3.237 \(\int \cos ^2(c+d x) (a \sin (c+d x)+b \tan (c+d x))^2 \, dx\)
Optimal. Leaf size=86 \[ -\frac{\left (a^2+4 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left (a^2+4 b^2\right )+\frac{5 a b \sin ^3(c+d x)}{12 d}+\frac{a \sin ^3(c+d x) (a \cos (c+d x)+b)}{4 d} \]
[Out]
((a^2 + 4*b^2)*x)/8 - ((a^2 + 4*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (5*a*b*Sin[c + d*x]^3)/(12*d) + (a*(b
+ a*Cos[c + d*x])*Sin[c + d*x]^3)/(4*d)
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Rubi [A] time = 0.189586, antiderivative size = 86, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used =
{4397, 2692, 2669, 2635, 8} \[ -\frac{\left (a^2+4 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left (a^2+4 b^2\right )+\frac{5 a b \sin ^3(c+d x)}{12 d}+\frac{a \sin ^3(c+d x) (a \cos (c+d x)+b)}{4 d} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^2*(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]
[Out]
((a^2 + 4*b^2)*x)/8 - ((a^2 + 4*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (5*a*b*Sin[c + d*x]^3)/(12*d) + (a*(b
+ a*Cos[c + d*x])*Sin[c + d*x]^3)/(4*d)
Rule 4397
Int[u_, x_Symbol] :> Int[TrigSimplify[u], x] /; TrigSimplifyQ[u]
Rule 2692
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(b*(g
*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1))/(f*g*(m + p)), x] + Dist[1/(m + p), Int[(g*Cos[e + f*x])^
p*(a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(m + p) + a*b*(2*m + p - 1)*Sin[e + f*x]), x], x] /; FreeQ[{
a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && NeQ[m + p, 0] && (IntegersQ[2*m, 2*p] || IntegerQ[m
])
Rule 2669
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(b*(g*Cos[
e + f*x])^(p + 1))/(f*g*(p + 1)), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x]
&& (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
Rule 2635
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]
Rule 8
Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
Rubi steps
\begin{align*} \int \cos ^2(c+d x) (a \sin (c+d x)+b \tan (c+d x))^2 \, dx &=\int (b+a \cos (c+d x))^2 \sin ^2(c+d x) \, dx\\ &=\frac{a (b+a \cos (c+d x)) \sin ^3(c+d x)}{4 d}+\frac{1}{4} \int \left (a^2+4 b^2+5 a b \cos (c+d x)\right ) \sin ^2(c+d x) \, dx\\ &=\frac{5 a b \sin ^3(c+d x)}{12 d}+\frac{a (b+a \cos (c+d x)) \sin ^3(c+d x)}{4 d}+\frac{1}{4} \left (a^2+4 b^2\right ) \int \sin ^2(c+d x) \, dx\\ &=-\frac{\left (a^2+4 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac{5 a b \sin ^3(c+d x)}{12 d}+\frac{a (b+a \cos (c+d x)) \sin ^3(c+d x)}{4 d}+\frac{1}{8} \left (a^2+4 b^2\right ) \int 1 \, dx\\ &=\frac{1}{8} \left (a^2+4 b^2\right ) x-\frac{\left (a^2+4 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac{5 a b \sin ^3(c+d x)}{12 d}+\frac{a (b+a \cos (c+d x)) \sin ^3(c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.222586, size = 82, normalized size = 0.95 \[ \frac{-3 a^2 \sin (4 (c+d x))+12 a^2 c+12 a^2 d x+48 a b \sin (c+d x)-16 a b \sin (3 (c+d x))-24 b^2 \sin (2 (c+d x))+48 b^2 c+48 b^2 d x}{96 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]^2*(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]
[Out]
(12*a^2*c + 48*b^2*c + 12*a^2*d*x + 48*b^2*d*x + 48*a*b*Sin[c + d*x] - 24*b^2*Sin[2*(c + d*x)] - 16*a*b*Sin[3*
(c + d*x)] - 3*a^2*Sin[4*(c + d*x)])/(96*d)
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Maple [A] time = 0.059, size = 86, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( -{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{8}}+{\frac{dx}{8}}+{\frac{c}{8}} \right ) +{\frac{2\,ab \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3}}+{b}^{2} \left ( -{\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)^2*(a*sin(d*x+c)+b*tan(d*x+c))^2,x)
[Out]
1/d*(a^2*(-1/4*sin(d*x+c)*cos(d*x+c)^3+1/8*cos(d*x+c)*sin(d*x+c)+1/8*d*x+1/8*c)+2/3*a*b*sin(d*x+c)^3+b^2*(-1/2
*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c))
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Maxima [A] time = 1.0166, size = 89, normalized size = 1.03 \begin{align*} \frac{64 \, a b \sin \left (d x + c\right )^{3} + 3 \,{\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2} + 24 \,{\left (2 \, d x + 2 \, c - \sin \left (2 \, d x + 2 \, c\right )\right )} b^{2}}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^2*(a*sin(d*x+c)+b*tan(d*x+c))^2,x, algorithm="maxima")
[Out]
1/96*(64*a*b*sin(d*x + c)^3 + 3*(4*d*x + 4*c - sin(4*d*x + 4*c))*a^2 + 24*(2*d*x + 2*c - sin(2*d*x + 2*c))*b^2
)/d
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Fricas [A] time = 0.498968, size = 178, normalized size = 2.07 \begin{align*} \frac{3 \,{\left (a^{2} + 4 \, b^{2}\right )} d x -{\left (6 \, a^{2} \cos \left (d x + c\right )^{3} + 16 \, a b \cos \left (d x + c\right )^{2} - 16 \, a b - 3 \,{\left (a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^2*(a*sin(d*x+c)+b*tan(d*x+c))^2,x, algorithm="fricas")
[Out]
