3.200 \(\int \csc ^6(c+b x) \sin (a+b x) \, dx\)
Optimal. Leaf size=94 \[ -\frac{3 \cos (a-c) \tanh ^{-1}(\cos (b x+c))}{8 b}-\frac{\sin (a-c) \csc ^5(b x+c)}{5 b}-\frac{\cos (a-c) \cot (b x+c) \csc ^3(b x+c)}{4 b}-\frac{3 \cos (a-c) \cot (b x+c) \csc (b x+c)}{8 b} \]
[Out]
(-3*ArcTanh[Cos[c + b*x]]*Cos[a - c])/(8*b) - (3*Cos[a - c]*Cot[c + b*x]*Csc[c + b*x])/(8*b) - (Cos[a - c]*Cot
[c + b*x]*Csc[c + b*x]^3)/(4*b) - (Csc[c + b*x]^5*Sin[a - c])/(5*b)
________________________________________________________________________________________
Rubi [A] time = 0.0582293, antiderivative size = 94, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used =
{4582, 2606, 30, 3768, 3770} \[ -\frac{3 \cos (a-c) \tanh ^{-1}(\cos (b x+c))}{8 b}-\frac{\sin (a-c) \csc ^5(b x+c)}{5 b}-\frac{\cos (a-c) \cot (b x+c) \csc ^3(b x+c)}{4 b}-\frac{3 \cos (a-c) \cot (b x+c) \csc (b x+c)}{8 b} \]
Antiderivative was successfully verified.
[In]
Int[Csc[c + b*x]^6*Sin[a + b*x],x]
[Out]
(-3*ArcTanh[Cos[c + b*x]]*Cos[a - c])/(8*b) - (3*Cos[a - c]*Cot[c + b*x]*Csc[c + b*x])/(8*b) - (Cos[a - c]*Cot
[c + b*x]*Csc[c + b*x]^3)/(4*b) - (Csc[c + b*x]^5*Sin[a - c])/(5*b)
Rule 4582
Int[Csc[w_]^(n_.)*Sin[v_], x_Symbol] :> Dist[Sin[v - w], Int[Cot[w]*Csc[w]^(n - 1), x], x] + Dist[Cos[v - w],
Int[Csc[w]^(n - 1), x], x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]
Rule 2606
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
Rule 30
Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]
Rule 3768
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]
Rule 3770
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Rubi steps
\begin{align*} \int \csc ^6(c+b x) \sin (a+b x) \, dx &=\cos (a-c) \int \csc ^5(c+b x) \, dx+\sin (a-c) \int \cot (c+b x) \csc ^5(c+b x) \, dx\\ &=-\frac{\cos (a-c) \cot (c+b x) \csc ^3(c+b x)}{4 b}+\frac{1}{4} (3 \cos (a-c)) \int \csc ^3(c+b x) \, dx-\frac{\sin (a-c) \operatorname{Subst}\left (\int x^4 \, dx,x,\csc (c+b x)\right )}{b}\\ &=-\frac{3 \cos (a-c) \cot (c+b x) \csc (c+b x)}{8 b}-\frac{\cos (a-c) \cot (c+b x) \csc ^3(c+b x)}{4 b}-\frac{\csc ^5(c+b x) \sin (a-c)}{5 b}+\frac{1}{8} (3 \cos (a-c)) \int \csc (c+b x) \, dx\\ &=-\frac{3 \tanh ^{-1}(\cos (c+b x)) \cos (a-c)}{8 b}-\frac{3 \cos (a-c) \cot (c+b x) \csc (c+b x)}{8 b}-\frac{\cos (a-c) \cot (c+b x) \csc ^3(c+b x)}{4 b}-\frac{\csc ^5(c+b x) \sin (a-c)}{5 b}\\ \end{align*}
Mathematica [A] time = 1.16511, size = 79, normalized size = 0.84 \[ -\frac{2 \csc ^5(b x+c) (5 \cos (a-c) (14 \sin (2 (b x+c))-3 \sin (4 (b x+c)))+64 \sin (a-c))+480 \cos (a-c) \tanh ^{-1}\left (\cos (c)-\sin (c) \tan \left (\frac{b x}{2}\right )\right )}{640 b} \]
Antiderivative was successfully verified.
[In]
Integrate[Csc[c + b*x]^6*Sin[a + b*x],x]
[Out]
-(480*ArcTanh[Cos[c] - Sin[c]*Tan[(b*x)/2]]*Cos[a - c] + 2*Csc[c + b*x]^5*(64*Sin[a - c] + 5*Cos[a - c]*(14*Si
n[2*(c + b*x)] - 3*Sin[4*(c + b*x)])))/(640*b)
________________________________________________________________________________________
Maple [B] time = 10.075, size = 97954, normalized size = 1042.1 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csc(b*x+c)^6*sin(b*x+a),x)
[Out]
result too large to display
________________________________________________________________________________________
Maxima [B] time = 2.23054, size = 5237, normalized size = 55.71 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+c)^6*sin(b*x+a),x, algorithm="maxima")
[Out]
1/80*(2*(15*cos(9*b*x + 2*a + 8*c) + 15*cos(9*b*x + 10*c) - 70*cos(7*b*x + 2*a + 6*c) - 70*cos(7*b*x + 8*c) -
128*cos(5*b*x + 2*a + 4*c) + 128*cos(5*b*x + 6*c) + 70*cos(3*b*x + 2*a + 2*c) + 70*cos(3*b*x + 4*c) - 15*cos(b
*x + 2*a) - 15*cos(b*x + 2*c))*cos(10*b*x + a + 10*c) - 30*(5*cos(8*b*x + a + 8*c) - 10*cos(6*b*x + a + 6*c) +
10*cos(4*b*x + a + 4*c) - 5*cos(2*b*x + a + 2*c) + cos(a))*cos(9*b*x + 2*a + 8*c) - 30*(5*cos(8*b*x + a + 8*c
) - 10*cos(6*b*x + a + 6*c) + 10*cos(4*b*x + a + 4*c) - 5*cos(2*b*x + a + 2*c) + cos(a))*cos(9*b*x + 10*c) + 1
0*(70*cos(7*b*x + 2*a + 6*c) + 70*cos(7*b*x + 8*c) + 128*cos(5*b*x + 2*a + 4*c) - 128*cos(5*b*x + 6*c) - 70*co
s(3*b*x + 2*a + 2*c) - 70*cos(3*b*x + 4*c) + 15*cos(b*x + 2*a) + 15*cos(b*x + 2*c))*cos(8*b*x + a + 8*c) - 140
*(10*cos(6*b*x + a + 6*c) - 10*cos(4*b*x + a + 4*c) + 5*cos(2*b*x + a + 2*c) - cos(a))*cos(7*b*x + 2*a + 6*c)
- 140*(10*cos(6*b*x + a + 6*c) - 10*cos(4*b*x + a + 4*c) + 5*cos(2*b*x + a + 2*c) - cos(a))*cos(7*b*x + 8*c) -
20*(128*cos(5*b*x + 2*a + 4*c) - 128*cos(5*b*x + 6*c) - 70*cos(3*b*x + 2*a + 2*c) - 70*cos(3*b*x + 4*c) + 15*
