\(\int \csc ^4(c+d x) \sin ^2(a+b x) \, dx\) [122]

Optimal result

Integrand size = 17, antiderivative size = 153 \[ \int \csc ^4(c+d x) \sin ^2(a+b x) \, dx=-\frac {\cot (c+d x)}{2 d}-\frac {\cot ^3(c+d x)}{6 d}-\frac {2 i e^{-2 i a-2 i b x+4 i (c+d x)} \operatorname {Hypergeometric2F1}\left (4,2-\frac {b}{d},3-\frac {b}{d},e^{2 i (c+d x)}\right )}{b-2 d}+\frac {2 i e^{2 i a+2 i b x+4 i (c+d x)} \operatorname {Hypergeometric2F1}\left (4,2+\frac {b}{d},3+\frac {b}{d},e^{2 i (c+d x)}\right )}{b+2 d} \] Output:

-1/2*cot(d*x+c)/d-1/6*cot(d*x+c)^3/d-2*I*exp(-2*I*a-2*I*b*x+4*I*(d*x+c))*h 
ypergeom([4, 2-b/d],[3-b/d],exp(2*I*(d*x+c)))/(b-2*d)+2*I*exp(2*I*a+2*I*b* 
x+4*I*(d*x+c))*hypergeom([4, 2+b/d],[3+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. \(563\) vs. \(2(153)=306\).

Time = 2.82 (sec) , antiderivative size = 563, normalized size of antiderivative = 3.68 \[ \int \csc ^4(c+d x) \sin ^2(a+b x) \, dx=-\frac {i \left (16 (b+d) e^{-2 i (a-c+2 b x-d x)} \left (b e^{2 i b x} \operatorname {Hypergeometric2F1}\left (1,1-\frac {b}{d},2-\frac {b}{d},e^{2 i (c+d x)}\right )-(b-d) e^{2 i (b-d) x} \operatorname {Hypergeometric2F1}\left (1,-\frac {b}{d},1-\frac {b}{d},e^{2 i (c+d x)}\right )\right )-16 (b-d) e^{2 i (a+c)} \left ((b+d) e^{2 i b x} \operatorname {Hypergeometric2F1}\left (1,\frac {b}{d},\frac {b+d}{d},e^{2 i (c+d x)}\right )-b e^{2 i (b+d) x} \operatorname {Hypergeometric2F1}\left (1,\frac {b+d}{d},2+\frac {b}{d},e^{2 i (c+d x)}\right )\right )-2 i \left (-1+e^{2 i c}\right ) \csc (c) \csc ^3(c+d x) (\cos (2 (a+b x))-i \sin (2 (a+b x))) (\cos (2 (a+b x))+i \sin (2 (a+b x))) \left (6 d^2 \sin (d x)+\left (b^2-d^2\right ) \sin (2 a-2 c+2 b x-3 d x)-2 b^2 \sin (2 a+2 b x-d x)-b d \sin (2 a+2 b x-d x)+3 d^2 \sin (2 a+2 b x-d x)-b^2 \sin (2 a-2 c+2 b x-d x)+b d \sin (2 a-2 c+2 b x-d x)+2 b^2 \sin (2 a+2 b x+d x)-b d \sin (2 a+2 b x+d x)-3 d^2 \sin (2 a+2 b x+d x)+b^2 \sin (2 a+2 c+2 b x+d x)+b d \sin (2 a+2 c+2 b x+d x)-2 d^2 \sin (2 c+3 d x)-b^2 \sin (2 a+2 c+2 b x+3 d x)+d^2 \sin (2 a+2 c+2 b x+3 d x)\right )\right )}{48 d^3 \left (1-e^{2 i c}\right )} \] Input:

Integrate[Csc[c + d*x]^4*Sin[a + b*x]^2,x]
 

Output:

((-1/48*I)*((16*(b + d)*(b*E^((2*I)*b*x)*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))]))/E^((2*I)*(a - c + 2*b*x - d*x) 
) - 16*(b - d)*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)*Hypergeome 
tric2F1[1, (b + d)/d, 2 + b/d, E^((2*I)*(c + d*x))]) - (2*I)*(-1 + E^((2*I 
)*c))*Csc[c]*Csc[c + d*x]^3*(Cos[2*(a + b*x)] - I*Sin[2*(a + b*x)])*(Cos[2 
*(a + b*x)] + I*Sin[2*(a + b*x)])*(6*d^2*Sin[d*x] + (b^2 - d^2)*Sin[2*a - 
2*c + 2*b*x - 3*d*x] - 2*b^2*Sin[2*a + 2*b*x - d*x] - b*d*Sin[2*a + 2*b*x 
- d*x] + 3*d^2*Sin[2*a + 2*b*x - d*x] - b^2*Sin[2*a - 2*c + 2*b*x - d*x] + 
 b*d*Sin[2*a - 2*c + 2*b*x - d*x] + 2*b^2*Sin[2*a + 2*b*x + d*x] - b*d*Sin 
[2*a + 2*b*x + d*x] - 3*d^2*Sin[2*a + 2*b*x + d*x] + b^2*Sin[2*a + 2*c + 2 
*b*x + d*x] + b*d*Sin[2*a + 2*c + 2*b*x + d*x] - 2*d^2*Sin[2*c + 3*d*x] - 
b^2*Sin[2*a + 2*c + 2*b*x + 3*d*x] + d^2*Sin[2*a + 2*c + 2*b*x + 3*d*x]))) 
/(d^3*(1 - E^((2*I)*c)))
 

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.