1/24*(3*(a^2 + 4*b^2)*d*x - (6*a^2*cos(d*x + c)^3 + 16*a*b*cos(d*x + c)^2 - 16*a*b - 3*(a^2 - 4*b^2)*cos(d*x +
c))*sin(d*x + c))/d
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \sin{\left (c + d x \right )} + b \tan{\left (c + d x \right )}\right )^{2} \cos ^{2}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)**2*(a*sin(d*x+c)+b*tan(d*x+c))**2,x)
[Out]
Integral((a*sin(c + d*x) + b*tan(c + d*x))**2*cos(c + d*x)**2, x)
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Giac [B] time = 7.74578, size = 6967, normalized size = 81.01 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^2*(a*sin(d*x+c)+b*tan(d*x+c))^2,x, algorithm="giac")
[Out]
1/8*a^2*x - 1/32*a^2*sin(4*d*x + 4*c)/d + 1/6*(3*b^2*d*x*tan(d*x)^2*tan(1/2*d*x)^6*tan(1/2*c)^6*tan(c)^2 + 3*b
^2*d*x*tan(d*x)^2*tan(1/2*d*x)^6*tan(1/2*c)^6 + 9*b^2*d*x*tan(d*x)^2*tan(1/2*d*x)^6*tan(1/2*c)^4*tan(c)^2 + 9*
b^2*d*x*tan(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c)^6*tan(c)^2 + 3*b^2*d*x*tan(1/2*d*x)^6*tan(1/2*c)^6*tan(c)^2 + 3*b
^2*tan(d*x)^2*tan(1/2*d*x)^6*tan(1/2*c)^6*tan(c) + 3*b^2*tan(d*x)*tan(1/2*d*x)^6*tan(1/2*c)^6*tan(c)^2 + 9*b^2
*d*x*tan(d*x)^2*tan(1/2*d*x)^6*tan(1/2*c)^4 + 9*b^2*d*x*tan(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c)^6 + 3*b^2*d*x*tan
(1/2*d*x)^6*tan(1/2*c)^6 + 9*b^2*d*x*tan(d*x)^2*tan(1/2*d*x)^6*tan(1/2*c)^2*tan(c)^2 + 27*b^2*d*x*tan(d*x)^2*t
an(1/2*d*x)^4*tan(1/2*c)^4*tan(c)^2 + 9*b^2*d*x*tan(1/2*d*x)^6*tan(1/2*c)^4*tan(c)^2 + 9*b^2*d*x*tan(d*x)^2*ta
n(1/2*d*x)^2*tan(1/2*c)^6*tan(c)^2 + 9*b^2*d*x*tan(1/2*d*x)^4*tan(1/2*c)^6*tan(c)^2 - 3*b^2*tan(d*x)*tan(1/2*d
*x)^6*tan(1/2*c)^6 + 9*b^2*tan(d*x)^2*tan(1/2*d*x)^6*tan(1/2*c)^4*tan(c) + 9*b^2*tan(d*x)^2*tan(1/2*d*x)^4*tan
(1/2*c)^6*tan(c) - 3*b^2*tan(1/2*d*x)^6*tan(1/2*c)^6*tan(c) - 32*a*b*tan(d*x)^2*tan(1/2*d*x)^6*tan(1/2*c)^3*ta
n(c)^2 - 96*a*b*tan(d*x)^2*tan(1/2*d*x)^5*tan(1/2*c)^4*tan(c)^2 + 9*b^2*tan(d*x)*tan(1/2*d*x)^6*tan(1/2*c)^4*t
an(c)^2 - 96*a*b*tan(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c)^5*tan(c)^2 - 32*a*b*tan(d*x)^2*tan(1/2*d*x)^3*tan(1/2*c)
^6*tan(c)^2 + 9*b^2*tan(d*x)*tan(1/2*d*x)^4*tan(1/2*c)^6*tan(c)^2 + 9*b^2*d*x*tan(d*x)^2*tan(1/2*d*x)^6*tan(1/
2*c)^2 + 27*b^2*d*x*tan(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c)^4 + 9*b^2*d*x*tan(1/2*d*x)^6*tan(1/2*c)^4 + 9*b^2*d*x
*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c)^6 + 9*b^2*d*x*tan(1/2*d*x)^4*tan(1/2*c)^6 + 3*b^2*d*x*tan(d*x)^2*tan(1/2