cos(b*x + 2*a) + 15*cos(b*x + 2*c))*cos(6*b*x + a + 6*c) + 256*(10*cos(4*b*x + a + 4*c) - 5*cos(2*b*x + a + 2*
c) + cos(a))*cos(5*b*x + 2*a + 4*c) - 256*(10*cos(4*b*x + a + 4*c) - 5*cos(2*b*x + a + 2*c) + cos(a))*cos(5*b*
x + 6*c) - 100*(14*cos(3*b*x + 2*a + 2*c) + 14*cos(3*b*x + 4*c) - 3*cos(b*x + 2*a) - 3*cos(b*x + 2*c))*cos(4*b
*x + a + 4*c) + 140*(5*cos(2*b*x + a + 2*c) - cos(a))*cos(3*b*x + 2*a + 2*c) + 140*(5*cos(2*b*x + a + 2*c) - c
os(a))*cos(3*b*x + 4*c) - 150*(cos(b*x + 2*a) + cos(b*x + 2*c))*cos(2*b*x + a + 2*c) + 30*cos(b*x + 2*a)*cos(a
) + 30*cos(b*x + 2*c)*cos(a) - 15*(cos(10*b*x + a + 10*c)^2*cos(-a + c) + 25*cos(8*b*x + a + 8*c)^2*cos(-a + c
) + 100*cos(6*b*x + a + 6*c)^2*cos(-a + c) + 100*cos(4*b*x + a + 4*c)^2*cos(-a + c) + 25*cos(2*b*x + a + 2*c)^
2*cos(-a + c) - 10*cos(2*b*x + a + 2*c)*cos(a)*cos(-a + c) + cos(-a + c)*sin(10*b*x + a + 10*c)^2 + 25*cos(-a
+ c)*sin(8*b*x + a + 8*c)^2 + 100*cos(-a + c)*sin(6*b*x + a + 6*c)^2 + 100*cos(-a + c)*sin(4*b*x + a + 4*c)^2
+ 25*cos(-a + c)*sin(2*b*x + a + 2*c)^2 - 10*cos(-a + c)*sin(2*b*x + a + 2*c)*sin(a) - 2*(5*cos(8*b*x + a + 8*
c)*cos(-a + c) - 10*cos(6*b*x + a + 6*c)*cos(-a + c) + 10*cos(4*b*x + a + 4*c)*cos(-a + c) - 5*cos(2*b*x + a +
2*c)*cos(-a + c) + cos(a)*cos(-a + c))*cos(10*b*x + a + 10*c) - 10*(10*cos(6*b*x + a + 6*c)*cos(-a + c) - 10*
cos(4*b*x + a + 4*c)*cos(-a + c) + 5*cos(2*b*x + a + 2*c)*cos(-a + c) - cos(a)*cos(-a + c))*cos(8*b*x + a + 8*
c) - 20*(10*cos(4*b*x + a + 4*c)*cos(-a + c) - 5*cos(2*b*x + a + 2*c)*cos(-a + c) + cos(a)*cos(-a + c))*cos(6*
b*x + a + 6*c) - 20*(5*cos(2*b*x + a + 2*c)*cos(-a + c) - cos(a)*cos(-a + c))*cos(4*b*x + a + 4*c) + (cos(a)^2
+ sin(a)^2)*cos(-a + c) - 2*(5*cos(-a + c)*sin(8*b*x + a + 8*c) - 10*cos(-a + c)*sin(6*b*x + a + 6*c) + 10*co
s(-a + c)*sin(4*b*x + a + 4*c) - 5*cos(-a + c)*sin(2*b*x + a + 2*c) + cos(-a + c)*sin(a))*sin(10*b*x + a + 10*
c) - 10*(10*cos(-a + c)*sin(6*b*x + a + 6*c) - 10*cos(-a + c)*sin(4*b*x + a + 4*c) + 5*cos(-a + c)*sin(2*b*x +
a + 2*c) - cos(-a + c)*sin(a))*sin(8*b*x + a + 8*c) - 20*(10*cos(-a + c)*sin(4*b*x + a + 4*c) - 5*cos(-a + c)
*sin(2*b*x + a + 2*c) + cos(-a + c)*sin(a))*sin(6*b*x + a + 6*c) - 20*(5*cos(-a + c)*sin(2*b*x + a + 2*c) - co
s(-a + c)*sin(a))*sin(4*b*x + a + 4*c))*log(cos(b*x)^2 + 2*cos(b*x)*cos(c) + cos(c)^2 + sin(b*x)^2 - 2*sin(b*x
)*sin(c) + sin(c)^2) + 15*(cos(10*b*x + a + 10*c)^2*cos(-a + c) + 25*cos(8*b*x + a + 8*c)^2*cos(-a + c) + 100*
cos(6*b*x + a + 6*c)^2*cos(-a + c) + 100*cos(4*b*x + a + 4*c)^2*cos(-a + c) + 25*cos(2*b*x + a + 2*c)^2*cos(-a
+ c) - 10*cos(2*b*x + a + 2*c)*cos(a)*cos(-a + c) + cos(-a + c)*sin(10*b*x + a + 10*c)^2 + 25*cos(-a + c)*sin
(8*b*x + a + 8*c)^2 + 100*cos(-a + c)*sin(6*b*x + a + 6*c)^2 + 100*cos(-a + c)*sin(4*b*x + a + 4*c)^2 + 25*cos
(-a + c)*sin(2*b*x + a + 2*c)^2 - 10*cos(-a + c)*sin(2*b*x + a + 2*c)*sin(a) - 2*(5*cos(8*b*x + a + 8*c)*cos(-
a + c) - 10*cos(6*b*x + a + 6*c)*cos(-a + c) + 10*cos(4*b*x + a + 4*c)*cos(-a + c) - 5*cos(2*b*x + a + 2*c)*co
s(-a + c) + cos(a)*cos(-a + c))*cos(10*b*x + a + 10*c) - 10*(10*cos(6*b*x + a + 6*c)*cos(-a + c) - 10*cos(4*b*
x + a + 4*c)*cos(-a + c) + 5*cos(2*b*x + a + 2*c)*cos(-a + c) - cos(a)*cos(-a + c))*cos(8*b*x + a + 8*c) - 20*
(10*cos(4*b*x + a + 4*c)*cos(-a + c) - 5*cos(2*b*x + a + 2*c)*cos(-a + c) + cos(a)*cos(-a + c))*cos(6*b*x + a
+ 6*c) - 20*(5*cos(2*b*x + a + 2*c)*cos(-a + c) - cos(a)*cos(-a + c))*cos(4*b*x + a + 4*c) + (cos(a)^2 + sin(a
)^2)*cos(-a + c) - 2*(5*cos(-a + c)*sin(8*b*x + a + 8*c) - 10*cos(-a + c)*sin(6*b*x + a + 6*c) + 10*cos(-a + c
)*sin(4*b*x + a + 4*c) - 5*cos(-a + c)*sin(2*b*x + a + 2*c) + cos(-a + c)*sin(a))*sin(10*b*x + a + 10*c) - 10*
(10*cos(-a + c)*sin(6*b*x + a + 6*c) - 10*cos(-a + c)*sin(4*b*x + a + 4*c) + 5*cos(-a + c)*sin(2*b*x + a + 2*c
) - cos(-a + c)*sin(a))*sin(8*b*x + a + 8*c) - 20*(10*cos(-a + c)*sin(4*b*x + a + 4*c) - 5*cos(-a + c)*sin(2*b
*x + a + 2*c) + cos(-a + c)*sin(a))*sin(6*b*x + a + 6*c) - 20*(5*cos(-a + c)*sin(2*b*x + a + 2*c) - cos(-a + c
)*sin(a))*sin(4*b*x + a + 4*c))*log(cos(b*x)^2 - 2*cos(b*x)*cos(c) + cos(c)^2 + sin(b*x)^2 + 2*sin(b*x)*sin(c)
+ sin(c)^2) + 2*(15*sin(9*b*x + 2*a + 8*c) + 15*sin(9*b*x + 10*c) - 70*sin(7*b*x + 2*a + 6*c) - 70*sin(7*b*x
+ 8*c) - 128*sin(5*b*x + 2*a + 4*c) + 128*sin(5*b*x + 6*c) + 70*sin(3*b*x + 2*a + 2*c) + 70*sin(3*b*x + 4*c) -
15*sin(b*x + 2*a) - 15*sin(b*x + 2*c))*sin(10*b*x + a + 10*c) - 30*(5*sin(8*b*x + a + 8*c) - 10*sin(6*b*x + a
