\(\int \csc ^6(c+b x) \sin (a+b x) \, dx\) [200]
Optimal result
Integrand size = 15, antiderivative size = 94 \[
\int \csc ^6(c+b x) \sin (a+b x) \, dx=-\frac {3 \text {arctanh}(\cos (c+b x)) \cos (a-c)}{8 b}-\frac {3 \cos (a-c) \cot (c+b x) \csc (c+b x)}{8 b}-\frac {\cos (a-c) \cot (c+b x) \csc ^3(c+b x)}{4 b}-\frac {\csc ^5(c+b x) \sin (a-c)}{5 b}
\]
[Out]
-3/8*arctanh(cos(b*x+c))*cos(a-c)/b-3/8*cos(a-c)*cot(b*x+c)*csc(b*x+c)/b-1/4*cos(a-c)*cot(b*x+c)*csc(b*x+c)^3/
b-1/5*csc(b*x+c)^5*sin(a-c)/b
Rubi [A] (verified)
Time = 0.08 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of
steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4678, 2686, 30, 3853, 3855}
\[
\int \csc ^6(c+b x) \sin (a+b x) \, dx=-\frac {3 \cos (a-c) \text {arctanh}(\cos (b x+c))}{8 b}-\frac {\sin (a-c) \csc ^5(b x+c)}{5 b}-\frac {\cos (a-c) \cot (b x+c) \csc ^3(b x+c)}{4 b}-\frac {3 \cos (a-c) \cot (b x+c) \csc (b x+c)}{8 b}
\]
[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 30
Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]
Rule 2686
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 3853
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 3855
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Rule 4678
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]
Rubi steps \begin{align*}
\text {integral}& = \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) \text {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 \text {arctanh}(\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] (verified)
Time = 1.35 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.84
\[
\int \csc ^6(c+b x) \sin (a+b x) \, dx=-\frac {480 \text {arctanh}\left (\cos (c)-\sin (c) \tan \left (\frac {b x}{2}\right )\right ) \cos (a-c)+2 \csc ^5(c+b x) (64 \sin (a-c)+5 \cos (a-c) (14 \sin (2 (c+b x))-3 \sin (4 (c+b x))))}{640 b}
\]
[In]
Integrate[Csc[c + b*x]^6*Sin[a + b*x],x]
[Out]
-1/640*(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*Sin[2*(c + b*x)] - 3*Sin[4*(c + b*x)])))/b
Maple [C] (verified)
Result contains complex when optimal does not.
Time = 13.88 (sec) , antiderivative size = 257, normalized size of antiderivative =
2.73
| | |
| method | result | size |
| | |
| risch |
\(\frac {-15 \,{\mathrm e}^{i \left (9 x b +11 a +8 c \right )}-15 \,{\mathrm e}^{i \left (9 x b +9 a +10 c \right )}+70 \,{\mathrm e}^{i \left (7 x b +11 a +6 c \right )}+70 \,{\mathrm e}^{i \left (7 x b +9 a +8 c \right )}+128 \,{\mathrm e}^{i \left (5 x b +11 a +4 c \right )}-128 \,{\mathrm e}^{i \left (5 x b +9 a +6 c \right )}-70 \,{\mathrm e}^{i \left (3 x b +11 a +2 c \right )}-70 \,{\mathrm e}^{i \left (3 x b +9 a +4 c \right )}+15 \,{\mathrm e}^{i \left (x b +11 a \right )}+15 \,{\mathrm e}^{i \left (x b +9 a +2 c \right )}}{40 b \left (-{\mathrm e}^{2 i \left (x b +a +c \right )}+{\mathrm e}^{2 i a}\right )^{5}}+\frac {3 \ln \left ({\mathrm e}^{i \left (x b +a \right )}-{\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (a -c \right )}{8 b}-\frac {3 \ln \left ({\mathrm e}^{i \left (x b +a \right )}+{\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (a -c \right )}{8 b}\) |
\(257\) |
| default |
\(\text {Expression too large to display}\) |
\(6750\) |
| | |
|
|
|
|
[In]
int(csc(b*x+c)^6*sin(b*x+a),x,method=_RETURNVERBOSE)
[Out]
1/40/b/(-exp(2*I*(b*x+a+c))+exp(2*I*a))^5*(-15*exp(I*(9*b*x+11*a+8*c))-15*exp(I*(9*b*x+9*a+10*c))+70*exp(I*(7*
b*x+11*a+6*c))+70*exp(I*(7*b*x+9*a+8*c))+128*exp(I*(5*b*x+11*a+4*c))-128*exp(I*(5*b*x+9*a+6*c))-70*exp(I*(3*b*
x+11*a+2*c))-70*exp(I*(3*b*x+9*a+4*c))+15*exp(I*(b*x+11*a))+15*exp(I*(b*x+9*a+2*c)))+3/8*ln(exp(I*(b*x+a))-exp
(I*(a-c)))/b*cos(a-c)-3/8*ln(exp(I*(b*x+a))+exp(I*(a-c)))/b*cos(a-c)
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (86) = 172\).
Time = 0.26 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.10
\[
\int \csc ^6(c+b x) \sin (a+b x) \, dx=-\frac {15 \, {\left (\cos \left (b x + c\right )^{4} \cos \left (-a + c\right ) - 2 \, \cos \left (b x + c\right )^{2} \cos \left (-a + c\right ) + \cos \left (-a + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (b x + c\right ) + \frac {1}{2}\right ) \sin \left (b x + c\right ) - 15 \, {\left (\cos \left (b x + c\right )^{4} \cos \left (-a + c\right ) - 2 \, \cos \left (b x + c\right )^{2} \cos \left (-a + c\right ) + \cos \left (-a + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + c\right ) + \frac {1}{2}\right ) \sin \left (b x + c\right ) - 10 \, {\left (3 \, \cos \left (b x + c\right )^{3} \cos \left (-a + c\right ) - 5 \, \cos \left (b x + c\right ) \cos \left (-a + c\right )\right )} \sin \left (b x + c\right ) - 16 \, \sin \left (-a + c\right )}{80 \, {\left (b \cos \left (b x + c\right )^{4} - 2 \, b \cos \left (b x + c\right )^{2} + b\right )} \sin \left (b x + c\right )}
\]
[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)]
Timed out. \[
\int \csc ^6(c+b x) \sin (a+b x) \, dx=\text {Timed out}
\]
[In]
integrate(csc(b*x+c)**6*sin(b*x+a),x)
[Out]
Timed out
Maxima [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 3879 vs. \(2 (86) = 172\).
Time = 0.43 (sec) , antiderivative size = 3879, normalized size of antiderivative = 41.27
\[
\int \csc ^6(c+b x) \sin (a+b x) \, dx=\text {Too large to display}
\]
[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))
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 8035 vs. \(2 (86) = 172\).
Time = 0.36 (sec) , antiderivative size = 8035, normalized size of antiderivative = 85.48
\[
\int \csc ^6(c+b x) \sin (a+b x) \, dx=\text {Too large to display}
\]
[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
Mupad [F(-1)]
Timed out. \[
\int \csc ^6(c+b x) \sin (a+b x) \, dx=\text {Hanged}
\]
[In]
int(sin(a + b*x)/sin(c + b*x)^6,x)
[Out]
\text{Hanged}