\(\int \csc ^4(c+b x) \sin (a+b x) \, dx\) [198]
Optimal result
Integrand size = 15, antiderivative size = 67 \[
\int \csc ^4(c+b x) \sin (a+b x) \, dx=-\frac {\text {arctanh}(\cos (c+b x)) \cos (a-c)}{2 b}-\frac {\cos (a-c) \cot (c+b x) \csc (c+b x)}{2 b}-\frac {\csc ^3(c+b x) \sin (a-c)}{3 b}
\]
[Out]
-1/2*arctanh(cos(b*x+c))*cos(a-c)/b-1/2*cos(a-c)*cot(b*x+c)*csc(b*x+c)/b-1/3*csc(b*x+c)^3*sin(a-c)/b
Rubi [A] (verified)
Time = 0.06 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of
steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4678, 2686, 30, 3853, 3855}
\[
\int \csc ^4(c+b x) \sin (a+b x) \, dx=-\frac {\cos (a-c) \text {arctanh}(\cos (b x+c))}{2 b}-\frac {\sin (a-c) \csc ^3(b x+c)}{3 b}-\frac {\cos (a-c) \cot (b x+c) \csc (b x+c)}{2 b}
\]
[In]
Int[Csc[c + b*x]^4*Sin[a + b*x],x]
[Out]
-1/2*(ArcTanh[Cos[c + b*x]]*Cos[a - c])/b - (Cos[a - c]*Cot[c + b*x]*Csc[c + b*x])/(2*b) - (Csc[c + b*x]^3*Sin
[a - c])/(3*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 ^3(c+b x) \, dx+\sin (a-c) \int \cot (c+b x) \csc ^3(c+b x) \, dx \\ & = -\frac {\cos (a-c) \cot (c+b x) \csc (c+b x)}{2 b}+\frac {1}{2} \cos (a-c) \int \csc (c+b x) \, dx-\frac {\sin (a-c) \text {Subst}\left (\int x^2 \, dx,x,\csc (c+b x)\right )}{b} \\ & = -\frac {\text {arctanh}(\cos (c+b x)) \cos (a-c)}{2 b}-\frac {\cos (a-c) \cot (c+b x) \csc (c+b x)}{2 b}-\frac {\csc ^3(c+b x) \sin (a-c)}{3 b} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.63 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00
\[
\int \csc ^4(c+b x) \sin (a+b x) \, dx=-\frac {6 \text {arctanh}\left (\cos (c)-\sin (c) \tan \left (\frac {b x}{2}\right )\right ) \cos (a-c)+3 \cos (a-c) \cot (c+b x) \csc (c+b x)+2 \csc ^3(c+b x) \sin (a-c)}{6 b}
\]
[In]
Integrate[Csc[c + b*x]^4*Sin[a + b*x],x]
[Out]
-1/6*(6*ArcTanh[Cos[c] - Sin[c]*Tan[(b*x)/2]]*Cos[a - c] + 3*Cos[a - c]*Cot[c + b*x]*Csc[c + b*x] + 2*Csc[c +
b*x]^3*Sin[a - c])/b
Maple [C] (verified)
Result contains complex when optimal does not.
Time = 3.85 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.82
| | |
| method | result | size |
| | |
| risch |
\(\frac {-3 \,{\mathrm e}^{i \left (5 x b +7 a +4 c \right )}-3 \,{\mathrm e}^{i \left (5 x b +5 a +6 c \right )}-8 \,{\mathrm e}^{i \left (3 x b +7 a +2 c \right )}+8 \,{\mathrm e}^{i \left (3 x b +5 a +4 c \right )}+3 \,{\mathrm e}^{i \left (x b +7 a \right )}+3 \,{\mathrm e}^{i \left (x b +5 a +2 c \right )}}{6 b \left (-{\mathrm e}^{2 i \left (x b +a +c \right )}+{\mathrm e}^{2 i a}\right )^{3}}-\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}+{\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (a -c \right )}{2 b}+\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}-{\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (a -c \right )}{2 b}\) |
\(189\) |
| default |
\(\text {Expression too large to display}\) |
\(1779\) |
| | |
|
|
|
|
[In]
int(csc(b*x+c)^4*sin(b*x+a),x,method=_RETURNVERBOSE)
[Out]
1/6/b/(-exp(2*I*(b*x+a+c))+exp(2*I*a))^3*(-3*exp(I*(5*b*x+7*a+4*c))-3*exp(I*(5*b*x+5*a+6*c))-8*exp(I*(3*b*x+7*
a+2*c))+8*exp(I*(3*b*x+5*a+4*c))+3*exp(I*(b*x+7*a))+3*exp(I*(b*x+5*a+2*c)))-1/2*ln(exp(I*(b*x+a))+exp(I*(a-c))
)/b*cos(a-c)+1/2*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. 141 vs. \(2 (61) = 122\).