+ 6*c) + 10*sin(4*b*x + a + 4*c) - 5*sin(2*b*x + a + 2*c) + sin(a))*sin(9*b*x + 2*a + 8*c) - 30*(5*sin(8*b*x
+ a + 8*c) - 10*sin(6*b*x + a + 6*c) + 10*sin(4*b*x + a + 4*c) - 5*sin(2*b*x + a + 2*c) + sin(a))*sin(9*b*x +
10*c) + 10*(70*sin(7*b*x + 2*a + 6*c) + 70*sin(7*b*x + 8*c) + 128*sin(5*b*x + 2*a + 4*c) - 128*sin(5*b*x + 6*c
) - 70*sin(3*b*x + 2*a + 2*c) - 70*sin(3*b*x + 4*c) + 15*sin(b*x + 2*a) + 15*sin(b*x + 2*c))*sin(8*b*x + a + 8
*c) - 140*(10*sin(6*b*x + a + 6*c) - 10*sin(4*b*x + a + 4*c) + 5*sin(2*b*x + a + 2*c) - sin(a))*sin(7*b*x + 2*
a + 6*c) - 140*(10*sin(6*b*x + a + 6*c) - 10*sin(4*b*x + a + 4*c) + 5*sin(2*b*x + a + 2*c) - sin(a))*sin(7*b*x
+ 8*c) - 20*(128*sin(5*b*x + 2*a + 4*c) - 128*sin(5*b*x + 6*c) - 70*sin(3*b*x + 2*a + 2*c) - 70*sin(3*b*x + 4
*c) + 15*sin(b*x + 2*a) + 15*sin(b*x + 2*c))*sin(6*b*x + a + 6*c) + 256*(10*sin(4*b*x + a + 4*c) - 5*sin(2*b*x
+ a + 2*c) + sin(a))*sin(5*b*x + 2*a + 4*c) - 256*(10*sin(4*b*x + a + 4*c) - 5*sin(2*b*x + a + 2*c) + sin(a))
*sin(5*b*x + 6*c) - 100*(14*sin(3*b*x + 2*a + 2*c) + 14*sin(3*b*x + 4*c) - 3*sin(b*x + 2*a) - 3*sin(b*x + 2*c)
)*sin(4*b*x + a + 4*c) + 140*(5*sin(2*b*x + a + 2*c) - sin(a))*sin(3*b*x + 2*a + 2*c) + 140*(5*sin(2*b*x + a +
2*c) - sin(a))*sin(3*b*x + 4*c) - 150*(sin(b*x + 2*a) + sin(b*x + 2*c))*sin(2*b*x + a + 2*c) + 30*sin(b*x + 2
*a)*sin(a) + 30*sin(b*x + 2*c)*sin(a))/(b*cos(10*b*x + a + 10*c)^2 + 25*b*cos(8*b*x + a + 8*c)^2 + 100*b*cos(6
*b*x + a + 6*c)^2 + 100*b*cos(4*b*x + a + 4*c)^2 + 25*b*cos(2*b*x + a + 2*c)^2 - 10*b*cos(2*b*x + a + 2*c)*cos
(a) + b*sin(10*b*x + a + 10*c)^2 + 25*b*sin(8*b*x + a + 8*c)^2 + 100*b*sin(6*b*x + a + 6*c)^2 + 100*b*sin(4*b*
x + a + 4*c)^2 + 25*b*sin(2*b*x + a + 2*c)^2 - 10*b*sin(2*b*x + a + 2*c)*sin(a) + (cos(a)^2 + sin(a)^2)*b - 2*
(5*b*cos(8*b*x + a + 8*c) - 10*b*cos(6*b*x + a + 6*c) + 10*b*cos(4*b*x + a + 4*c) - 5*b*cos(2*b*x + a + 2*c) +
b*cos(a))*cos(10*b*x + a + 10*c) - 10*(10*b*cos(6*b*x + a + 6*c) - 10*b*cos(4*b*x + a + 4*c) + 5*b*cos(2*b*x
+ a + 2*c) - b*cos(a))*cos(8*b*x + a + 8*c) - 20*(10*b*cos(4*b*x + a + 4*c) - 5*b*cos(2*b*x + a + 2*c) + b*cos
(a))*cos(6*b*x + a + 6*c) - 20*(5*b*cos(2*b*x + a + 2*c) - b*cos(a))*cos(4*b*x + a + 4*c) - 2*(5*b*sin(8*b*x +
a + 8*c) - 10*b*sin(6*b*x + a + 6*c) + 10*b*sin(4*b*x + a + 4*c) - 5*b*sin(2*b*x + a + 2*c) + b*sin(a))*sin(1
0*b*x + a + 10*c) - 10*(10*b*sin(6*b*x + a + 6*c) - 10*b*sin(4*b*x + a + 4*c) + 5*b*sin(2*b*x + a + 2*c) - b*s
in(a))*sin(8*b*x + a + 8*c) - 20*(10*b*sin(4*b*x + a + 4*c) - 5*b*sin(2*b*x + a + 2*c) + b*sin(a))*sin(6*b*x +
a + 6*c) - 20*(5*b*sin(2*b*x + a + 2*c) - b*sin(a))*sin(4*b*x + a + 4*c))
________________________________________________________________________________________
Fricas [B] time = 0.532326, size = 543, normalized size = 5.78 \begin{align*} -\frac{15 \,{\left (\cos \left (b x + c\right )^{4} \cos \left (-a + c\right ) - 2 \, \cos \left (b x + c\right )^{2} \cos \left (-a + c\right ) + \cos \left (-a + c\right )\right )} \log \left (\frac{1}{2} \, \cos \left (b x + c\right ) + \frac{1}{2}\right ) \sin \left (b x + c\right ) - 15 \,{\left (\cos \left (b x + c\right )^{4} \cos \left (-a + c\right ) - 2 \, \cos \left (b x + c\right )^{2} \cos \left (-a + c\right ) + \cos \left (-a + c\right )\right )} \log \left (-\frac{1}{2} \, \cos \left (b x + c\right ) + \frac{1}{2}\right ) \sin \left (b x + c\right ) - 10 \,{\left (3 \, \cos \left (b x + c\right )^{3} \cos \left (-a + c\right ) - 5 \, \cos \left (b x + c\right ) \cos \left (-a + c\right )\right )} \sin \left (b x + c\right ) - 16 \, \sin \left (-a + c\right )}{80 \,{\left (b \cos \left (b x + c\right )^{4} - 2 \, b \cos \left (b x + c\right )^{2} + b\right )} \sin \left (b x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+c)^6*sin(b*x+a),x, algorithm="fricas")
[Out]
-1/80*(15*(cos(b*x + c)^4*cos(-a + c) - 2*cos(b*x + c)^2*cos(-a + c) + cos(-a + c))*log(1/2*cos(b*x + c) + 1/2
)*sin(b*x + c) - 15*(cos(b*x + c)^4*cos(-a + c) - 2*cos(b*x + c)^2*cos(-a + c) + cos(-a + c))*log(-1/2*cos(b*x
+ c) + 1/2)*sin(b*x + c) - 10*(3*cos(b*x + c)^3*cos(-a + c) - 5*cos(b*x + c)*cos(-a + c))*sin(b*x + c) - 16*s
in(-a + c))/((b*cos(b*x + c)^4 - 2*b*cos(b*x + c)^2 + b)*sin(b*x + c))
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+c)**6*sin(b*x+a),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.31999, size = 10847, normalized size = 115.39 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+c)^6*sin(b*x+a),x, algorithm="giac")
