3.21 \(\int (a+a \sin (c+d x))^2 \tan ^2(c+d x) \, dx\)
Optimal. Leaf size=71 \[ \frac{2 a^2 \cos (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{2 a^2 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac{5 a^2 x}{2} \]
[Out]
(-5*a^2*x)/2 + (2*a^2*Cos[c + d*x])/d + (2*a^2*Cos[c + d*x])/(d*(1 - Sin[c + d*x])) + (a^2*Cos[c + d*x]*Sin[c
+ d*x])/(2*d)
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Rubi [A] time = 0.0889338, antiderivative size = 71, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used =
{2709, 2648, 2638, 2635, 8} \[ \frac{2 a^2 \cos (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{2 a^2 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac{5 a^2 x}{2} \]
Antiderivative was successfully verified.
[In]
Int[(a + a*Sin[c + d*x])^2*Tan[c + d*x]^2,x]
[Out]
(-5*a^2*x)/2 + (2*a^2*Cos[c + d*x])/d + (2*a^2*Cos[c + d*x])/(d*(1 - Sin[c + d*x])) + (a^2*Cos[c + d*x]*Sin[c
+ d*x])/(2*d)
Rule 2709
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*tan[(e_.) + (f_.)*(x_)]^(p_), x_Symbol] :> Dist[a^p, Int[Expan
dIntegrand[(Sin[e + f*x]^p*(a + b*Sin[e + f*x])^(m - p/2))/(a - b*Sin[e + f*x])^(p/2), x], x], x] /; FreeQ[{a,
b, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p/2] && (LtQ[p, 0] || GtQ[m - p/2, 0])
Rule 2648
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Simp[Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Rule 2638
Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]
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 (a+a \sin (c+d x))^2 \tan ^2(c+d x) \, dx &=a^2 \int \left (-2-\frac{2}{-1+\sin (c+d x)}-2 \sin (c+d x)-\sin ^2(c+d x)\right ) \, dx\\ &=-2 a^2 x-a^2 \int \sin ^2(c+d x) \, dx-\left (2 a^2\right ) \int \frac{1}{-1+\sin (c+d x)} \, dx-\left (2 a^2\right ) \int \sin (c+d x) \, dx\\ &=-2 a^2 x+\frac{2 a^2 \cos (c+d x)}{d}+\frac{2 a^2 \cos (c+d x)}{d (1-\sin (c+d x))}+\frac{a^2 \cos (c+d x) \sin (c+d x)}{2 d}-\frac{1}{2} a^2 \int 1 \, dx\\ &=-\frac{5 a^2 x}{2}+\frac{2 a^2 \cos (c+d x)}{d}+\frac{2 a^2 \cos (c+d x)}{d (1-\sin (c+d x))}+\frac{a^2 \cos (c+d x) \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [B] time = 0.437947, size = 145, normalized size = 2.04 \[ -\frac{a^2 (\sin (c+d x)+1)^2 \left (\cos \left (\frac{1}{2} (c+d x)\right ) (10 (c+d x)-\sin (2 (c+d x))-8 \cos (c+d x))+\sin \left (\frac{1}{2} (c+d x)\right ) (-2 (5 c+5 d x+8)+\sin (2 (c+d x))+8 \cos (c+d x))\right )}{4 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + a*Sin[c + d*x])^2*Tan[c + d*x]^2,x]
