\(\int \cos ^2(a+b x) \csc (c+d x) \, dx\) [204]

Optimal result

Integrand size = 15, antiderivative size = 149 \[ \int \cos ^2(a+b x) \csc (c+d x) \, dx=-\frac {\text {arctanh}(\cos (c+d x))}{2 d}+\frac {e^{-2 i a-2 i b x+i (c+d x)} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}-\frac {b}{d},\frac {3}{2}-\frac {b}{d},e^{2 i (c+d x)}\right )}{2 (2 b-d)}-\frac {e^{2 i a+2 i b x+i (c+d x)} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+\frac {b}{d},\frac {3}{2}+\frac {b}{d},e^{2 i (c+d x)}\right )}{2 (2 b+d)} \] 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 = 13.36 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.43 \[ \int \cos ^2(a+b x) \csc (c+d x) \, dx=\frac {e^{-2 i (a+b x)} \left (-d \operatorname {Hypergeometric2F1}\left (1,-\frac {2 b}{d},1-\frac {2 b}{d},-e^{i (c+d x)}\right )+d \operatorname {Hypergeometric2F1}\left (1,-\frac {2 b}{d},1-\frac {2 b}{d},e^{i (c+d x)}\right )+d e^{4 i (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {2 b}{d},1+\frac {2 b}{d},-e^{i (c+d x)}\right )-d e^{4 i (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {2 b}{d},1+\frac {2 b}{d},e^{i (c+d x)}\right )+4 b e^{2 i (a+b x)} \left (\log \left (1-e^{i (c+d x)}\right )-\log \left (1+e^{i (c+d x)}\right )\right )\right )}{8 b d} \] Input:

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

Output:

(-(d*Hypergeometric2F1[1, (-2*b)/d, 1 - (2*b)/d, -E^(I*(c + d*x))]) + d*Hy 
pergeometric2F1[1, (-2*b)/d, 1 - (2*b)/d, E^(I*(c + d*x))] + d*E^((4*I)*(a 
 + b*x))*Hypergeometric2F1[1, (2*b)/d, 1 + (2*b)/d, -E^(I*(c + d*x))] - d* 
E^((4*I)*(a + b*x))*Hypergeometric2F1[1, (2*b)/d, 1 + (2*b)/d, E^(I*(c + d 
*x))] + 4*b*E^((2*I)*(a + b*x))*(Log[1 - E^(I*(c + d*x))] - Log[1 + E^(I*( 
c + d*x))]))/(8*b*d*E^((2*I)*(a + b*x)))
 

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.

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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