[Out]
1/320*(120*(tan(1/2*a)^2*tan(1/2*c)^2 - tan(1/2*a)^2 + 4*tan(1/2*a)*tan(1/2*c) - tan(1/2*c)^2 + 1)*log(abs(tan
(1/2*b*x + 1/2*c)))/(tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*a)^2 + tan(1/2*c)^2 + 1) - (4*tan(1/2*b*x + 1/2*c)^5*
tan(1/2*a)^10*tan(1/2*c)^9 - 4*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^9*tan(1/2*c)^10 - 5*tan(1/2*b*x + 1/2*c)^4*ta
n(1/2*a)^10*tan(1/2*c)^10 + 16*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^10*tan(1/2*c)^7 - 12*tan(1/2*b*x + 1/2*c)^5*t
an(1/2*a)^9*tan(1/2*c)^8 - 15*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^10*tan(1/2*c)^8 + 12*tan(1/2*b*x + 1/2*c)^5*ta
n(1/2*a)^8*tan(1/2*c)^9 - 20*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^9*tan(1/2*c)^9 + 20*tan(1/2*b*x + 1/2*c)^3*tan(
1/2*a)^10*tan(1/2*c)^9 - 16*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^7*tan(1/2*c)^10 - 15*tan(1/2*b*x + 1/2*c)^4*tan(
1/2*a)^8*tan(1/2*c)^10 - 20*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^9*tan(1/2*c)^10 - 40*tan(1/2*b*x + 1/2*c)^2*tan(
1/2*a)^10*tan(1/2*c)^10 + 24*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^10*tan(1/2*c)^5 - 8*tan(1/2*b*x + 1/2*c)^5*tan(
1/2*a)^9*tan(1/2*c)^6 - 10*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^10*tan(1/2*c)^6 + 48*tan(1/2*b*x + 1/2*c)^5*tan(1
/2*a)^8*tan(1/2*c)^7 - 80*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^9*tan(1/2*c)^7 + 80*tan(1/2*b*x + 1/2*c)^3*tan(1/2
*a)^10*tan(1/2*c)^7 - 48*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^7*tan(1/2*c)^8 - 45*tan(1/2*b*x + 1/2*c)^4*tan(1/2*
a)^8*tan(1/2*c)^8 - 60*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^9*tan(1/2*c)^8 - 120*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a
)^10*tan(1/2*c)^8 + 8*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^6*tan(1/2*c)^9 - 80*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^
7*tan(1/2*c)^9 + 60*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^8*tan(1/2*c)^9 - 160*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^9
*tan(1/2*c)^9 + 40*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^10*tan(1/2*c)^9 - 24*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^5*ta
n(1/2*c)^10 - 10*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^6*tan(1/2*c)^10 - 80*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^7*ta
n(1/2*c)^10 - 120*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^8*tan(1/2*c)^10 - 40*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^9*tan
(1/2*c)^10 + 16*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^10*tan(1/2*c)^3 + 8*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^9*tan(
1/2*c)^4 + 10*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^10*tan(1/2*c)^4 + 72*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^8*tan(1
/2*c)^5 - 120*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^9*tan(1/2*c)^5 + 120*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^10*tan(
1/2*c)^5 - 32*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^7*tan(1/2*c)^6 - 30*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^8*tan(1/
2*c)^6 - 40*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^9*tan(1/2*c)^6 - 80*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^10*tan(1/2
*c)^6 + 32*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^6*tan(1/2*c)^7 - 320*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^7*tan(1/2*
c)^7 + 240*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^8*tan(1/2*c)^7 - 640*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^9*tan(1/2*
c)^7 + 160*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^10*tan(1/2*c)^7 - 72*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^5*tan(1/2*c)
^8 - 30*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^6*tan(1/2*c)^8 - 240*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^7*tan(1/2*c)^
8 - 360*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^8*tan(1/2*c)^8 - 120*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^9*tan(1/2*c)^8
- 8*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^4*tan(1/2*c)^9 - 120*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^5*tan(1/2*c)^9 +
40*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^6*tan(1/2*c)^9 - 640*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^7*tan(1/2*c)^9 + 1
20*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^8*tan(1/2*c)^9 - 16*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^3*tan(1/2*c)^10 + 10*
tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^4*tan(1/2*c)^10 - 120*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^5*tan(1/2*c)^10 - 80
*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^6*tan(1/2*c)^10 - 160*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^7*tan(1/2*c)^10 + 4*t
an(1/2*b*x + 1/2*c)^5*tan(1/2*a)^10*tan(1/2*c) + 12*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^9*tan(1/2*c)^2 + 15*tan(
1/2*b*x + 1/2*c)^4*tan(1/2*a)^10*tan(1/2*c)^2 + 48*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^8*tan(1/2*c)^3 - 80*tan(1
/2*b*x + 1/2*c)^4*tan(1/2*a)^9*tan(1/2*c)^3 + 80*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^10*tan(1/2*c)^3 + 32*tan(1/
2*b*x + 1/2*c)^5*tan(1/2*a)^7*tan(1/2*c)^4 + 30*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^8*tan(1/2*c)^4 + 40*tan(1/2*
b*x + 1/2*c)^3*tan(1/2*a)^9*tan(1/2*c)^4 + 80*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^10*tan(1/2*c)^4 + 48*tan(1/2*b
*x + 1/2*c)^5*tan(1/2*a)^6*tan(1/2*c)^5 - 480*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^7*tan(1/2*c)^5 + 360*tan(1/2*b
*x + 1/2*c)^3*tan(1/2*a)^8*tan(1/2*c)^5 - 960*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^9*tan(1/2*c)^5 + 240*tan(1/2*b
*x + 1/2*c)*tan(1/2*a)^10*tan(1/2*c)^5 - 48*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^5*tan(1/2*c)^6 - 20*tan(1/2*b*x
+ 1/2*c)^4*tan(1/2*a)^6*tan(1/2*c)^6 - 160*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^7*tan(1/2*c)^6 - 240*tan(1/2*b*x
+ 1/2*c)^2*tan(1/2*a)^8*tan(1/2*c)^6 - 80*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^9*tan(1/2*c)^6 - 32*tan(1/2*b*x + 1/
2*c)^5*tan(1/2*a)^4*tan(1/2*c)^7 - 480*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^5*tan(1/2*c)^7 + 160*tan(1/2*b*x + 1/
2*c)^3*tan(1/2*a)^6*tan(1/2*c)^7 - 2560*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^7*tan(1/2*c)^7 + 480*tan(1/2*b*x + 1
/2*c)*tan(1/2*a)^8*tan(1/2*c)^7 - 48*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^3*tan(1/2*c)^8 + 30*tan(1/2*b*x + 1/2*c
)^4*tan(1/2*a)^4*tan(1/2*c)^8 - 360*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^5*tan(1/2*c)^8 - 240*tan(1/2*b*x + 1/2*c
)^2*tan(1/2*a)^6*tan(1/2*c)^8 - 480*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^7*tan(1/2*c)^8 - 12*tan(1/2*b*x + 1/2*c)^5
*tan(1/2*a)^2*tan(1/2*c)^9 - 80*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^3*tan(1/2*c)^9 - 40*tan(1/2*b*x + 1/2*c)^3*t
an(1/2*a)^4*tan(1/2*c)^9 - 960*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^5*tan(1/2*c)^9 + 80*tan(1/2*b*x + 1/2*c)*tan(
1/2*a)^6*tan(1/2*c)^9 - 4*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)*tan(1/2*c)^10 + 15*tan(1/2*b*x + 1/2*c)^4*tan(1/2*
a)^2*tan(1/2*c)^10 - 80*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^3*tan(1/2*c)^10 + 80*tan(1/2*b*x + 1/2*c)^2*tan(1/2*
a)^4*tan(1/2*c)^10 - 240*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^5*tan(1/2*c)^10 + 4*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)
^9 + 5*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^10 + 12*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^8*tan(1/2*c) - 20*tan(1/2*b
*x + 1/2*c)^4*tan(1/2*a)^9*tan(1/2*c) + 20*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^10*tan(1/2*c) + 48*tan(1/2*b*x +
1/2*c)^5*tan(1/2*a)^7*tan(1/2*c)^2 + 45*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^8*tan(1/2*c)^2 + 60*tan(1/2*b*x + 1/
2*c)^3*tan(1/2*a)^9*tan(1/2*c)^2 + 120*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^10*tan(1/2*c)^2 + 32*tan(1/2*b*x + 1/
2*c)^5*tan(1/2*a)^6*tan(1/2*c)^3 - 320*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^7*tan(1/2*c)^3 + 240*tan(1/2*b*x + 1/
2*c)^3*tan(1/2*a)^8*tan(1/2*c)^3 - 640*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^9*tan(1/2*c)^3 + 160*tan(1/2*b*x + 1/
2*c)*tan(1/2*a)^10*tan(1/2*c)^3 + 48*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^5*tan(1/2*c)^4 + 20*tan(1/2*b*x + 1/2*c
)^4*tan(1/2*a)^6*tan(1/2*c)^4 + 160*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^7*tan(1/2*c)^4 + 240*tan(1/2*b*x + 1/2*c
)^2*tan(1/2*a)^8*tan(1/2*c)^4 + 80*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^9*tan(1/2*c)^4 - 48*tan(1/2*b*x + 1/2*c)^5*
tan(1/2*a)^4*tan(1/2*c)^5 - 720*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^5*tan(1/2*c)^5 + 240*tan(1/2*b*x + 1/2*c)^3*
tan(1/2*a)^6*tan(1/2*c)^5 - 3840*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^7*tan(1/2*c)^5 + 720*tan(1/2*b*x + 1/2*c)*t