[Out]
-(a^2*(1 + Sin[c + d*x])^2*(Cos[(c + d*x)/2]*(10*(c + d*x) - 8*Cos[c + d*x] - Sin[2*(c + d*x)]) + Sin[(c + d*x
)/2]*(-2*(8 + 5*c + 5*d*x) + 8*Cos[c + d*x] + Sin[2*(c + d*x)])))/(4*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(
Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)
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Maple [A] time = 0.049, size = 117, normalized size = 1.7 \begin{align*}{\frac{1}{d} \left ({a}^{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 ) +2\,{a}^{2} \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 ) +{a}^{2} \left ( \tan \left ( dx+c \right ) -dx-c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+a*sin(d*x+c))^2*tan(d*x+c)^2,x)
[Out]
1/d*(a^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)+2*a^2*(sin(d*x+c)^4/
cos(d*x+c)+(2+sin(d*x+c)^2)*cos(d*x+c))+a^2*(tan(d*x+c)-d*x-c))
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Maxima [A] time = 1.61458, size = 113, normalized size = 1.59 \begin{align*} -\frac{{\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 )} a^{2} + 2 \,{\left (d x + c - \tan \left (d x + c\right )\right )} a^{2} - 4 \, a^{2}{\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+a*sin(d*x+c))^2*tan(d*x+c)^2,x, algorithm="maxima")
[Out]
-1/2*((3*d*x + 3*c - tan(d*x + c)/(tan(d*x + c)^2 + 1) - 2*tan(d*x + c))*a^2 + 2*(d*x + c - tan(d*x + c))*a^2
- 4*a^2*(1/cos(d*x + c) + cos(d*x + c)))/d
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Fricas [A] time = 1.53691, size = 296, normalized size = 4.17 \begin{align*} \frac{a^{2} \cos \left (d x + c\right )^{3} - 5 \, a^{2} d x + 4 \, a^{2} \cos \left (d x + c\right )^{2} + 4 \, a^{2} -{\left (5 \, a^{2} d x - 7 \, a^{2}\right )} \cos \left (d x + c\right ) +{\left (5 \, a^{2} d x + a^{2} \cos \left (d x + c\right )^{2} - 3 \, a^{2} \cos \left (d x + c\right ) + 4 \, a^{2}\right )} \sin \left (d x + c\right )}{2 \,{\left (d \cos \left (d x + c\right ) - d \sin \left (d x + c\right ) + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(d*x+c))^2*tan(d*x+c)^2,x, algorithm="fricas")
[Out]
1/2*(a^2*cos(d*x + c)^3 - 5*a^2*d*x + 4*a^2*cos(d*x + c)^2 + 4*a^2 - (5*a^2*d*x - 7*a^2)*cos(d*x + c) + (5*a^2
*d*x + a^2*cos(d*x + c)^2 - 3*a^2*cos(d*x + c) + 4*a^2)*sin(d*x + c))/(d*cos(d*x + c) - d*sin(d*x + c) + d)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a^{2} \left (\int 2 \sin{\left (c + d x \right )} \tan ^{2}{\left (c + d x \right )}\, dx + \int \sin ^{2}{\left (c + d x \right )} \tan ^{2}{\left (c + d x \right )}\, dx + \int \tan ^{2}{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(d*x+c))**2*tan(d*x+c)**2,x)
