\(\int \csc ^3(c+d x) \sin (a+b x) \, dx\) [115]

Optimal result

Integrand size = 15, antiderivative size = 141 \[ \int \csc ^3(c+d x) \sin (a+b x) \, dx=-\frac {4 i e^{-i a-i b x+3 i (c+d x)} \operatorname {Hypergeometric2F1}\left (3,\frac {1}{2} \left (3-\frac {b}{d}\right ),\frac {1}{2} \left (5-\frac {b}{d}\right ),e^{2 i (c+d x)}\right )}{b-3 d}-\frac {4 i e^{i a+i b x+3 i (c+d x)} \operatorname {Hypergeometric2F1}\left (3,\frac {1}{2} \left (3+\frac {b}{d}\right ),\frac {1}{2} \left (5+\frac {b}{d}\right ),e^{2 i (c+d x)}\right )}{b+3 d} \] Output:

-4*I*exp(-I*a-I*b*x+3*I*(d*x+c))*hypergeom([3, 3/2-1/2*b/d],[5/2-1/2*b/d], 
exp(2*I*(d*x+c)))/(b-3*d)-4*I*exp(I*a+I*b*x+3*I*(d*x+c))*hypergeom([3, 3/2 
+1/2*b/d],[5/2+1/2*b/d],exp(2*I*(d*x+c)))/(b+3*d)
 

Mathematica [A] (verified)

Time = 1.92 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.21 \[ \int \csc ^3(c+d x) \sin (a+b x) \, dx=-\frac {8 i (b+d) e^{-i (a-c+(b-d) x)} \operatorname {Hypergeometric2F1}\left (1,\frac {-b+d}{2 d},\frac {3}{2}-\frac {b}{2 d},e^{2 i (c+d x)}\right )+8 i (b-d) e^{i (a+c+(b+d) x)} \operatorname {Hypergeometric2F1}\left (1,\frac {b+d}{2 d},\frac {1}{2} \left (3+\frac {b}{d}\right ),e^{2 i (c+d x)}\right )+4 \csc ^2(c+d x) ((-b+d) \sin (a-c+b x-d x)+(b+d) \sin (a+c+(b+d) x))}{16 d^2} \] Input:

Integrate[Csc[c + d*x]^3*Sin[a + b*x],x]
 

Output:

-1/16*(((8*I)*(b + d)*Hypergeometric2F1[1, (-b + d)/(2*d), 3/2 - b/(2*d), 
E^((2*I)*(c + d*x))])/E^(I*(a - c + (b - d)*x)) + (8*I)*(b - d)*E^(I*(a + 
c + (b + d)*x))*Hypergeometric2F1[1, (b + d)/(2*d), (3 + b/d)/2, E^((2*I)* 
(c + d*x))] + 4*Csc[c + d*x]^2*((-b + d)*Sin[a - c + b*x - d*x] + (b + d)* 
Sin[a + c + (b + d)*x]))/d^2
 

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.

\(\displaystyle \int \sin (a+b x) \csc ^3(c+d x) \, dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \sin (a+b x) \csc ^3(c+d x)dx\)

Input:

Int[Csc[c + d*x]^3*Sin[a + b*x],x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

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

Maple [F]

\[\int \csc \left (d x +c \right )^{3} \sin \left (b x +a \right )d x\]

Input:

int(csc(d*x+c)^3*sin(b*x+a),x)
 

Output:

int(csc(d*x+c)^3*sin(b*x+a),x)
 

Fricas [F]

\[ \int \csc ^3(c+d x) \sin (a+b x) \, dx=\int { \csc \left (d x + c\right )^{3} \sin \left (b x + a\right ) \,d x } \] Input:

integrate(csc(d*x+c)^3*sin(b*x+a),x, algorithm="fricas")
 

Output:

integral(csc(d*x + c)^3*sin(b*x + a), x)
 

Sympy [F(-1)]

Timed out. \[ \int \csc ^3(c+d x) \sin (a+b x) \, dx=\text {Timed out} \] Input:

integrate(csc(d*x+c)**3*sin(b*x+a),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \csc ^3(c+d x) \sin (a+b x) \, dx=\int { \csc \left (d x + c\right )^{3} \sin \left (b x + a\right ) \,d x } \] Input:

integrate(csc(d*x+c)^3*sin(b*x+a),x, algorithm="maxima")
 