an(1/2*a)^8*tan(1/2*c)^5 - 32*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^3*tan(1/2*c)^6 + 20*tan(1/2*b*x + 1/2*c)^4*tan
(1/2*a)^4*tan(1/2*c)^6 - 240*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^5*tan(1/2*c)^6 - 160*tan(1/2*b*x + 1/2*c)^2*tan
(1/2*a)^6*tan(1/2*c)^6 - 320*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^7*tan(1/2*c)^6 - 48*tan(1/2*b*x + 1/2*c)^5*tan(1/
2*a)^2*tan(1/2*c)^7 - 320*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^3*tan(1/2*c)^7 - 160*tan(1/2*b*x + 1/2*c)^3*tan(1/
2*a)^4*tan(1/2*c)^7 - 3840*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^5*tan(1/2*c)^7 + 320*tan(1/2*b*x + 1/2*c)*tan(1/2
*a)^6*tan(1/2*c)^7 - 12*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)*tan(1/2*c)^8 + 45*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^
2*tan(1/2*c)^8 - 240*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^3*tan(1/2*c)^8 + 240*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^
4*tan(1/2*c)^8 - 720*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^5*tan(1/2*c)^8 - 4*tan(1/2*b*x + 1/2*c)^5*tan(1/2*c)^9 -
20*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)*tan(1/2*c)^9 - 60*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^2*tan(1/2*c)^9 - 640*
tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^3*tan(1/2*c)^9 - 80*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^4*tan(1/2*c)^9 + 5*tan(1
/2*b*x + 1/2*c)^4*tan(1/2*c)^10 - 20*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)*tan(1/2*c)^10 + 120*tan(1/2*b*x + 1/2*c
)^2*tan(1/2*a)^2*tan(1/2*c)^10 - 160*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^3*tan(1/2*c)^10 + 16*tan(1/2*b*x + 1/2*c)
^5*tan(1/2*a)^7 + 15*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^8 + 20*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^9 + 40*tan(1/2
*b*x + 1/2*c)^2*tan(1/2*a)^10 + 8*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^6*tan(1/2*c) - 80*tan(1/2*b*x + 1/2*c)^4*t
an(1/2*a)^7*tan(1/2*c) + 60*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^8*tan(1/2*c) - 160*tan(1/2*b*x + 1/2*c)^2*tan(1/
2*a)^9*tan(1/2*c) + 40*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^10*tan(1/2*c) + 72*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^5*
tan(1/2*c)^2 + 30*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^6*tan(1/2*c)^2 + 240*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^7*t
an(1/2*c)^2 + 360*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^8*tan(1/2*c)^2 + 120*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^9*tan
(1/2*c)^2 - 32*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^4*tan(1/2*c)^3 - 480*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^5*tan(
1/2*c)^3 + 160*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^6*tan(1/2*c)^3 - 2560*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^7*tan
(1/2*c)^3 + 480*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^8*tan(1/2*c)^3 + 32*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^3*tan(1/
2*c)^4 - 20*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^4*tan(1/2*c)^4 + 240*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^5*tan(1/2
*c)^4 + 160*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^6*tan(1/2*c)^4 + 320*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^7*tan(1/2*c
)^4 - 72*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^2*tan(1/2*c)^5 - 480*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^3*tan(1/2*c)
^5 - 240*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^4*tan(1/2*c)^5 - 5760*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^5*tan(1/2*c
)^5 + 480*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^6*tan(1/2*c)^5 - 8*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)*tan(1/2*c)^6 +
30*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^2*tan(1/2*c)^6 - 160*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^3*tan(1/2*c)^6 + 1
60*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^4*tan(1/2*c)^6 - 480*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^5*tan(1/2*c)^6 - 16*
tan(1/2*b*x + 1/2*c)^5*tan(1/2*c)^7 - 80*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)*tan(1/2*c)^7 - 240*tan(1/2*b*x + 1/
2*c)^3*tan(1/2*a)^2*tan(1/2*c)^7 - 2560*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^3*tan(1/2*c)^7 - 320*tan(1/2*b*x + 1
/2*c)*tan(1/2*a)^4*tan(1/2*c)^7 + 15*tan(1/2*b*x + 1/2*c)^4*tan(1/2*c)^8 - 60*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a