[Out]
a**2*(Integral(2*sin(c + d*x)*tan(c + d*x)**2, x) + Integral(sin(c + d*x)**2*tan(c + d*x)**2, x) + Integral(ta
n(c + d*x)**2, x))
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Giac [B] time = 64.9824, size = 7250, normalized size = 102.11 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(d*x+c))^2*tan(d*x+c)^2,x, algorithm="giac")
[Out]
-1/2*(5*a^2*d*x*tan(d*x)^3*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c)^3 + 5*a^2*d*x*tan(d*x)^3*tan(1/2*d*x)^4*tan(1/2*
c)^4*tan(c) - 5*a^2*d*x*tan(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c)^2 - 20*a^2*d*x*tan(d*x)^3*tan(1/2*d*x)^3
*tan(1/2*c)^3*tan(c)^3 + 5*a^2*d*x*tan(d*x)*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c)^3 - 8*a^2*tan(d*x)^3*tan(1/2*d*
x)^4*tan(1/2*c)^4*tan(c)^3 + 5*a^2*tan(d*x)^3*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c)^2 + 5*a^2*tan(d*x)^2*tan(1/2*
d*x)^4*tan(1/2*c)^4*tan(c)^3 - 5*a^2*d*x*tan(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c)^4 - 20*a^2*d*x*tan(d*x)^3*tan(1/
2*d*x)^3*tan(1/2*c)^3*tan(c) + 5*a^2*d*x*tan(d*x)*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)^4*tan(1/2*c)^4*tan(c) + 20*a^2*d*x*tan(d*x)^2*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c)^2 - 5*a^2*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)^4*tan(1/2*c)^4*tan(c)^2 - 5*a^2*d*x*tan(d*x)^3*ta
n(1/2*d*x)^4*tan(c)^3 - 20*a^2*d*x*tan(d*x)^3*tan(1/2*d*x)^3*tan(1/2*c)*tan(c)^3 - 20*a^2*d*x*tan(d*x)^3*tan(1
/2*d*x)*tan(1/2*c)^3*tan(c)^3 - 20*a^2*d*x*tan(d*x)*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c)^3 + 32*a^2*tan(d*x)^3*t
an(1/2*d*x)^3*tan(1/2*c)^3*tan(c)^3 - 5*a^2*d*x*tan(d*x)^3*tan(1/2*c)^4*tan(c)^3 - 8*a^2*tan(d*x)*tan(1/2*d*x)
^4*tan(1/2*c)^4*tan(c)^3 + 4*a^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) - 20*a^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 - 20*a^2*tan(d*x)^2*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c)^3 + 4*a^2*tan(1/2*d*x)^4*tan(1/2*c)^4*
tan(c)^3 + 20*a^2*d*x*tan(d*x)^2*tan(1/2*d*x)^3*tan(1/2*c)^3 - 5*a^2*d*x*tan(1/2*d*x)^4*tan(1/2*c)^4 + 8*a^2*t
an(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c)^4 - 5*a^2*d*x*tan(d*x)^3*tan(1/2*d*x)^4*tan(c) - 20*a^2*d*x*tan(d*x)^3*tan
(1/2*d*x)^3*tan(1/2*c)*tan(c) - 20*a^2*d*x*tan(d*x)^3*tan(1/2*d*x)*tan(1/2*c)^3*tan(c) - 20*a^2*d*x*tan(d*x)*t