Time = 0.26 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.10
\[
\int \csc ^4(c+b x) \sin (a+b x) \, dx=\frac {6 \, \cos \left (b x + c\right ) \cos \left (-a + c\right ) \sin \left (b x + c\right ) - 3 \, {\left (\cos \left (b x + c\right )^{2} \cos \left (-a + c\right ) - \cos \left (-a + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (b x + c\right ) + \frac {1}{2}\right ) \sin \left (b x + c\right ) + 3 \, {\left (\cos \left (b x + c\right )^{2} \cos \left (-a + c\right ) - \cos \left (-a + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + c\right ) + \frac {1}{2}\right ) \sin \left (b x + c\right ) - 4 \, \sin \left (-a + c\right )}{12 \, {\left (b \cos \left (b x + c\right )^{2} - b\right )} \sin \left (b x + c\right )}
\]
[In]
integrate(csc(b*x+c)^4*sin(b*x+a),x, algorithm="fricas")
[Out]
1/12*(6*cos(b*x + c)*cos(-a + c)*sin(b*x + c) - 3*(cos(b*x + c)^2*cos(-a + c) - cos(-a + c))*log(1/2*cos(b*x +
c) + 1/2)*sin(b*x + c) + 3*(cos(b*x + c)^2*cos(-a + c) - cos(-a + c))*log(-1/2*cos(b*x + c) + 1/2)*sin(b*x +
c) - 4*sin(-a + c))/((b*cos(b*x + c)^2 - b)*sin(b*x + c))
Sympy [F(-1)]
Timed out. \[
\int \csc ^4(c+b x) \sin (a+b x) \, dx=\text {Timed out}
\]
[In]
integrate(csc(b*x+c)**4*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. 1773 vs. \(2 (61) = 122\).
Time = 0.30 (sec) , antiderivative size = 1773, normalized size of antiderivative = 26.46
\[
\int \csc ^4(c+b x) \sin (a+b x) \, dx=\text {Too large to display}
\]
[In]
integrate(csc(b*x+c)^4*sin(b*x+a),x, algorithm="maxima")
[Out]
1/12*(2*(3*cos(5*b*x + 2*a + 4*c) + 3*cos(5*b*x + 6*c) + 8*cos(3*b*x + 2*a + 2*c) - 8*cos(3*b*x + 4*c) - 3*cos
(b*x + 2*a) - 3*cos(b*x + 2*c))*cos(6*b*x + a + 6*c) - 6*(3*cos(4*b*x + a + 4*c) - 3*cos(2*b*x + a + 2*c) + co
s(a))*cos(5*b*x + 2*a + 4*c) - 6*(3*cos(4*b*x + a + 4*c) - 3*cos(2*b*x + a + 2*c) + cos(a))*cos(5*b*x + 6*c) -
6*(8*cos(3*b*x + 2*a + 2*c) - 8*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)
+ 16*(3*cos(2*b*x + a + 2*c) - cos(a))*cos(3*b*x + 2*a + 2*c) - 16*(3*cos(2*b*x + a + 2*c) - cos(a))*cos(3*b*x
+ 4*c) - 18*(cos(b*x + 2*a) + cos(b*x + 2*c))*cos(2*b*x + a + 2*c) + 6*cos(b*x + 2*a)*cos(a) + 6*cos(b*x + 2*
c)*cos(a) - 3*(cos(6*b*x + a + 6*c)^2*cos(-a + c) + 9*cos(4*b*x + a + 4*c)^2*cos(-a + c) + 9*cos(2*b*x + a + 2
*c)^2*cos(-a + c) - 6*cos(2*b*x + a + 2*c)*cos(a)*cos(-a + c) + cos(-a + c)*sin(6*b*x + a + 6*c)^2 + 9*cos(-a