Output:

-1/2*((b - d)*cos(b*x + a)*sin((2*b + d)*x + 2*a + c) - (b - d)*cos((2*b + 
 d)*x + 2*a + c)*sin(b*x + a) + (b - d)*cos(3*d*x + 3*c)*sin(b*x + a) - (b 
 + d)*cos(d*x + c)*sin(b*x + a) - (b - d)*cos(b*x + a)*sin(3*d*x + 3*c) + 
(b + d)*cos(b*x + a)*sin(d*x + c) - (2*(b + d)*sin((b + 2*d)*x + a + 2*c) 
- (b + d)*sin(b*x + a))*cos((2*b + 3*d)*x + 2*a + 3*c) - ((b + d)*sin((2*b 
 + 3*d)*x + 2*a + 3*c) - (b - d)*sin((2*b + d)*x + 2*a + c) + (b - d)*sin( 
3*d*x + 3*c) - (b + d)*sin(d*x + c))*cos((b + 4*d)*x + a + 4*c) - 2*((b - 
d)*sin((2*b + d)*x + 2*a + c) - (b - d)*sin(3*d*x + 3*c) + (b + d)*sin(d*x 
 + c))*cos((b + 2*d)*x + a + 2*c) - 2*(d^2*cos((b + 4*d)*x + a + 4*c)^2 + 
4*d^2*cos((b + 2*d)*x + a + 2*c)^2 - 4*d^2*cos((b + 2*d)*x + a + 2*c)*cos( 
b*x + a) + d^2*cos(b*x + a)^2 + d^2*sin((b + 4*d)*x + a + 4*c)^2 + 4*d^2*s 
in((b + 2*d)*x + a + 2*c)^2 - 4*d^2*sin((b + 2*d)*x + a + 2*c)*sin(b*x + a 
) + d^2*sin(b*x + a)^2 - 2*(2*d^2*cos((b + 2*d)*x + a + 2*c) - d^2*cos(b*x 
 + a))*cos((b + 4*d)*x + a + 4*c) - 2*(2*d^2*sin((b + 2*d)*x + a + 2*c) - 
d^2*sin(b*x + a))*sin((b + 4*d)*x + a + 4*c))*integrate(-1/4*((b^2 - d^2)* 
cos(2*b*x + 2*a)*cos(b*x + a) + (b^2 - d^2)*sin((b + d)*x + a + c)*sin(2*b 
*x + 2*a) + (b^2 - d^2)*sin(2*b*x + 2*a)*sin(b*x + a) - (b^2 - d^2 - (b^2 
- d^2)*cos(2*b*x + 2*a))*cos((b + d)*x + a + c) - (b^2 - d^2)*cos(b*x + a) 
)/(d^2*cos((b + d)*x + a + c)^2 + 2*d^2*cos((b + d)*x + a + c)*cos(b*x + a 
) + d^2*cos(b*x + a)^2 + d^2*sin((b + d)*x + a + c)^2 + 2*d^2*sin((b + ...
 

Giac [F]

\[ \int \csc ^3(c+d x) \sin (a+b x) \, dx=\int { \csc \left (d x + c\right )^{3} \sin \left (b x + a\right ) \,d x } \] Input:

integrate(csc(d*x+c)^3*sin(b*x+a),x, algorithm="giac")
 

Output:

integrate(csc(d*x + c)^3*sin(b*x + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \csc ^3(c+d x) \sin (a+b x) \, dx=\int \frac {\sin \left (a+b\,x\right )}{{\sin \left (c+d\,x\right )}^3} \,d x \] Input:

int(sin(a + b*x)/sin(c + d*x)^3,x)
 

Output:

int(sin(a + b*x)/sin(c + d*x)^3, x)
 

Reduce [F]

\[ \int \csc ^3(c+d x) \sin (a+b x) \, dx=\int \csc \left (d x +c \right )^{3} \sin \left (b x +a \right )d x \] Input:

int(csc(d*x+c)^3*sin(b*x+a),x)
 

Output:

int(csc(c + d*x)**3*sin(a + b*x),x)