an(1/2*d*x)^3*tan(1/2*c)^3*tan(c) + 32*a^2*tan(d*x)^3*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c) - 5*a^2*d*x*tan(d*x)^
3*tan(1/2*c)^4*tan(c) - 8*a^2*tan(d*x)*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c) + 5*a^2*d*x*tan(d*x)^2*tan(1/2*d*x)^
4*tan(c)^2 + 20*a^2*d*x*tan(d*x)^2*tan(1/2*d*x)^3*tan(1/2*c)*tan(c)^2 + 20*a^2*d*x*tan(d*x)^2*tan(1/2*d*x)*tan
(1/2*c)^3*tan(c)^2 + 20*a^2*d*x*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c)^2 - 32*a^2*tan(d*x)^2*tan(1/2*d*x)^3*tan(1/
2*c)^3*tan(c)^2 + 5*a^2*d*x*tan(d*x)^2*tan(1/2*c)^4*tan(c)^2 + 8*a^2*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c)^2 - 5*
a^2*d*x*tan(d*x)*tan(1/2*d*x)^4*tan(c)^3 - 8*a^2*tan(d*x)^3*tan(1/2*d*x)^4*tan(c)^3 - 20*a^2*d*x*tan(d*x)^3*ta
n(1/2*d*x)*tan(1/2*c)*tan(c)^3 - 20*a^2*d*x*tan(d*x)*tan(1/2*d*x)^3*tan(1/2*c)*tan(c)^3 - 32*a^2*tan(d*x)^3*ta
n(1/2*d*x)^3*tan(1/2*c)*tan(c)^3 - 96*a^2*tan(d*x)^3*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(c)^3 - 20*a^2*d*x*tan(d*x
)*tan(1/2*d*x)*tan(1/2*c)^3*tan(c)^3 - 32*a^2*tan(d*x)^3*tan(1/2*d*x)*tan(1/2*c)^3*tan(c)^3 + 32*a^2*tan(d*x)*
tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c)^3 - 5*a^2*d*x*tan(d*x)*tan(1/2*c)^4*tan(c)^3 - 8*a^2*tan(d*x)^3*tan(1/2*c)^
4*tan(c)^3 - 16*a^2*tan(d*x)^3*tan(1/2*d*x)^3*tan(1/2*c)^3 + 5*a^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) + 5*a^2*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c) - 5*a^2*tan(d*x)^3*
tan(1/2*d*x)^4*tan(c)^2 - 20*a^2*tan(d*x)^3*tan(1/2*d*x)^3*tan(1/2*c)*tan(c)^2 - 20*a^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 - 5*a^2*tan(d*x)^3*tan(1/2*c)^4*
tan(c)^2 - 5*a^2*tan(d*x)^2*tan(1/2*d*x)^4*tan(c)^3 - 20*a^2*tan(d*x)^2*tan(1/2*d*x)^3*tan(1/2*c)*tan(c)^3 - 2
0*a^2*tan(d*x)^2*tan(1/2*d*x)*tan(1/2*c)^3*tan(c)^3 - 16*a^2*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c)^3 - 5*a^2*tan(
d*x)^2*tan(1/2*c)^4*tan(c)^3 + 5*a^2*d*x*tan(d*x)^2*tan(1/2*d*x)^4 + 20*a^2*d*x*tan(d*x)^2*tan(1/2*d*x)^3*tan(
1/2*c) + 20*a^2*d*x*tan(d*x)^2*tan(1/2*d*x)*tan(1/2*c)^3 + 20*a^2*d*x*tan(1/2*d*x)^3*tan(1/2*c)^3 - 32*a^2*tan
(d*x)^2*tan(1/2*d*x)^3*tan(1/2*c)^3 + 5*a^2*d*x*tan(d*x)^2*tan(1/2*c)^4 + 8*a^2*tan(1/2*d*x)^4*tan(1/2*c)^4 -
5*a^2*d*x*tan(d*x)*tan(1/2*d*x)^4*tan(c) - 8*a^2*tan(d*x)^3*tan(1/2*d*x)^4*tan(c) - 20*a^2*d*x*tan(d*x)^3*tan(
1/2*d*x)*tan(1/2*c)*tan(c) - 20*a^2*d*x*tan(d*x)*tan(1/2*d*x)^3*tan(1/2*c)*tan(c) - 32*a^2*tan(d*x)^3*tan(1/2*
d*x)^3*tan(1/2*c)*tan(c) - 96*a^2*tan(d*x)^3*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(c) - 20*a^2*d*x*tan(d*x)*tan(1/2*