+ c)*sin(4*b*x + a + 4*c)^2 + 9*cos(-a + c)*sin(2*b*x + a + 2*c)^2 - 6*cos(-a + c)*sin(2*b*x + a + 2*c)*sin(a)
- 2*(3*cos(4*b*x + a + 4*c)*cos(-a + c) - 3*cos(2*b*x + a + 2*c)*cos(-a + c) + cos(a)*cos(-a + c))*cos(6*b*x
+ a + 6*c) - 6*(3*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 + si
n(a)^2)*cos(-a + c) - 2*(3*cos(-a + c)*sin(4*b*x + a + 4*c) - 3*cos(-a + c)*sin(2*b*x + a + 2*c) + cos(-a + c)
*sin(a))*sin(6*b*x + a + 6*c) - 6*(3*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) + 3*(cos(6*b*x
+ a + 6*c)^2*cos(-a + c) + 9*cos(4*b*x + a + 4*c)^2*cos(-a + c) + 9*cos(2*b*x + a + 2*c)^2*cos(-a + c) - 6*cos
(2*b*x + a + 2*c)*cos(a)*cos(-a + c) + cos(-a + c)*sin(6*b*x + a + 6*c)^2 + 9*cos(-a + c)*sin(4*b*x + a + 4*c)
^2 + 9*cos(-a + c)*sin(2*b*x + a + 2*c)^2 - 6*cos(-a + c)*sin(2*b*x + a + 2*c)*sin(a) - 2*(3*cos(4*b*x + a + 4
*c)*cos(-a + c) - 3*cos(2*b*x + a + 2*c)*cos(-a + c) + cos(a)*cos(-a + c))*cos(6*b*x + a + 6*c) - 6*(3*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*(
3*cos(-a + c)*sin(4*b*x + a + 4*c) - 3*cos(-a + c)*sin(2*b*x + a + 2*c) + cos(-a + c)*sin(a))*sin(6*b*x + a +
6*c) - 6*(3*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*co
s(b*x)*cos(c) + cos(c)^2 + sin(b*x)^2 + 2*sin(b*x)*sin(c) + sin(c)^2) + 2*(3*sin(5*b*x + 2*a + 4*c) + 3*sin(5*
b*x + 6*c) + 8*sin(3*b*x + 2*a + 2*c) - 8*sin(3*b*x + 4*c) - 3*sin(b*x + 2*a) - 3*sin(b*x + 2*c))*sin(6*b*x +
a + 6*c) - 6*(3*sin(4*b*x + a + 4*c) - 3*sin(2*b*x + a + 2*c) + sin(a))*sin(5*b*x + 2*a + 4*c) - 6*(3*sin(4*b*
x + a + 4*c) - 3*sin(2*b*x + a + 2*c) + sin(a))*sin(5*b*x + 6*c) - 6*(8*sin(3*b*x + 2*a + 2*c) - 8*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) + 16*(3*sin(2*b*x + a + 2*c) - sin(a))*sin(3
*b*x + 2*a + 2*c) - 16*(3*sin(2*b*x + a + 2*c) - sin(a))*sin(3*b*x + 4*c) - 18*(sin(b*x + 2*a) + sin(b*x + 2*c
))*sin(2*b*x + a + 2*c) + 6*sin(b*x + 2*a)*sin(a) + 6*sin(b*x + 2*c)*sin(a))/(b*cos(6*b*x + a + 6*c)^2 + 9*b*c
os(4*b*x + a + 4*c)^2 + 9*b*cos(2*b*x + a + 2*c)^2 - 6*b*cos(2*b*x + a + 2*c)*cos(a) + b*sin(6*b*x + a + 6*c)^
2 + 9*b*sin(4*b*x + a + 4*c)^2 + 9*b*sin(2*b*x + a + 2*c)^2 - 6*b*sin(2*b*x + a + 2*c)*sin(a) + (cos(a)^2 + si
n(a)^2)*b - 2*(3*b*cos(4*b*x + a + 4*c) - 3*b*cos(2*b*x + a + 2*c) + b*cos(a))*cos(6*b*x + a + 6*c) - 6*(3*b*c
os(2*b*x + a + 2*c) - b*cos(a))*cos(4*b*x + a + 4*c) - 2*(3*b*sin(4*b*x + a + 4*c) - 3*b*sin(2*b*x + a + 2*c)
+ b*sin(a))*sin(6*b*x + a + 6*c) - 6*(3*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. 2221 vs. \(2 (61) = 122\).
