Integrand size = 13, antiderivative size = 138 \[ \int \csc (c+d x) \sin (a+b x) \, dx=\frac {i e^{-i a-i b x+i (c+d x)} \operatorname {Hypergeometric2F1}\left (1,-\frac {b-d}{2 d},\frac {1}{2} \left (3-\frac {b}{d}\right ),e^{2 i (c+d x)}\right )}{b-d}+\frac {i e^{i a+i b x+i (c+d x)} \operatorname {Hypergeometric2F1}\left (1,\frac {b+d}{2 d},\frac {1}{2} \left (3+\frac {b}{d}\right ),e^{2 i (c+d x)}\right )}{b+d} \] Output:
I*exp(-I*a-I*b*x+I*(d*x+c))*hypergeom([1, -1/2*(b-d)/d],[3/2-1/2*b/d],exp( 2*I*(d*x+c)))/(b-d)+I*exp(I*a+I*b*x+I*(d*x+c))*hypergeom([1, 1/2*(b+d)/d], [3/2+1/2*b/d],exp(2*I*(d*x+c)))/(b+d)
Time = 0.66 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.90 \[ \int \csc (c+d x) \sin (a+b x) \, dx=\frac {i e^{-i (a-c+(b-d) x)} \operatorname {Hypergeometric2F1}\left (1,\frac {-b+d}{2 d},\frac {3}{2}-\frac {b}{2 d},e^{2 i (c+d x)}\right )}{b-d}+\frac {i e^{i (a+c+(b+d) x)} \operatorname {Hypergeometric2F1}\left (1,\frac {b+d}{2 d},\frac {1}{2} \left (3+\frac {b}{d}\right ),e^{2 i (c+d x)}\right )}{b+d} \] Input:
Integrate[Csc[c + d*x]*Sin[a + b*x],x]
Output:
(I*Hypergeometric2F1[1, (-b + d)/(2*d), 3/2 - b/(2*d), E^((2*I)*(c + d*x)) ])/((b - d)*E^(I*(a - c + (b - d)*x))) + (I*E^(I*(a + c + (b + d)*x))*Hype rgeometric2F1[1, (b + d)/(2*d), (3 + b/d)/2, E^((2*I)*(c + d*x))])/(b + d)
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.
\(\displaystyle \int \sin (a+b x) \csc (c+d x) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \sin (a+b x) \csc (c+d x)dx\) |
Input:
Int[Csc[c + d*x]*Sin[a + b*x],x]
Output:
$Aborted
\[\int \csc \left (d x +c \right ) \sin \left (b x +a \right )d x\]
Input:
int(csc(d*x+c)*sin(b*x+a),x)
Output:
int(csc(d*x+c)*sin(b*x+a),x)
\[ \int \csc (c+d x) \sin (a+b x) \, dx=\int { \csc \left (d x + c\right ) \sin \left (b x + a\right ) \,d x } \] Input:
integrate(csc(d*x+c)*sin(b*x+a),x, algorithm="fricas")
Output:
integral(csc(d*x + c)*sin(b*x + a), x)
\[ \int \csc (c+d x) \sin (a+b x) \, dx=\int \sin {\left (a + b x \right )} \csc {\left (c + d x \right )}\, dx \] Input:
integrate(csc(d*x+c)*sin(b*x+a),x)
Output:
Integral(sin(a + b*x)*csc(c + d*x), x)
\[ \int \csc (c+d x) \sin (a+b x) \, dx=\int { \csc \left (d x + c\right ) \sin \left (b x + a\right ) \,d x } \] Input:
integrate(csc(d*x+c)*sin(b*x+a),x, algorithm="maxima")
Output:
integrate(csc(d*x + c)*sin(b*x + a), x)
\[ \int \csc (c+d x) \sin (a+b x) \, dx=\int { \csc \left (d x + c\right ) \sin \left (b x + a\right ) \,d x } \] Input:
integrate(csc(d*x+c)*sin(b*x+a),x, algorithm="giac")
Output:
integrate(csc(d*x + c)*sin(b*x + a), x)
Timed out. \[ \int \csc (c+d x) \sin (a+b x) \, dx=\int \frac {\sin \left (a+b\,x\right )}{\sin \left (c+d\,x\right )} \,d x \] Input:
int(sin(a + b*x)/sin(c + d*x),x)
Output:
int(sin(a + b*x)/sin(c + d*x), x)
\[ \int \csc (c+d x) \sin (a+b x) \, dx=\int \csc \left (d x +c \right ) \sin \left (b x +a \right )d x \] Input:
int(csc(d*x+c)*sin(b*x+a),x)
Output:
int(csc(c + d*x)*sin(a + b*x),x)