d*x)*tan(1/2*c)^3*tan(c) - 32*a^2*tan(d*x)^3*tan(1/2*d*x)*tan(1/2*c)^3*tan(c) + 32*a^2*tan(d*x)*tan(1/2*d*x)^3
*tan(1/2*c)^3*tan(c) - 5*a^2*d*x*tan(d*x)*tan(1/2*c)^4*tan(c) - 8*a^2*tan(d*x)^3*tan(1/2*c)^4*tan(c) + 5*a^2*d
*x*tan(1/2*d*x)^4*tan(c)^2 + 8*a^2*tan(d*x)^2*tan(1/2*d*x)^4*tan(c)^2 + 20*a^2*d*x*tan(d*x)^2*tan(1/2*d*x)*tan
(1/2*c)*tan(c)^2 + 20*a^2*d*x*tan(1/2*d*x)^3*tan(1/2*c)*tan(c)^2 + 32*a^2*tan(d*x)^2*tan(1/2*d*x)^3*tan(1/2*c)
*tan(c)^2 + 96*a^2*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(c)^2 + 20*a^2*d*x*tan(1/2*d*x)*tan(1/2*c)^3*tan(
c)^2 + 32*a^2*tan(d*x)^2*tan(1/2*d*x)*tan(1/2*c)^3*tan(c)^2 - 32*a^2*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c)^2 + 5*
a^2*d*x*tan(1/2*c)^4*tan(c)^2 + 8*a^2*tan(d*x)^2*tan(1/2*c)^4*tan(c)^2 + 5*a^2*d*x*tan(d*x)^3*tan(c)^3 - 8*a^2
*tan(d*x)*tan(1/2*d*x)^4*tan(c)^3 - 20*a^2*d*x*tan(d*x)*tan(1/2*d*x)*tan(1/2*c)*tan(c)^3 + 32*a^2*tan(d*x)^3*t
an(1/2*d*x)*tan(1/2*c)*tan(c)^3 - 32*a^2*tan(d*x)*tan(1/2*d*x)^3*tan(1/2*c)*tan(c)^3 - 96*a^2*tan(d*x)*tan(1/2
*d*x)^2*tan(1/2*c)^2*tan(c)^3 - 32*a^2*tan(d*x)*tan(1/2*d*x)*tan(1/2*c)^3*tan(c)^3 - 8*a^2*tan(d*x)*tan(1/2*c)
^4*tan(c)^3 - 4*a^2*tan(d*x)^3*tan(1/2*d*x)^4 - 16*a^2*tan(d*x)^3*tan(1/2*d*x)^3*tan(1/2*c) - 16*a^2*tan(d*x)^
3*tan(1/2*d*x)*tan(1/2*c)^3 - 20*a^2*tan(d*x)*tan(1/2*d*x)^3*tan(1/2*c)^3 - 4*a^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*ta
n(1/2*d*x)*tan(1/2*c)^3*tan(c) - 20*a^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 - 20*a^2*tan(d*x)^3*tan(1/2*d*x)*tan(1/2*c)*tan(c)^2 - 8*a^2*tan(
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)*t
an(1/2*c)^4*tan(c)^2 - 4*a^2*tan(1/2*d*x)^4*tan(c)^3 - 20*a^2*tan(d*x)^2*tan(1/2*d*x)*tan(1/2*c)*tan(c)^3 - 16
*a^2*tan(1/2*d*x)^3*tan(1/2*c)*tan(c)^3 - 16*a^2*tan(1/2*d*x)*tan(1/2*c)^3*tan(c)^3 - 4*a^2*tan(1/2*c)^4*tan(c
)^3 + 5*a^2*d*x*tan(1/2*d*x)^4 + 8*a^2*tan(d*x)^2*tan(1/2*d*x)^4 + 20*a^2*d*x*tan(d*x)^2*tan(1/2*d*x)*tan(1/2*
c) + 20*a^2*d*x*tan(1/2*d*x)^3*tan(1/2*c) + 32*a^2*tan(d*x)^2*tan(1/2*d*x)^3*tan(1/2*c) + 96*a^2*tan(d*x)^2*ta
n(1/2*d*x)^2*tan(1/2*c)^2 + 20*a^2*d*x*tan(1/2*d*x)*tan(1/2*c)^3 + 32*a^2*tan(d*x)^2*tan(1/2*d*x)*tan(1/2*c)^3
- 32*a^2*tan(1/2*d*x)^3*tan(1/2*c)^3 + 5*a^2*d*x*tan(1/2*c)^4 + 8*a^2*tan(d*x)^2*tan(1/2*c)^4 + 5*a^2*d*x*tan
(d*x)^3*tan(c) - 8*a^2*tan(d*x)*tan(1/2*d*x)^4*tan(c) - 20*a^2*d*x*tan(d*x)*tan(1/2*d*x)*tan(1/2*c)*tan(c) + 3
2*a^2*tan(d*x)^3*tan(1/2*d*x)*tan(1/2*c)*tan(c) - 32*a^2*tan(d*x)*tan(1/2*d*x)^3*tan(1/2*c)*tan(c) - 96*a^2*ta