Time = 0.32 (sec) , antiderivative size = 2221, normalized size of antiderivative = 33.15
\[
\int \csc ^4(c+b x) \sin (a+b x) \, dx=\text {Too large to display}
\]
[In]
integrate(csc(b*x+c)^4*sin(b*x+a),x, algorithm="giac")
[Out]
1/24*(12*(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) - (2*tan(1/2*b*x + 1/2*c)^3*ta
n(1/2*a)^6*tan(1/2*c)^5 - 2*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^5*tan(1/2*c)^6 - 3*tan(1/2*b*x + 1/2*c)^2*tan(1/
2*a)^6*tan(1/2*c)^6 + 4*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^6*tan(1/2*c)^3 - 2*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)
^5*tan(1/2*c)^4 - 3*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^6*tan(1/2*c)^4 + 2*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^4*t
an(1/2*c)^5 - 12*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^5*tan(1/2*c)^5 + 6*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^6*tan(1/
2*c)^5 - 4*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^3*tan(1/2*c)^6 - 3*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^4*tan(1/2*c)
^6 - 6*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^5*tan(1/2*c)^6 + 2*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^6*tan(1/2*c) + 2*t
an(1/2*b*x + 1/2*c)^3*tan(1/2*a)^5*tan(1/2*c)^2 + 3*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^6*tan(1/2*c)^2 + 4*tan(1
/2*b*x + 1/2*c)^3*tan(1/2*a)^4*tan(1/2*c)^3 - 24*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^5*tan(1/2*c)^3 + 12*tan(1/2
*b*x + 1/2*c)*tan(1/2*a)^6*tan(1/2*c)^3 - 4*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^3*tan(1/2*c)^4 - 3*tan(1/2*b*x +
1/2*c)^2*tan(1/2*a)^4*tan(1/2*c)^4 - 6*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^5*tan(1/2*c)^4 - 2*tan(1/2*b*x + 1/2*c
)^3*tan(1/2*a)^2*tan(1/2*c)^5 - 24*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^3*tan(1/2*c)^5 + 6*tan(1/2*b*x + 1/2*c)*t
an(1/2*a)^4*tan(1/2*c)^5 - 2*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)*tan(1/2*c)^6 + 3*tan(1/2*b*x + 1/2*c)^2*tan(1/2
*a)^2*tan(1/2*c)^6 - 12*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^3*tan(1/2*c)^6 + 2*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^5
+ 3*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^6 + 2*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^4*tan(1/2*c) - 12*tan(1/2*b*x +
1/2*c)^2*tan(1/2*a)^5*tan(1/2*c) + 6*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^6*tan(1/2*c) + 4*tan(1/2*b*x + 1/2*c)^3*
tan(1/2*a)^3*tan(1/2*c)^2 + 3*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^4*tan(1/2*c)^2 + 6*tan(1/2*b*x + 1/2*c)*tan(1/
2*a)^5*tan(1/2*c)^2 - 4*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^2*tan(1/2*c)^3 - 48*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a
)^3*tan(1/2*c)^3 + 12*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^4*tan(1/2*c)^3 - 2*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)*tan
(1/2*c)^4 + 3*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^2*tan(1/2*c)^4 - 12*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^3*tan(1/2*