*d*x)^6*tan(c)^2 + 27*b^2*d*x*tan(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c)^2*tan(c)^2 + 9*b^2*d*x*tan(1/2*d*x)^6*tan(1
/2*c)^2*tan(c)^2 + 27*b^2*d*x*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c)^4*tan(c)^2 + 27*b^2*d*x*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*c)^6*tan(c)^2 + 9*b^2*d*x*tan(1/2*d*x)^2*tan(1/2*c)^6*tan(c)^
2 - 32*a*b*tan(d*x)^2*tan(1/2*d*x)^6*tan(1/2*c)^3 - 96*a*b*tan(d*x)^2*tan(1/2*d*x)^5*tan(1/2*c)^4 - 9*b^2*tan(
d*x)*tan(1/2*d*x)^6*tan(1/2*c)^4 - 96*a*b*tan(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c)^5 - 32*a*b*tan(d*x)^2*tan(1/2*d
*x)^3*tan(1/2*c)^6 - 9*b^2*tan(d*x)*tan(1/2*d*x)^4*tan(1/2*c)^6 + 9*b^2*tan(d*x)^2*tan(1/2*d*x)^6*tan(1/2*c)^2
*tan(c) + 27*b^2*tan(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c) - 9*b^2*tan(1/2*d*x)^6*tan(1/2*c)^4*tan(c) + 9*
b^2*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c)^6*tan(c) - 9*b^2*tan(1/2*d*x)^4*tan(1/2*c)^6*tan(c) + 96*a*b*tan(d*x)
^2*tan(1/2*d*x)^5*tan(1/2*c)^2*tan(c)^2 + 9*b^2*tan(d*x)*tan(1/2*d*x)^6*tan(1/2*c)^2*tan(c)^2 + 288*a*b*tan(d*
x)^2*tan(1/2*d*x)^4*tan(1/2*c)^3*tan(c)^2 - 32*a*b*tan(1/2*d*x)^6*tan(1/2*c)^3*tan(c)^2 + 288*a*b*tan(d*x)^2*t
an(1/2*d*x)^3*tan(1/2*c)^4*tan(c)^2 + 27*b^2*tan(d*x)*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c)^2 - 96*a*b*tan(1/2*d*
x)^5*tan(1/2*c)^4*tan(c)^2 + 96*a*b*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c)^5*tan(c)^2 - 96*a*b*tan(1/2*d*x)^4*ta
n(1/2*c)^5*tan(c)^2 + 9*b^2*tan(d*x)*tan(1/2*d*x)^2*tan(1/2*c)^6*tan(c)^2 - 32*a*b*tan(1/2*d*x)^3*tan(1/2*c)^6
*tan(c)^2 + 3*b^2*d*x*tan(d*x)^2*tan(1/2*d*x)^6 + 27*b^2*d*x*tan(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c)^2 + 9*b^2*d*
x*tan(1/2*d*x)^6*tan(1/2*c)^2 + 27*b^2*d*x*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c)^4 + 27*b^2*d*x*tan(1/2*d*x)^4*
tan(1/2*c)^4 + 3*b^2*d*x*tan(d*x)^2*tan(1/2*c)^6 + 9*b^2*d*x*tan(1/2*d*x)^2*tan(1/2*c)^6 + 9*b^2*d*x*tan(d*x)^
2*tan(1/2*d*x)^4*tan(c)^2 + 3*b^2*d*x*tan(1/2*d*x)^6*tan(c)^2 + 27*b^2*d*x*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c
)^2*tan(c)^2 + 27*b^2*d*x*tan(1/2*d*x)^4*tan(1/2*c)^2*tan(c)^2 + 9*b^2*d*x*tan(d*x)^2*tan(1/2*c)^4*tan(c)^2 +
27*b^2*d*x*tan(1/2*d*x)^2*tan(1/2*c)^4*tan(c)^2 + 3*b^2*d*x*tan(1/2*c)^6*tan(c)^2 + 96*a*b*tan(d*x)^2*tan(1/2*
d*x)^5*tan(1/2*c)^2 - 9*b^2*tan(d*x)*tan(1/2*d*x)^6*tan(1/2*c)^2 + 288*a*b*tan(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c
)^3 - 32*a*b*tan(1/2*d*x)^6*tan(1/2*c)^3 + 288*a*b*tan(d*x)^2*tan(1/2*d*x)^3*tan(1/2*c)^4 - 27*b^2*tan(d*x)*ta
