\(\int \csc ^2(c+d x) \sin (a+b x) \, dx\) [114]

Optimal result

Integrand size = 15, antiderivative size = 131 \[ \int \csc ^2(c+d x) \sin (a+b x) \, dx=\frac {2 e^{-i a-i b x+2 i (c+d x)} \operatorname {Hypergeometric2F1}\left (2,1-\frac {b}{2 d},2-\frac {b}{2 d},e^{2 i (c+d x)}\right )}{b-2 d}+\frac {2 e^{i a+i b x+2 i (c+d x)} \operatorname {Hypergeometric2F1}\left (2,1+\frac {b}{2 d},2+\frac {b}{2 d},e^{2 i (c+d x)}\right )}{b+2 d} \] Output:

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

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(273\) vs. \(2(131)=262\).

Time = 4.32 (sec) , antiderivative size = 273, normalized size of antiderivative = 2.08 \[ \int \csc ^2(c+d x) \sin (a+b x) \, dx=\frac {\frac {e^{-i (a-2 c+b x)} \left (-b e^{2 i d x} \operatorname {Hypergeometric2F1}\left (1,1-\frac {b}{2 d},2-\frac {b}{2 d},e^{2 i (c+d x)}\right )+(b-2 d) \operatorname {Hypergeometric2F1}\left (1,-\frac {b}{2 d},1-\frac {b}{2 d},e^{2 i (c+d x)}\right )\right )}{(b-2 d) \left (-1+e^{2 i c}\right )}+\frac {e^{i (a+2 c+b x)} \left (b e^{2 i d x} \operatorname {Hypergeometric2F1}\left (1,1+\frac {b}{2 d},2+\frac {b}{2 d},e^{2 i (c+d x)}\right )-(b+2 d) \operatorname {Hypergeometric2F1}\left (1,\frac {b}{2 d},1+\frac {b}{2 d},e^{2 i (c+d x)}\right )\right )}{(b+2 d) \left (-1+e^{2 i c}\right )}+\cos (b x) \csc (c) \csc (c+d x) \sin (a) \sin (d x)+\cos (a) \csc (c) \csc (c+d x) \sin (b x) \sin (d x)}{d} \] Input:

Integrate[Csc[c + d*x]^2*Sin[a + b*x],x]
 

Output:

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

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.

\(\displaystyle \int \sin (a+b x) \csc ^2(c+d x) \, dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \sin (a+b x) \csc ^2(c+d x)dx\)

Input:

Int[Csc[c + d*x]^2*Sin[a + b*x],x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

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

Maple [F]

\[\int \csc \left (d x +c \right )^{2} \sin \left (b x +a \right )d x\]

Input:

int(csc(d*x+c)^2*sin(b*x+a),x)
 

Output:

int(csc(d*x+c)^2*sin(b*x+a),x)
 

Fricas [F]

\[ \int \csc ^2(c+d x) \sin (a+b x) \, dx=\int { \csc \left (d x + c\right )^{2} \sin \left (b x + a\right ) \,d x } \] Input:

integrate(csc(d*x+c)^2*sin(b*x+a),x, algorithm="fricas")
 

Output:

integral(csc(d*x + c)^2*sin(b*x + a), x)
 

Sympy [F(-1)]

Timed out. \[ \int \csc ^2(c+d x) \sin (a+b x) \, dx=\text {Timed out} \] Input:

integrate(csc(d*x+c)**2*sin(b*x+a),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \csc ^2(c+d x) \sin (a+b x) \, dx=\int { \csc \left (d x + c\right )^{2} \sin \left (b x + a\right ) \,d x } \] Input:

integrate(csc(d*x+c)^2*sin(b*x+a),x, algorithm="maxima")
 

Output:

-((cos(2*b*x + 2*a) - 1)*cos((b + 2*d)*x + a + 2*c) - cos(2*b*x + 2*a)*cos 
(b*x + a) + (d*cos((b + 2*d)*x + a + 2*c)^2 - 2*d*cos((b + 2*d)*x + a + 2* 
c)*cos(b*x + a) + d*cos(b*x + a)^2 + d*sin((b + 2*d)*x + a + 2*c)^2 - 2*d* 
sin((b + 2*d)*x + a + 2*c)*sin(b*x + a) + d*sin(b*x + a)^2)*integrate(-1/2 
*(b*cos((b + d)*x + a + c)*sin(2*b*x + 2*a) + b*cos(b*x + a)*sin(2*b*x + 2 
*a) - b*cos(2*b*x + 2*a)*sin(b*x + a) - (b*cos(2*b*x + 2*a) + b)*sin((b + 
d)*x + a + c) - b*sin(b*x + a))/(d*cos((b + d)*x + a + c)^2 + 2*d*cos((b + 
 d)*x + a + c)*cos(b*x + a) + d*cos(b*x + a)^2 + d*sin((b + d)*x + a + c)^ 
2 + 2*d*sin((b + d)*x + a + c)*sin(b*x + a) + d*sin(b*x + a)^2), x) - (d*c 
os((b + 2*d)*x + a + 2*c)^2 - 2*d*cos((b + 2*d)*x + a + 2*c)*cos(b*x + a) 
+ d*cos(b*x + a)^2 + d*sin((b + 2*d)*x + a + 2*c)^2 - 2*d*sin((b + 2*d)*x 
+ a + 2*c)*sin(b*x + a) + d*sin(b*x + a)^2)*integrate(-1/2*(b*cos((b + d)* 
x + a + c)*sin(2*b*x + 2*a) - b*cos(b*x + a)*sin(2*b*x + 2*a) + b*cos(2*b* 
x + 2*a)*sin(b*x + a) - (b*cos(2*b*x + 2*a) + b)*sin((b + d)*x + a + c) + 
b*sin(b*x + a))/(d*cos((b + d)*x + a + c)^2 - 2*d*cos((b + d)*x + a + c)*c 
os(b*x + a) + d*cos(b*x + a)^2 + d*sin((b + d)*x + a + c)^2 - 2*d*sin((b + 
 d)*x + a + c)*sin(b*x + a) + d*sin(b*x + a)^2), x) + sin((b + 2*d)*x + a 
+ 2*c)*sin(2*b*x + 2*a) - sin(2*b*x + 2*a)*sin(b*x + a) + cos(b*x + a))/(d 
*cos((b + 2*d)*x + a + 2*c)^2 - 2*d*cos((b + 2*d)*x + a + 2*c)*cos(b*x + a 
) + d*cos(b*x + a)^2 + d*sin((b + 2*d)*x + a + 2*c)^2 - 2*d*sin((b + 2*...
 

Giac [F]

\[ \int \csc ^2(c+d x) \sin (a+b x) \, dx=\int { \csc \left (d x + c\right )^{2} \sin \left (b x + a\right ) \,d x } \] Input:

integrate(csc(d*x+c)^2*sin(b*x+a),x, algorithm="giac")
 

Output:

integrate(csc(d*x + c)^2*sin(b*x + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \csc ^2(c+d x) \sin (a+b x) \, dx=\int \frac {\sin \left (a+b\,x\right )}{{\sin \left (c+d\,x\right )}^2} \,d x \] Input:

int(sin(a + b*x)/sin(c + d*x)^2,x)
 

Output:

int(sin(a + b*x)/sin(c + d*x)^2, x)
 

Reduce [F]

\[ \int \csc ^2(c+d x) \sin (a+b x) \, dx=\int \csc \left (d x +c \right )^{2} \sin \left (b x +a \right )d x \] Input:

int(csc(d*x+c)^2*sin(b*x+a),x)
 

Output:

int(csc(c + d*x)**2*sin(a + b*x),x)