\(\int \csc (c+d x) \sin ^2(a+b x) \, dx\) [119]

Optimal result

Integrand size = 15, antiderivative size = 149 \[ \int \csc (c+d x) \sin ^2(a+b x) \, dx=-\frac {\text {arctanh}(\cos (c+d x))}{2 d}-\frac {e^{-2 i a-2 i b x+i (c+d x)} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}-\frac {b}{d},\frac {3}{2}-\frac {b}{d},e^{2 i (c+d x)}\right )}{2 (2 b-d)}+\frac {e^{2 i a+2 i b x+i (c+d x)} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+\frac {b}{d},\frac {3}{2}+\frac {b}{d},e^{2 i (c+d x)}\right )}{2 (2 b+d)} \] Output:

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

Mathematica [A] (verified)

Time = 0.86 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.07 \[ \int \csc (c+d x) \sin ^2(a+b x) \, dx=-\frac {e^{-i (2 a-c+2 b x-d x)} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}-\frac {b}{d},\frac {3}{2}-\frac {b}{d},e^{2 i (c+d x)}\right )}{2 (2 b-d)}+\frac {e^{i (2 a+c+2 b x+d x)} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+\frac {b}{d},\frac {3}{2}+\frac {b}{d},e^{2 i (c+d x)}\right )}{2 (2 b+d)}+\frac {-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d} \] Input:

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

Output:

-1/2*Hypergeometric2F1[1, 1/2 - b/d, 3/2 - b/d, E^((2*I)*(c + d*x))]/((2*b 
 - d)*E^(I*(2*a - c + 2*b*x - d*x))) + (E^(I*(2*a + c + 2*b*x + d*x))*Hype 
rgeometric2F1[1, 1/2 + b/d, 3/2 + b/d, E^((2*I)*(c + d*x))])/(2*(2*b + d)) 
 + (-Log[Cos[(c + d*x)/2]] + Log[Sin[(c + d*x)/2]])/(2*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.

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(-(cos(b*x + a)^2 - 1)*csc(d*x + c), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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