\(\int \csc ^4(c-b x) \sin (a+b x) \, dx\) [284]

Optimal result

Integrand size = 16, antiderivative size = 65 \[ \int \csc ^4(c-b x) \sin (a+b x) \, dx=-\frac {\text {arctanh}(\cos (c-b x)) \cos (a+c)}{2 b}-\frac {\cos (a+c) \cot (c-b x) \csc (c-b x)}{2 b}+\frac {\csc ^3(c-b x) \sin (a+c)}{3 b} \] Output:

-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
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.92 \[ \int \csc ^4(c-b x) \sin (a+b x) \, dx=\frac {-12 \text {arctanh}\left (\cos (c)+\sin (c) \tan \left (\frac {b x}{2}\right )\right ) \cos (a+c)+\csc ^3(c-b x) (4 \sin (a+c)-3 \cos (a+c) \sin (2 (c-b x)))}{12 b} \] Input:

Integrate[Csc[c - b*x]^4*Sin[a + b*x],x]
 

Output:

(-12*ArcTanh[Cos[c] + Sin[c]*Tan[(b*x)/2]]*Cos[a + c] + Csc[c - b*x]^3*(4* 
Sin[a + c] - 3*Cos[a + c]*Sin[2*(c - b*x)]))/(12*b)
 

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \sin (a+b x) \csc ^4(c-b x) \, dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \sin (a+b x) \csc ^4(c-b x)dx\)

Input:

Int[Csc[c - b*x]^4*Sin[a + b*x],x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 4.42 (sec) , antiderivative size = 179, normalized size of antiderivative = 2.75

method	result	size
risch	\(\frac {3 \,{\mathrm e}^{i \left (b x +7 a +6 c \right )}-8 \,{\mathrm e}^{i \left (3 b x +7 a +4 c \right )}+3 \,{\mathrm e}^{i \left (b x +5 a +4 c \right )}-3 \,{\mathrm e}^{i \left (5 b x +7 a +2 c \right )}+8 \,{\mathrm e}^{i \left (3 b x +5 a +2 c \right )}-3 \,{\mathrm e}^{5 i \left (b x +a \right )}}{6 b \left ({\mathrm e}^{2 i \left (a +c \right )}-{\mathrm e}^{2 i \left (b x +a \right )}\right )^{3}}-\frac {\ln \left ({\mathrm e}^{i \left (a +c \right )}+{\mathrm e}^{i \left (b x +a \right )}\right ) \cos \left (a +c \right )}{2 b}+\frac {\ln \left (-{\mathrm e}^{i \left (a +c \right )}+{\mathrm e}^{i \left (b x +a \right )}\right ) \cos \left (a +c \right )}{2 b}\)	\(179\)
default	\(\text {Expression too large to display}\)	\(1749\)

Input:

int(csc(b*x-c)^4*sin(b*x+a),x,method=_RETURNVERBOSE)
 

Output:

1/6/b/(exp(2*I*(a+c))-exp(2*I*(b*x+a)))^3*(3*exp(I*(b*x+7*a+6*c))-8*exp(I* 
(3*b*x+7*a+4*c))+3*exp(I*(b*x+5*a+4*c))-3*exp(I*(5*b*x+7*a+2*c))+8*exp(I*( 
3*b*x+5*a+2*c))-3*exp(5*I*(b*x+a)))-1/2*ln(exp(I*(a+c))+exp(I*(b*x+a)))/b* 
cos(a+c)+1/2*ln(-exp(I*(a+c))+exp(I*(b*x+a)))/b*cos(a+c)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (63) = 126\).

Time = 0.10 (sec) , antiderivative size = 398, normalized size of antiderivative = 6.12 \[ \int \csc ^4(c-b x) \sin (a+b x) \, dx=-\frac {6 \, {\left (2 \, \cos \left (a + c\right )^{3} - \cos \left (a + c\right )\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 3 \, {\left ({\left (\cos \left (a + c\right )^{4} - {\left (4 \, \cos \left (a + c\right )^{4} - 3 \, \cos \left (a + c\right )^{2}\right )} \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right ) + {\left ({\left (4 \, \cos \left (a + c\right )^{3} - \cos \left (a + c\right )\right )} \cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right ) \cos \left (a + c\right )^{3}\right )} \sin \left (a + c\right )\right )} \log \left (\frac {\cos \left (b x + a\right ) \cos \left (a + c\right ) + \sin \left (b x + a\right ) \sin \left (a + c\right ) + 1}{\cos \left (a + c\right ) + 1}\right ) - 3 \, {\left ({\left (\cos \left (a + c\right )^{4} - {\left (4 \, \cos \left (a + c\right )^{4} - 3 \, \cos \left (a + c\right )^{2}\right )} \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right ) + {\left ({\left (4 \, \cos \left (a + c\right )^{3} - \cos \left (a + c\right )\right )} \cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right ) \cos \left (a + c\right )^{3}\right )} \sin \left (a + c\right )\right )} \log \left (-\frac {\cos \left (b x + a\right ) \cos \left (a + c\right ) + \sin \left (b x + a\right ) \sin \left (a + c\right ) - 1}{\cos \left (a + c\right ) + 1}\right ) - 2 \, {\left (6 \, \cos \left (b x + a\right )^{2} \cos \left (a + c\right )^{2} - 3 \, \cos \left (a + c\right )^{2} - 2\right )} \sin \left (a + c\right )}{12 \, {\left ({\left (b \cos \left (a + c\right )^{3} - {\left (4 \, b \cos \left (a + c\right )^{3} - 3 \, b \cos \left (a + c\right )\right )} \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right ) + {\left ({\left (4 \, b \cos \left (a + c\right )^{2} - b\right )} \cos \left (b x + a\right )^{3} - 3 \, b \cos \left (b x + a\right ) \cos \left (a + c\right )^{2}\right )} \sin \left (a + c\right )\right )}} \] Input:

integrate(csc(b*x-c)^4*sin(b*x+a),x, algorithm="fricas")
 

Output:

-1/12*(6*(2*cos(a + c)^3 - cos(a + c))*cos(b*x + a)*sin(b*x + a) + 3*((cos 
(a + c)^4 - (4*cos(a + c)^4 - 3*cos(a + c)^2)*cos(b*x + a)^2)*sin(b*x + a) 
 + ((4*cos(a + c)^3 - cos(a + c))*cos(b*x + a)^3 - 3*cos(b*x + a)*cos(a + 
c)^3)*sin(a + c))*log((cos(b*x + a)*cos(a + c) + sin(b*x + a)*sin(a + c) + 
 1)/(cos(a + c) + 1)) - 3*((cos(a + c)^4 - (4*cos(a + c)^4 - 3*cos(a + c)^ 
2)*cos(b*x + a)^2)*sin(b*x + a) + ((4*cos(a + c)^3 - cos(a + c))*cos(b*x + 
 a)^3 - 3*cos(b*x + a)*cos(a + c)^3)*sin(a + c))*log(-(cos(b*x + a)*cos(a 
+ c) + sin(b*x + a)*sin(a + c) - 1)/(cos(a + c) + 1)) - 2*(6*cos(b*x + a)^ 
2*cos(a + c)^2 - 3*cos(a + c)^2 - 2)*sin(a + c))/((b*cos(a + c)^3 - (4*b*c 
os(a + c)^3 - 3*b*cos(a + c))*cos(b*x + a)^2)*sin(b*x + a) + ((4*b*cos(a + 
 c)^2 - b)*cos(b*x + a)^3 - 3*b*cos(b*x + a)*cos(a + c)^2)*sin(a + c))
 

Sympy [F(-1)]

Timed out. \[ \int \csc ^4(c-b x) \sin (a+b x) \, dx=\text {Timed out} \] Input:

integrate(csc(b*x-c)**4*sin(b*x+a),x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1798 vs. \(2 (63) = 126\).

Time = 0.09 (sec) , antiderivative size = 1798, normalized size of antiderivative = 27.66 \[ \int \csc ^4(c-b x) \sin (a+b x) \, dx=\text {Too large to display} \] Input:

integrate(csc(b*x-c)^4*sin(b*x+a),x, algorithm="maxima")
 

Output:

1/12*(2*(3*cos(5*b*x) + 3*cos(5*b*x + 2*a + 2*c) + 8*cos(3*b*x + 2*a + 4*c 
) - 8*cos(3*b*x + 2*c) - 3*cos(b*x + 2*a + 6*c) - 3*cos(b*x + 4*c))*cos(6* 
b*x + a) - 6*(3*cos(4*b*x + a + 2*c) - 3*cos(2*b*x + a + 4*c) + cos(a + 6* 
c))*cos(5*b*x + 2*a + 2*c) - 6*(3*cos(5*b*x) + 8*cos(3*b*x + 2*a + 4*c) - 
8*cos(3*b*x + 2*c) - 3*cos(b*x + 2*a + 6*c) - 3*cos(b*x + 4*c))*cos(4*b*x 
+ a + 2*c) + 16*(3*cos(2*b*x + a + 4*c) - cos(a + 6*c))*cos(3*b*x + 2*a + 
4*c) - 16*(3*cos(2*b*x + a + 4*c) - cos(a + 6*c))*cos(3*b*x + 2*c) + 18*(c 
os(5*b*x) - cos(b*x + 2*a + 6*c) - cos(b*x + 4*c))*cos(2*b*x + a + 4*c) - 
6*cos(5*b*x)*cos(a + 6*c) + 6*cos(b*x + 2*a + 6*c)*cos(a + 6*c) + 6*cos(b* 
x + 4*c)*cos(a + 6*c) - 3*(cos(6*b*x + a)^2*cos(a + c) + 9*cos(4*b*x + a + 
 2*c)^2*cos(a + c) + 9*cos(2*b*x + a + 4*c)^2*cos(a + c) - 6*cos(2*b*x + a 
 + 4*c)*cos(a + 6*c)*cos(a + c) + cos(a + 6*c)^2*cos(a + c) + cos(a + c)*s 
in(6*b*x + a)^2 + 9*cos(a + c)*sin(4*b*x + a + 2*c)^2 + 9*cos(a + c)*sin(2 
*b*x + a + 4*c)^2 - 6*cos(a + c)*sin(2*b*x + a + 4*c)*sin(a + 6*c) + cos(a 
 + c)*sin(a + 6*c)^2 - 2*(3*cos(4*b*x + a + 2*c)*cos(a + c) - 3*cos(2*b*x 
+ a + 4*c)*cos(a + c) + cos(a + 6*c)*cos(a + c))*cos(6*b*x + a) - 6*(3*cos 
(2*b*x + a + 4*c)*cos(a + c) - cos(a + 6*c)*cos(a + c))*cos(4*b*x + a + 2* 
c) - 2*(3*cos(a + c)*sin(4*b*x + a + 2*c) - 3*cos(a + c)*sin(2*b*x + a + 4 
*c) + cos(a + c)*sin(a + 6*c))*sin(6*b*x + a) - 6*(3*cos(a + c)*sin(2*b*x 
+ a + 4*c) - cos(a + c)*sin(a + 6*c))*sin(4*b*x + a + 2*c))*log(cos(b*x...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2220 vs. \(2 (63) = 126\).