)*tan(1/2*c)^8 + 360*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^2*tan(1/2*c)^8 - 480*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^3*
tan(1/2*c)^8 - 20*tan(1/2*b*x + 1/2*c)^3*tan(1/2*c)^9 - 160*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)*tan(1/2*c)^9 - 1
20*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^2*tan(1/2*c)^9 + 40*tan(1/2*b*x + 1/2*c)^2*tan(1/2*c)^10 - 40*tan(1/2*b*x +
1/2*c)*tan(1/2*a)*tan(1/2*c)^10 + 24*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^5 + 10*tan(1/2*b*x + 1/2*c)^4*tan(1/2*
a)^6 + 80*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^7 + 120*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^8 + 40*tan(1/2*b*x + 1/2
*c)*tan(1/2*a)^9 - 8*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^4*tan(1/2*c) - 120*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^5*
tan(1/2*c) + 40*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^6*tan(1/2*c) - 640*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^7*tan(1
/2*c) + 120*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^8*tan(1/2*c) + 48*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^3*tan(1/2*c)^2
- 30*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^4*tan(1/2*c)^2 + 360*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^5*tan(1/2*c)^2
+ 240*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^6*tan(1/2*c)^2 + 480*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^7*tan(1/2*c)^2 -
48*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^2*tan(1/2*c)^3 - 320*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^3*tan(1/2*c)^3 - 1
60*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^4*tan(1/2*c)^3 - 3840*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^5*tan(1/2*c)^3 +
320*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^6*tan(1/2*c)^3 + 8*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)*tan(1/2*c)^4 - 30*tan
(1/2*b*x + 1/2*c)^4*tan(1/2*a)^2*tan(1/2*c)^4 + 160*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^3*tan(1/2*c)^4 - 160*tan
(1/2*b*x + 1/2*c)^2*tan(1/2*a)^4*tan(1/2*c)^4 + 480*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^5*tan(1/2*c)^4 - 24*tan(1/
2*b*x + 1/2*c)^5*tan(1/2*c)^5 - 120*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)*tan(1/2*c)^5 - 360*tan(1/2*b*x + 1/2*c)^
3*tan(1/2*a)^2*tan(1/2*c)^5 - 3840*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^3*tan(1/2*c)^5 - 480*tan(1/2*b*x + 1/2*c)
*tan(1/2*a)^4*tan(1/2*c)^5 + 10*tan(1/2*b*x + 1/2*c)^4*tan(1/2*c)^6 - 40*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)*tan
(1/2*c)^6 + 240*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^2*tan(1/2*c)^6 - 320*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^3*tan(1
/2*c)^6 - 80*tan(1/2*b*x + 1/2*c)^3*tan(1/2*c)^7 - 640*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)*tan(1/2*c)^7 - 480*ta
n(1/2*b*x + 1/2*c)*tan(1/2*a)^2*tan(1/2*c)^7 + 120*tan(1/2*b*x + 1/2*c)^2*tan(1/2*c)^8 - 120*tan(1/2*b*x + 1/2
*c)*tan(1/2*a)*tan(1/2*c)^8 - 40*tan(1/2*b*x + 1/2*c)*tan(1/2*c)^9 + 16*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^3 -
10*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^4 + 120*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^5 + 80*tan(1/2*b*x + 1/2*c)^2*t
an(1/2*a)^6 + 160*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^7 - 12*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^2*tan(1/2*c) - 80*t
an(1/2*b*x + 1/2*c)^4*tan(1/2*a)^3*tan(1/2*c) - 40*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^4*tan(1/2*c) - 960*tan(1/
2*b*x + 1/2*c)^2*tan(1/2*a)^5*tan(1/2*c) + 80*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^6*tan(1/2*c) + 12*tan(1/2*b*x +
1/2*c)^5*tan(1/2*a)*tan(1/2*c)^2 - 45*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^2*tan(1/2*c)^2 + 240*tan(1/2*b*x + 1/2
*c)^3*tan(1/2*a)^3*tan(1/2*c)^2 - 240*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^4*tan(1/2*c)^2 + 720*tan(1/2*b*x + 1/2
*c)*tan(1/2*a)^5*tan(1/2*c)^2 - 16*tan(1/2*b*x + 1/2*c)^5*tan(1/2*c)^3 - 80*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)*
tan(1/2*c)^3 - 240*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^2*tan(1/2*c)^3 - 2560*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^3
*tan(1/2*c)^3 - 320*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^4*tan(1/2*c)^3 - 10*tan(1/2*b*x + 1/2*c)^4*tan(1/2*c)^4 +
40*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)*tan(1/2*c)^4 - 240*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^2*tan(1/2*c)^4 + 320