n(1/2*d*x)^4*tan(1/2*c)^4 - 96*a*b*tan(1/2*d*x)^5*tan(1/2*c)^4 + 96*a*b*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c)^5
- 96*a*b*tan(1/2*d*x)^4*tan(1/2*c)^5 - 9*b^2*tan(d*x)*tan(1/2*d*x)^2*tan(1/2*c)^6 - 32*a*b*tan(1/2*d*x)^3*tan
(1/2*c)^6 + 3*b^2*tan(d*x)^2*tan(1/2*d*x)^6*tan(c) + 27*b^2*tan(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c)^2*tan(c) - 9*
b^2*tan(1/2*d*x)^6*tan(1/2*c)^2*tan(c) + 27*b^2*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c)^4*tan(c) - 27*b^2*tan(1/2
*d*x)^4*tan(1/2*c)^4*tan(c) + 3*b^2*tan(d*x)^2*tan(1/2*c)^6*tan(c) - 9*b^2*tan(1/2*d*x)^2*tan(1/2*c)^6*tan(c)
+ 3*b^2*tan(d*x)*tan(1/2*d*x)^6*tan(c)^2 - 96*a*b*tan(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c)*tan(c)^2 - 288*a*b*tan(
d*x)^2*tan(1/2*d*x)^3*tan(1/2*c)^2*tan(c)^2 + 27*b^2*tan(d*x)*tan(1/2*d*x)^4*tan(1/2*c)^2*tan(c)^2 + 96*a*b*ta
n(1/2*d*x)^5*tan(1/2*c)^2*tan(c)^2 - 288*a*b*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c)^3*tan(c)^2 + 288*a*b*tan(1/2
*d*x)^4*tan(1/2*c)^3*tan(c)^2 - 96*a*b*tan(d*x)^2*tan(1/2*d*x)*tan(1/2*c)^4*tan(c)^2 + 27*b^2*tan(d*x)*tan(1/2
*d*x)^2*tan(1/2*c)^4*tan(c)^2 + 288*a*b*tan(1/2*d*x)^3*tan(1/2*c)^4*tan(c)^2 + 96*a*b*tan(1/2*d*x)^2*tan(1/2*c
)^5*tan(c)^2 + 3*b^2*tan(d*x)*tan(1/2*c)^6*tan(c)^2 + 9*b^2*d*x*tan(d*x)^2*tan(1/2*d*x)^4 + 3*b^2*d*x*tan(1/2*
d*x)^6 + 27*b^2*d*x*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c)^2 + 27*b^2*d*x*tan(1/2*d*x)^4*tan(1/2*c)^2 + 9*b^2*d*
x*tan(d*x)^2*tan(1/2*c)^4 + 27*b^2*d*x*tan(1/2*d*x)^2*tan(1/2*c)^4 + 3*b^2*d*x*tan(1/2*c)^6 + 9*b^2*d*x*tan(d*
x)^2*tan(1/2*d*x)^2*tan(c)^2 + 9*b^2*d*x*tan(1/2*d*x)^4*tan(c)^2 + 9*b^2*d*x*tan(d*x)^2*tan(1/2*c)^2*tan(c)^2
+ 27*b^2*d*x*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(c)^2 + 9*b^2*d*x*tan(1/2*c)^4*tan(c)^2 - 3*b^2*tan(d*x)*tan(1/2*d
*x)^6 - 96*a*b*tan(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c) - 288*a*b*tan(d*x)^2*tan(1/2*d*x)^3*tan(1/2*c)^2 - 27*b^2*
tan(d*x)*tan(1/2*d*x)^4*tan(1/2*c)^2 + 96*a*b*tan(1/2*d*x)^5*tan(1/2*c)^2 - 288*a*b*tan(d*x)^2*tan(1/2*d*x)^2*
tan(1/2*c)^3 + 288*a*b*tan(1/2*d*x)^4*tan(1/2*c)^3 - 96*a*b*tan(d*x)^2*tan(1/2*d*x)*tan(1/2*c)^4 - 27*b^2*tan(
d*x)*tan(1/2*d*x)^2*tan(1/2*c)^4 + 288*a*b*tan(1/2*d*x)^3*tan(1/2*c)^4 + 96*a*b*tan(1/2*d*x)^2*tan(1/2*c)^5 -
3*b^2*tan(d*x)*tan(1/2*c)^6 + 9*b^2*tan(d*x)^2*tan(1/2*d*x)^4*tan(c) - 3*b^2*tan(1/2*d*x)^6*tan(c) + 27*b^2*ta