c)^4 - 2*tan(1/2*b*x + 1/2*c)^3*tan(1/2*c)^5 - 12*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)*tan(1/2*c)^5 - 6*tan(1/2*b
*x + 1/2*c)*tan(1/2*a)^2*tan(1/2*c)^5 + 3*tan(1/2*b*x + 1/2*c)^2*tan(1/2*c)^6 - 6*tan(1/2*b*x + 1/2*c)*tan(1/2
*a)*tan(1/2*c)^6 + 4*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^3 + 3*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^4 + 6*tan(1/2*b
*x + 1/2*c)*tan(1/2*a)^5 - 2*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^2*tan(1/2*c) - 24*tan(1/2*b*x + 1/2*c)^2*tan(1/
2*a)^3*tan(1/2*c) + 6*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^4*tan(1/2*c) + 2*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)*tan(1
/2*c)^2 - 3*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^2*tan(1/2*c)^2 + 12*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^3*tan(1/2*c)
^2 - 4*tan(1/2*b*x + 1/2*c)^3*tan(1/2*c)^3 - 24*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)*tan(1/2*c)^3 - 12*tan(1/2*b*
x + 1/2*c)*tan(1/2*a)^2*tan(1/2*c)^3 + 3*tan(1/2*b*x + 1/2*c)^2*tan(1/2*c)^4 - 6*tan(1/2*b*x + 1/2*c)*tan(1/2*
a)*tan(1/2*c)^4 - 6*tan(1/2*b*x + 1/2*c)*tan(1/2*c)^5 + 2*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a) - 3*tan(1/2*b*x +
1/2*c)^2*tan(1/2*a)^2 + 12*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^3 - 2*tan(1/2*b*x + 1/2*c)^3*tan(1/2*c) - 12*tan(1/
2*b*x + 1/2*c)^2*tan(1/2*a)*tan(1/2*c) - 6*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^2*tan(1/2*c) - 3*tan(1/2*b*x + 1/2*
c)^2*tan(1/2*c)^2 + 6*tan(1/2*b*x + 1/2*c)*tan(1/2*a)*tan(1/2*c)^2 - 12*tan(1/2*b*x + 1/2*c)*tan(1/2*c)^3 - 3*
tan(1/2*b*x + 1/2*c)^2 + 6*tan(1/2*b*x + 1/2*c)*tan(1/2*a) - 6*tan(1/2*b*x + 1/2*c)*tan(1/2*c))/(tan(1/2*a)^6*
tan(1/2*c)^6 + 3*tan(1/2*a)^6*tan(1/2*c)^4 + 3*tan(1/2*a)^4*tan(1/2*c)^6 + 3*tan(1/2*a)^6*tan(1/2*c)^2 + 9*tan
(1/2*a)^4*tan(1/2*c)^4 + 3*tan(1/2*a)^2*tan(1/2*c)^6 + tan(1/2*a)^6 + 9*tan(1/2*a)^4*tan(1/2*c)^2 + 9*tan(1/2*
a)^2*tan(1/2*c)^4 + tan(1/2*c)^6 + 3*tan(1/2*a)^4 + 9*tan(1/2*a)^2*tan(1/2*c)^2 + 3*tan(1/2*c)^4 + 3*tan(1/2*a
)^2 + 3*tan(1/2*c)^2 + 1) - (22*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^2*tan(1/2*c)^2 - 22*tan(1/2*b*x + 1/2*c)^3*t
an(1/2*a)^2 + 88*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)*tan(1/2*c) + 6*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^2*tan(1/2*
c) - 22*tan(1/2*b*x + 1/2*c)^3*tan(1/2*c)^2 - 6*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)*tan(1/2*c)^2 + 3*tan(1/2*b*x
+ 1/2*c)*tan(1/2*a)^2*tan(1/2*c)^2 + 22*tan(1/2*b*x + 1/2*c)^3 + 6*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a) - 3*tan(
1/2*b*x + 1/2*c)*tan(1/2*a)^2 - 6*tan(1/2*b*x + 1/2*c)^2*tan(1/2*c) + 12*tan(1/2*b*x + 1/2*c)*tan(1/2*a)*tan(1
/2*c) + 2*tan(1/2*a)^2*tan(1/2*c) - 3*tan(1/2*b*x + 1/2*c)*tan(1/2*c)^2 - 2*tan(1/2*a)*tan(1/2*c)^2 + 3*tan(1/
2*b*x + 1/2*c) + 2*tan(1/2*a) - 2*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)*t
an(1/2*b*x + 1/2*c)^3))/b
Mupad [F(-1)]
Timed out. \[
\int \csc ^4(c+b x) \sin (a+b x) \, dx=\text {Hanged}
\]
[In]
int(sin(a + b*x)/sin(c + b*x)^4,x)
[Out]
\text{Hanged}