*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^3*tan(1/2*c)^4 - 120*tan(1/2*b*x + 1/2*c)^3*tan(1/2*c)^5 - 960*tan(1/2*b*x +
1/2*c)^2*tan(1/2*a)*tan(1/2*c)^5 - 720*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^2*tan(1/2*c)^5 + 80*tan(1/2*b*x + 1/2*c
)^2*tan(1/2*c)^6 - 80*tan(1/2*b*x + 1/2*c)*tan(1/2*a)*tan(1/2*c)^6 - 160*tan(1/2*b*x + 1/2*c)*tan(1/2*c)^7 + 4
*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a) - 15*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^2 + 80*tan(1/2*b*x + 1/2*c)^3*tan(1/
2*a)^3 - 80*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^4 + 240*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^5 - 4*tan(1/2*b*x + 1/2*
c)^5*tan(1/2*c) - 20*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)*tan(1/2*c) - 60*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^2*tan
(1/2*c) - 640*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^3*tan(1/2*c) - 80*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^4*tan(1/2*c)
- 15*tan(1/2*b*x + 1/2*c)^4*tan(1/2*c)^2 + 60*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)*tan(1/2*c)^2 - 360*tan(1/2*b*
x + 1/2*c)^2*tan(1/2*a)^2*tan(1/2*c)^2 + 480*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^3*tan(1/2*c)^2 - 80*tan(1/2*b*x +
1/2*c)^3*tan(1/2*c)^3 - 640*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)*tan(1/2*c)^3 - 480*tan(1/2*b*x + 1/2*c)*tan(1/2
*a)^2*tan(1/2*c)^3 - 80*tan(1/2*b*x + 1/2*c)^2*tan(1/2*c)^4 + 80*tan(1/2*b*x + 1/2*c)*tan(1/2*a)*tan(1/2*c)^4
- 240*tan(1/2*b*x + 1/2*c)*tan(1/2*c)^5 - 5*tan(1/2*b*x + 1/2*c)^4 + 20*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a) - 12
0*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^2 + 160*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^3 - 20*tan(1/2*b*x + 1/2*c)^3*tan(
1/2*c) - 160*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)*tan(1/2*c) - 120*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^2*tan(1/2*c) -
120*tan(1/2*b*x + 1/2*c)^2*tan(1/2*c)^2 + 120*tan(1/2*b*x + 1/2*c)*tan(1/2*a)*tan(1/2*c)^2 - 160*tan(1/2*b*x
+ 1/2*c)*tan(1/2*c)^3 - 40*tan(1/2*b*x + 1/2*c)^2 + 40*tan(1/2*b*x + 1/2*c)*tan(1/2*a) - 40*tan(1/2*b*x + 1/2*
c)*tan(1/2*c))/(tan(1/2*a)^10*tan(1/2*c)^10 + 5*tan(1/2*a)^10*tan(1/2*c)^8 + 5*tan(1/2*a)^8*tan(1/2*c)^10 + 10
*tan(1/2*a)^10*tan(1/2*c)^6 + 25*tan(1/2*a)^8*tan(1/2*c)^8 + 10*tan(1/2*a)^6*tan(1/2*c)^10 + 10*tan(1/2*a)^10*
tan(1/2*c)^4 + 50*tan(1/2*a)^8*tan(1/2*c)^6 + 50*tan(1/2*a)^6*tan(1/2*c)^8 + 10*tan(1/2*a)^4*tan(1/2*c)^10 + 5
*tan(1/2*a)^10*tan(1/2*c)^2 + 50*tan(1/2*a)^8*tan(1/2*c)^4 + 100*tan(1/2*a)^6*tan(1/2*c)^6 + 50*tan(1/2*a)^4*t
an(1/2*c)^8 + 5*tan(1/2*a)^2*tan(1/2*c)^10 + tan(1/2*a)^10 + 25*tan(1/2*a)^8*tan(1/2*c)^2 + 100*tan(1/2*a)^6*t
an(1/2*c)^4 + 100*tan(1/2*a)^4*tan(1/2*c)^6 + 25*tan(1/2*a)^2*tan(1/2*c)^8 + tan(1/2*c)^10 + 5*tan(1/2*a)^8 +
50*tan(1/2*a)^6*tan(1/2*c)^2 + 100*tan(1/2*a)^4*tan(1/2*c)^4 + 50*tan(1/2*a)^2*tan(1/2*c)^6 + 5*tan(1/2*c)^8 +
10*tan(1/2*a)^6 + 50*tan(1/2*a)^4*tan(1/2*c)^2 + 50*tan(1/2*a)^2*tan(1/2*c)^4 + 10*tan(1/2*c)^6 + 10*tan(1/2*
a)^4 + 25*tan(1/2*a)^2*tan(1/2*c)^2 + 10*tan(1/2*c)^4 + 5*tan(1/2*a)^2 + 5*tan(1/2*c)^2 + 1) - (274*tan(1/2*b*
x + 1/2*c)^5*tan(1/2*a)^2*tan(1/2*c)^2 - 274*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^2 + 1096*tan(1/2*b*x + 1/2*c)^5
*tan(1/2*a)*tan(1/2*c) + 40*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^2*tan(1/2*c) - 274*tan(1/2*b*x + 1/2*c)^5*tan(1/
2*c)^2 - 40*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)*tan(1/2*c)^2 + 40*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^2*tan(1/2*c)
^2 + 274*tan(1/2*b*x + 1/2*c)^5 + 40*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a) - 40*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^
2 - 40*tan(1/2*b*x + 1/2*c)^4*tan(1/2*c) + 160*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)*tan(1/2*c) + 20*tan(1/2*b*x +
1/2*c)^2*tan(1/2*a)^2*tan(1/2*c) - 40*tan(1/2*b*x + 1/2*c)^3*tan(1/2*c)^2 - 20*tan(1/2*b*x + 1/2*c)^2*tan(1/2
*a)*tan(1/2*c)^2 + 5*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^2*tan(1/2*c)^2 + 40*tan(1/2*b*x + 1/2*c)^3 + 20*tan(1/2*b
*x + 1/2*c)^2*tan(1/2*a) - 5*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^2 - 20*tan(1/2*b*x + 1/2*c)^2*tan(1/2*c) + 20*tan
(1/2*b*x + 1/2*c)*tan(1/2*a)*tan(1/2*c) + 4*tan(1/2*a)^2*tan(1/2*c) - 5*tan(1/2*b*x + 1/2*c)*tan(1/2*c)^2 - 4*
tan(1/2*a)*tan(1/2*c)^2 + 5*tan(1/2*b*x + 1/2*c) + 4*tan(1/2*a) - 4*tan(1/2*c))/((tan(1/2*a)^2*tan(1/2*c)^2 +
tan(1/2*a)^2 + tan(1/2*c)^2 + 1)*tan(1/2*b*x + 1/2*c)^5))/b