n(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(c) - 27*b^2*tan(1/2*d*x)^4*tan(1/2*c)^2*tan(c) + 9*b^2*tan(d*x)^2*tan
(1/2*c)^4*tan(c) - 27*b^2*tan(1/2*d*x)^2*tan(1/2*c)^4*tan(c) - 3*b^2*tan(1/2*c)^6*tan(c) + 32*a*b*tan(d*x)^2*t
an(1/2*d*x)^3*tan(c)^2 + 9*b^2*tan(d*x)*tan(1/2*d*x)^4*tan(c)^2 + 96*a*b*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c)*
tan(c)^2 - 96*a*b*tan(1/2*d*x)^4*tan(1/2*c)*tan(c)^2 + 96*a*b*tan(d*x)^2*tan(1/2*d*x)*tan(1/2*c)^2*tan(c)^2 +
27*b^2*tan(d*x)*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(c)^2 - 288*a*b*tan(1/2*d*x)^3*tan(1/2*c)^2*tan(c)^2 + 32*a*b*t
an(d*x)^2*tan(1/2*c)^3*tan(c)^2 - 288*a*b*tan(1/2*d*x)^2*tan(1/2*c)^3*tan(c)^2 + 9*b^2*tan(d*x)*tan(1/2*c)^4*t
an(c)^2 - 96*a*b*tan(1/2*d*x)*tan(1/2*c)^4*tan(c)^2 + 9*b^2*d*x*tan(d*x)^2*tan(1/2*d*x)^2 + 9*b^2*d*x*tan(1/2*
d*x)^4 + 9*b^2*d*x*tan(d*x)^2*tan(1/2*c)^2 + 27*b^2*d*x*tan(1/2*d*x)^2*tan(1/2*c)^2 + 9*b^2*d*x*tan(1/2*c)^4 +
3*b^2*d*x*tan(d*x)^2*tan(c)^2 + 9*b^2*d*x*tan(1/2*d*x)^2*tan(c)^2 + 9*b^2*d*x*tan(1/2*c)^2*tan(c)^2 + 32*a*b*
tan(d*x)^2*tan(1/2*d*x)^3 - 9*b^2*tan(d*x)*tan(1/2*d*x)^4 + 96*a*b*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c) - 96*a
*b*tan(1/2*d*x)^4*tan(1/2*c) + 96*a*b*tan(d*x)^2*tan(1/2*d*x)*tan(1/2*c)^2 - 27*b^2*tan(d*x)*tan(1/2*d*x)^2*ta
n(1/2*c)^2 - 288*a*b*tan(1/2*d*x)^3*tan(1/2*c)^2 + 32*a*b*tan(d*x)^2*tan(1/2*c)^3 - 288*a*b*tan(1/2*d*x)^2*tan
(1/2*c)^3 - 9*b^2*tan(d*x)*tan(1/2*c)^4 - 96*a*b*tan(1/2*d*x)*tan(1/2*c)^4 + 9*b^2*tan(d*x)^2*tan(1/2*d*x)^2*t
an(c) - 9*b^2*tan(1/2*d*x)^4*tan(c) + 9*b^2*tan(d*x)^2*tan(1/2*c)^2*tan(c) - 27*b^2*tan(1/2*d*x)^2*tan(1/2*c)^
2*tan(c) - 9*b^2*tan(1/2*c)^4*tan(c) + 9*b^2*tan(d*x)*tan(1/2*d*x)^2*tan(c)^2 + 32*a*b*tan(1/2*d*x)^3*tan(c)^2
+ 96*a*b*tan(1/2*d*x)^2*tan(1/2*c)*tan(c)^2 + 9*b^2*tan(d*x)*tan(1/2*c)^2*tan(c)^2 + 96*a*b*tan(1/2*d*x)*tan(
1/2*c)^2*tan(c)^2 + 32*a*b*tan(1/2*c)^3*tan(c)^2 + 3*b^2*d*x*tan(d*x)^2 + 9*b^2*d*x*tan(1/2*d*x)^2 + 9*b^2*d*x
*tan(1/2*c)^2 + 3*b^2*d*x*tan(c)^2 - 9*b^2*tan(d*x)*tan(1/2*d*x)^2 + 32*a*b*tan(1/2*d*x)^3 + 96*a*b*tan(1/2*d*
x)^2*tan(1/2*c) - 9*b^2*tan(d*x)*tan(1/2*c)^2 + 96*a*b*tan(1/2*d*x)*tan(1/2*c)^2 + 32*a*b*tan(1/2*c)^3 + 3*b^2
*tan(d*x)^2*tan(c) - 9*b^2*tan(1/2*d*x)^2*tan(c) - 9*b^2*tan(1/2*c)^2*tan(c) + 3*b^2*tan(d*x)*tan(c)^2 + 3*b^2
*d*x - 3*b^2*tan(d*x) - 3*b^2*tan(c))/(d*tan(d*x)^2*tan(1/2*d*x)^6*tan(1/2*c)^6*tan(c)^2 + d*tan(d*x)^2*tan(1/
