*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c) - d*tan(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c)^
2 - 4*d*tan(d*x)^3*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c)^3 + d*tan(d*x)*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c)^3 - d*
tan(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*d*tan(d*x)^3*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c) + d*tan(d*x)*tan(1/
2*d*x)^4*tan(1/2*c)^4*tan(c) + 4*d*tan(d*x)^2*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c)^2 - d*tan(1/2*d*x)^4*tan(1/2*
c)^4*tan(c)^2 - d*tan(d*x)^3*tan(1/2*d*x)^4*tan(c)^3 - 4*d*tan(d*x)^3*tan(1/2*d*x)^3*tan(1/2*c)*tan(c)^3 - 4*d
*tan(d*x)^3*tan(1/2*d*x)*tan(1/2*c)^3*tan(c)^3 - 4*d*tan(d*x)*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c)^3 - d*tan(d*x
)^3*tan(1/2*c)^4*tan(c)^3 + 4*d*tan(d*x)^2*tan(1/2*d*x)^3*tan(1/2*c)^3 - d*tan(1/2*d*x)^4*tan(1/2*c)^4 - d*tan
(d*x)^3*tan(1/2*d*x)^4*tan(c) - 4*d*tan(d*x)^3*tan(1/2*d*x)^3*tan(1/2*c)*tan(c) - 4*d*tan(d*x)^3*tan(1/2*d*x)*
tan(1/2*c)^3*tan(c) - 4*d*tan(d*x)*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c) - d*tan(d*x)^3*tan(1/2*c)^4*tan(c) + d*t
an(d*x)^2*tan(1/2*d*x)^4*tan(c)^2 + 4*d*tan(d*x)^2*tan(1/2*d*x)^3*tan(1/2*c)*tan(c)^2 + 4*d*tan(d*x)^2*tan(1/2
*d*x)*tan(1/2*c)^3*tan(c)^2 + 4*d*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c)^2 + d*tan(d*x)^2*tan(1/2*c)^4*tan(c)^2 -
d*tan(d*x)*tan(1/2*d*x)^4*tan(c)^3 - 4*d*tan(d*x)^3*tan(1/2*d*x)*tan(1/2*c)*tan(c)^3 - 4*d*tan(d*x)*tan(1/2*d*
x)^3*tan(1/2*c)*tan(c)^3 - 4*d*tan(d*x)*tan(1/2*d*x)*tan(1/2*c)^3*tan(c)^3 - d*tan(d*x)*tan(1/2*c)^4*tan(c)^3
+ d*tan(d*x)^2*tan(1/2*d*x)^4 + 4*d*tan(d*x)^2*tan(1/2*d*x)^3*tan(1/2*c) + 4*d*tan(d*x)^2*tan(1/2*d*x)*tan(1/2
*c)^3 + 4*d*tan(1/2*d*x)^3*tan(1/2*c)^3 + d*tan(d*x)^2*tan(1/2*c)^4 - d*tan(d*x)*tan(1/2*d*x)^4*tan(c) - 4*d*t
an(d*x)^3*tan(1/2*d*x)*tan(1/2*c)*tan(c) - 4*d*tan(d*x)*tan(1/2*d*x)^3*tan(1/2*c)*tan(c) - 4*d*tan(d*x)*tan(1/
2*d*x)*tan(1/2*c)^3*tan(c) - d*tan(d*x)*tan(1/2*c)^4*tan(c) + d*tan(1/2*d*x)^4*tan(c)^2 + 4*d*tan(d*x)^2*tan(1
/2*d*x)*tan(1/2*c)*tan(c)^2 + 4*d*tan(1/2*d*x)^3*tan(1/2*c)*tan(c)^2 + 4*d*tan(1/2*d*x)*tan(1/2*c)^3*tan(c)^2
+ d*tan(1/2*c)^4*tan(c)^2 + d*tan(d*x)^3*tan(c)^3 - 4*d*tan(d*x)*tan(1/2*d*x)*tan(1/2*c)*tan(c)^3 + d*tan(1/2*
d*x)^4 + 4*d*tan(d*x)^2*tan(1/2*d*x)*tan(1/2*c) + 4*d*tan(1/2*d*x)^3*tan(1/2*c) + 4*d*tan(1/2*d*x)*tan(1/2*c)^
3 + d*tan(1/2*c)^4 + d*tan(d*x)^3*tan(c) - 4*d*tan(d*x)*tan(1/2*d*x)*tan(1/2*c)*tan(c) - d*tan(d*x)^2*tan(c)^2
+ 4*d*tan(1/2*d*x)*tan(1/2*c)*tan(c)^2 + d*tan(d*x)*tan(c)^3 - d*tan(d*x)^2 + 4*d*tan(1/2*d*x)*tan(1/2*c) + d
*tan(d*x)*tan(c) - d*tan(c)^2 - d)