n(d*x)*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(c) - 32*a^2*tan(d*x)*tan(1/2*d*x)*tan(1/2*c)^3*tan(c) - 8*a^2*tan(d*x)*
tan(1/2*c)^4*tan(c) - 5*a^2*d*x*tan(d*x)^2*tan(c)^2 + 8*a^2*tan(1/2*d*x)^4*tan(c)^2 + 20*a^2*d*x*tan(1/2*d*x)*
tan(1/2*c)*tan(c)^2 - 32*a^2*tan(d*x)^2*tan(1/2*d*x)*tan(1/2*c)*tan(c)^2 + 32*a^2*tan(1/2*d*x)^3*tan(1/2*c)*ta
n(c)^2 + 96*a^2*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(c)^2 + 32*a^2*tan(1/2*d*x)*tan(1/2*c)^3*tan(c)^2 + 8*a^2*tan(1
/2*c)^4*tan(c)^2 + 5*a^2*d*x*tan(d*x)*tan(c)^3 - 8*a^2*tan(d*x)^3*tan(c)^3 + 32*a^2*tan(d*x)*tan(1/2*d*x)*tan(
1/2*c)*tan(c)^3 - 5*a^2*tan(d*x)*tan(1/2*d*x)^4 - 16*a^2*tan(d*x)^3*tan(1/2*d*x)*tan(1/2*c) - 20*a^2*tan(d*x)*
tan(1/2*d*x)^3*tan(1/2*c) - 20*a^2*tan(d*x)*tan(1/2*d*x)*tan(1/2*c)^3 - 5*a^2*tan(d*x)*tan(1/2*c)^4 - 5*a^2*ta
n(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) - 20*a^2*tan(1/2*d*x)^3*tan(1/2*c)*tan(c
) - 20*a^2*tan(1/2*d*x)*tan(1/2*c)^3*tan(c) - 5*a^2*tan(1/2*c)^4*tan(c) + 5*a^2*tan(d*x)^3*tan(c)^2 - 8*a^2*ta
n(d*x)*tan(1/2*d*x)*tan(1/2*c)*tan(c)^2 + 5*a^2*tan(d*x)^2*tan(c)^3 - 16*a^2*tan(1/2*d*x)*tan(1/2*c)*tan(c)^3
- 5*a^2*d*x*tan(d*x)^2 + 8*a^2*tan(1/2*d*x)^4 + 20*a^2*d*x*tan(1/2*d*x)*tan(1/2*c) - 32*a^2*tan(d*x)^2*tan(1/2
*d*x)*tan(1/2*c) + 32*a^2*tan(1/2*d*x)^3*tan(1/2*c) + 96*a^2*tan(1/2*d*x)^2*tan(1/2*c)^2 + 32*a^2*tan(1/2*d*x)
*tan(1/2*c)^3 + 8*a^2*tan(1/2*c)^4 + 5*a^2*d*x*tan(d*x)*tan(c) - 8*a^2*tan(d*x)^3*tan(c) + 32*a^2*tan(d*x)*tan
(1/2*d*x)*tan(1/2*c)*tan(c) - 5*a^2*d*x*tan(c)^2 + 8*a^2*tan(d*x)^2*tan(c)^2 - 32*a^2*tan(1/2*d*x)*tan(1/2*c)*
tan(c)^2 - 8*a^2*tan(d*x)*tan(c)^3 + 4*a^2*tan(d*x)^3 - 20*a^2*tan(d*x)*tan(1/2*d*x)*tan(1/2*c) + 2*a^2*tan(d*
x)^2*tan(c) - 20*a^2*tan(1/2*d*x)*tan(1/2*c)*tan(c) + 2*a^2*tan(d*x)*tan(c)^2 + 4*a^2*tan(c)^3 - 5*a^2*d*x + 8
*a^2*tan(d*x)^2 - 32*a^2*tan(1/2*d*x)*tan(1/2*c) - 8*a^2*tan(d*x)*tan(c) + 8*a^2*tan(c)^2 + 5*a^2*tan(d*x) + 5
*a^2*tan(c) + 8*a^2)/(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*t
an(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*ta
n(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*t
an(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*tan(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*t
an(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*tan(d*x)^3*tan(1/2*d*x)*tan(1/2*c)*tan(c) - 4*d*t
an(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*t
an(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)