2*d*x)^6*tan(1/2*c)^6 + 3*d*tan(d*x)^2*tan(1/2*d*x)^6*tan(1/2*c)^4*tan(c)^2 + 3*d*tan(d*x)^2*tan(1/2*d*x)^4*ta
n(1/2*c)^6*tan(c)^2 + d*tan(1/2*d*x)^6*tan(1/2*c)^6*tan(c)^2 + 3*d*tan(d*x)^2*tan(1/2*d*x)^6*tan(1/2*c)^4 + 3*
d*tan(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c)^6 + d*tan(1/2*d*x)^6*tan(1/2*c)^6 + 3*d*tan(d*x)^2*tan(1/2*d*x)^6*tan(1
/2*c)^2*tan(c)^2 + 9*d*tan(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c)^2 + 3*d*tan(1/2*d*x)^6*tan(1/2*c)^4*tan(c
)^2 + 3*d*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c)^6*tan(c)^2 + 3*d*tan(1/2*d*x)^4*tan(1/2*c)^6*tan(c)^2 + 3*d*tan
(d*x)^2*tan(1/2*d*x)^6*tan(1/2*c)^2 + 9*d*tan(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c)^4 + 3*d*tan(1/2*d*x)^6*tan(1/2*
c)^4 + 3*d*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c)^6 + 3*d*tan(1/2*d*x)^4*tan(1/2*c)^6 + d*tan(d*x)^2*tan(1/2*d*x
)^6*tan(c)^2 + 9*d*tan(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c)^2*tan(c)^2 + 3*d*tan(1/2*d*x)^6*tan(1/2*c)^2*tan(c)^2
+ 9*d*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c)^4*tan(c)^2 + 9*d*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c)^2 + d*tan(d*x)^
2*tan(1/2*c)^6*tan(c)^2 + 3*d*tan(1/2*d*x)^2*tan(1/2*c)^6*tan(c)^2 + d*tan(d*x)^2*tan(1/2*d*x)^6 + 9*d*tan(d*x
)^2*tan(1/2*d*x)^4*tan(1/2*c)^2 + 3*d*tan(1/2*d*x)^6*tan(1/2*c)^2 + 9*d*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c)^4
+ 9*d*tan(1/2*d*x)^4*tan(1/2*c)^4 + d*tan(d*x)^2*tan(1/2*c)^6 + 3*d*tan(1/2*d*x)^2*tan(1/2*c)^6 + 3*d*tan(d*x
)^2*tan(1/2*d*x)^4*tan(c)^2 + d*tan(1/2*d*x)^6*tan(c)^2 + 9*d*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(c)^2
+ 9*d*tan(1/2*d*x)^4*tan(1/2*c)^2*tan(c)^2 + 3*d*tan(d*x)^2*tan(1/2*c)^4*tan(c)^2 + 9*d*tan(1/2*d*x)^2*tan(1/2
*c)^4*tan(c)^2 + d*tan(1/2*c)^6*tan(c)^2 + 3*d*tan(d*x)^2*tan(1/2*d*x)^4 + d*tan(1/2*d*x)^6 + 9*d*tan(d*x)^2*t
an(1/2*d*x)^2*tan(1/2*c)^2 + 9*d*tan(1/2*d*x)^4*tan(1/2*c)^2 + 3*d*tan(d*x)^2*tan(1/2*c)^4 + 9*d*tan(1/2*d*x)^
2*tan(1/2*c)^4 + d*tan(1/2*c)^6 + 3*d*tan(d*x)^2*tan(1/2*d*x)^2*tan(c)^2 + 3*d*tan(1/2*d*x)^4*tan(c)^2 + 3*d*t
an(d*x)^2*tan(1/2*c)^2*tan(c)^2 + 9*d*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(c)^2 + 3*d*tan(1/2*c)^4*tan(c)^2 + 3*d*t
an(d*x)^2*tan(1/2*d*x)^2 + 3*d*tan(1/2*d*x)^4 + 3*d*tan(d*x)^2*tan(1/2*c)^2 + 9*d*tan(1/2*d*x)^2*tan(1/2*c)^2
+ 3*d*tan(1/2*c)^4 + d*tan(d*x)^2*tan(c)^2 + 3*d*tan(1/2*d*x)^2*tan(c)^2 + 3*d*tan(1/2*c)^2*tan(c)^2 + d*tan(d
*x)^2 + 3*d*tan(1/2*d*x)^2 + 3*d*tan(1/2*c)^2 + d*tan(c)^2 + d)