\(\int \csc ^2(c+d x) \sin ^2(a+b x) \, dx\) [120]

Optimal result

Integrand size = 17, antiderivative size = 140 \[ \int \csc ^2(c+d x) \sin ^2(a+b x) \, dx=-\frac {\cot (c+d x)}{2 d}+\frac {i e^{-2 i a-2 i b x+2 i (c+d x)} \operatorname {Hypergeometric2F1}\left (2,1-\frac {b}{d},2-\frac {b}{d},e^{2 i (c+d x)}\right )}{2 (b-d)}-\frac {i e^{2 i a+2 i b x+2 i (c+d x)} \operatorname {Hypergeometric2F1}\left (2,\frac {b+d}{d},2+\frac {b}{d},e^{2 i (c+d x)}\right )}{2 (b+d)} \] Output:

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

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(290\) vs. \(2(140)=280\).

Time = 3.30 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.07 \[ \int \csc ^2(c+d x) \sin ^2(a+b x) \, dx=\frac {-\frac {i b e^{-2 i (a-c)} \left (\frac {e^{-2 i (b-d) x} \operatorname {Hypergeometric2F1}\left (1,1-\frac {b}{d},2-\frac {b}{d},e^{2 i (c+d x)}\right )}{b-d}-\frac {e^{-2 i b x} \operatorname {Hypergeometric2F1}\left (1,-\frac {b}{d},1-\frac {b}{d},e^{2 i (c+d x)}\right )}{b}\right )}{-1+e^{2 i c}}+\frac {i e^{2 i (a+c)} \left ((b+d) e^{2 i b x} \operatorname {Hypergeometric2F1}\left (1,\frac {b}{d},\frac {b+d}{d},e^{2 i (c+d x)}\right )-b e^{2 i (b+d) x} \operatorname {Hypergeometric2F1}\left (1,\frac {b+d}{d},2+\frac {b}{d},e^{2 i (c+d x)}\right )\right )}{(b+d) \left (-1+e^{2 i c}\right )}+\csc (c) \csc (c+d x) \sin (d x)-\cos (2 a) \cos (2 b x) \csc (c) \csc (c+d x) \sin (d x)+\csc (c) \csc (c+d x) \sin (2 a) \sin (2 b x) \sin (d x)}{2 d} \] Input:

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

Output:

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

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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