Time = 0.18 (sec) , antiderivative size = 2220, normalized size of antiderivative = 34.15 \[ \int \csc ^4(c-b x) \sin (a+b x) \, dx=\text {Too large to display} \] Input:

integrate(csc(b*x-c)^4*sin(b*x+a),x, algorithm="giac")
 

Output:

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*tan( 
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*t 
an(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*tan(1/2*c)^5 - 12*tan(1/2*b*x - 1/2*c)^2*ta 
n(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*tan(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)*tan(1/2*a)^4*tan(1/2*c)^5 + 2*tan(1/2*b*x - 1/2*...
 

Mupad [F(-1)]

Timed out. \[ \int \csc ^4(c-b x) \sin (a+b x) \, dx=\text {Hanged} \] Input:

int(sin(a + b*x)/sin(c - b*x)^4,x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \csc ^4(c-b x) \sin (a+b x) \, dx=\frac {-2 \cos \left (b x -c \right ) \cos \left (b x +a \right ) \sin \left (b x -c \right )-4 \cos \left (b x -c \right ) \sin \left (b x -c \right )^{2} \sin \left (b x +a \right )-2 \cos \left (b x -c \right ) \sin \left (b x -c \right )-4 \cos \left (b x -c \right ) \sin \left (b x +a \right )-4 \cos \left (b x +a \right ) \sin \left (b x -c \right )-6 \left (\int \frac {\tan \left (\frac {b x}{2}-\frac {c}{2}\right )}{\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+1}d x \right ) \sin \left (b x -c \right )^{3} b -2 \left (\int \frac {1}{\tan \left (\frac {b x}{2}-\frac {c}{2}\right )^{3} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+\tan \left (\frac {b x}{2}-\frac {c}{2}\right )^{3}}d x \right ) \sin \left (b x -c \right )^{3} b +4 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}-\frac {c}{2}\right )^{2}+1\right ) \sin \left (b x -c \right )^{3}-2 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}-\frac {c}{2}\right )\right ) \sin \left (b x -c \right )^{3}+3 \sin \left (b x -c \right )^{3}+2 \sin \left (b x -c \right )^{2} \sin \left (b x +a \right )-2 \sin \left (b x -c \right )}{12 \sin \left (b x -c \right )^{3} b} \] Input:

int(csc(b*x-c)^4*sin(b*x+a),x)
 

Output:

( - 2*cos(b*x - c)*cos(a + b*x)*sin(b*x - c) - 4*cos(b*x - c)*sin(b*x - c) 
**2*sin(a + b*x) - 2*cos(b*x - c)*sin(b*x - c) - 4*cos(b*x - c)*sin(a + b* 
x) - 4*cos(a + b*x)*sin(b*x - c) - 6*int(tan((b*x - c)/2)/(tan((a + b*x)/2 
)**2 + 1),x)*sin(b*x - c)**3*b - 2*int(1/(tan((b*x - c)/2)**3*tan((a + b*x 
)/2)**2 + tan((b*x - c)/2)**3),x)*sin(b*x - c)**3*b + 4*log(tan((b*x - c)/ 
2)**2 + 1)*sin(b*x - c)**3 - 2*log(tan((b*x - c)/2))*sin(b*x - c)**3 + 3*s 
in(b*x - c)**3 + 2*sin(b*x - c)**2*sin(a + b*x) - 2*sin(b*x - c))/(12*sin( 
b*x - c)**3*b)