\(\displaystyle \int \sin ^2(a+b x) \csc ^4(c+d x) \, dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \sin ^2(a+b x) \csc ^4(c+d x)dx\)

Input:

Int[Csc[c + d*x]^4*Sin[a + b*x]^2,x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

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

Maple [F]

\[\int \csc \left (d x +c \right )^{4} \sin \left (b x +a \right )^{2}d x\]

Input:

int(csc(d*x+c)^4*sin(b*x+a)^2,x)
 

Output:

int(csc(d*x+c)^4*sin(b*x+a)^2,x)
 

Fricas [F]

\[ \int \csc ^4(c+d x) \sin ^2(a+b x) \, dx=\int { \csc \left (d x + c\right )^{4} \sin \left (b x + a\right )^{2} \,d x } \] Input:

integrate(csc(d*x+c)^4*sin(b*x+a)^2,x, algorithm="fricas")
 

Output:

integral(-(cos(b*x + a)^2 - 1)*csc(d*x + c)^4, x)
 

Sympy [F(-1)]

Timed out. \[ \int \csc ^4(c+d x) \sin ^2(a+b x) \, dx=\text {Timed out} \] Input:

integrate(csc(d*x+c)**4*sin(b*x+a)**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \csc ^4(c+d x) \sin ^2(a+b x) \, dx=\int { \csc \left (d x + c\right )^{4} \sin \left (b x + a\right )^{2} \,d x } \] Input:

integrate(csc(d*x+c)^4*sin(b*x+a)^2,x, algorithm="maxima")
 

Output:

-1/3*((b^2 - d^2)*cos(2*b*x + 2*a)*sin(4*b*x + 4*a) - (b^2 - d^2)*cos(4*b* 
x + 4*a)*sin(2*b*x + 2*a) - (b^2 - b*d)*cos(4*d*x + 4*c)*sin(2*b*x + 2*a) 
+ (2*b^2 - b*d - 3*d^2)*cos(2*d*x + 2*c)*sin(2*b*x + 2*a) + (b^2 - b*d)*co 
s(2*b*x + 2*a)*sin(4*d*x + 4*c) - (2*b^2 - b*d - 3*d^2)*cos(2*b*x + 2*a)*s 
in(2*d*x + 2*c) - (3*(2*b^2 + b*d - 3*d^2)*sin(2*(b + d)*x + 2*a + 2*c) - 
(2*b^2 + b*d - 3*d^2)*sin(2*b*x + 2*a))*cos(2*(2*b + d)*x + 4*a + 2*c) + ( 
6*d^2*sin(2*(b + d)*x + 2*a + 2*c) - 2*d^2*sin(2*b*x + 2*a) + (2*b^2 + b*d 
 - 3*d^2)*sin(2*(2*b + d)*x + 4*a + 2*c) - (b^2 + b*d)*sin(4*(b + d)*x + 4 
*a + 4*c) - (b^2 - d^2)*sin(4*b*x + 4*a) - (b^2 - b*d)*sin(4*d*x + 4*c) + 
(2*b^2 - b*d - 3*d^2)*sin(2*d*x + 2*c))*cos(2*(b + 3*d)*x + 2*a + 6*c) - 3 
*(6*d^2*sin(2*(b + d)*x + 2*a + 2*c) - 2*d^2*sin(2*b*x + 2*a) + (2*b^2 + b 
*d - 3*d^2)*sin(2*(2*b + d)*x + 4*a + 2*c) - (b^2 - d^2)*sin(4*b*x + 4*a) 
- (b^2 - b*d)*sin(4*d*x + 4*c) + (2*b^2 - b*d - 3*d^2)*sin(2*d*x + 2*c))*c 
os(2*(b + 2*d)*x + 2*a + 4*c) - (3*(b^2 + b*d)*sin(2*(b + 2*d)*x + 2*a + 4 
*c) - 3*(b^2 + b*d)*sin(2*(b + d)*x + 2*a + 2*c) + (b^2 + b*d)*sin(2*b*x + 
 2*a))*cos(4*(b + d)*x + 4*a + 4*c) - 3*((b^2 - d^2)*sin(4*b*x + 4*a) + (b 
^2 - b*d)*sin(4*d*x + 4*c) - (2*b^2 - b*d - 3*d^2)*sin(2*d*x + 2*c))*cos(2 
*(b + d)*x + 2*a + 2*c) + 3*(d^3*cos(2*(b + 3*d)*x + 2*a + 6*c)^2 + 9*d^3* 
cos(2*(b + 2*d)*x + 2*a + 4*c)^2 + 9*d^3*cos(2*(b + d)*x + 2*a + 2*c)^2 - 
6*d^3*cos(2*(b + d)*x + 2*a + 2*c)*cos(2*b*x + 2*a) + d^3*cos(2*b*x + 2...
 

Giac [F]

\[ \int \csc ^4(c+d x) \sin ^2(a+b x) \, dx=\int { \csc \left (d x + c\right )^{4} \sin \left (b x + a\right )^{2} \,d x } \] Input:

integrate(csc(d*x+c)^4*sin(b*x+a)^2,x, algorithm="giac")
 

Output:

integrate(csc(d*x + c)^4*sin(b*x + a)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \csc ^4(c+d x) \sin ^2(a+b x) \, dx=\int \frac {{\sin \left (a+b\,x\right )}^2}{{\sin \left (c+d\,x\right )}^4} \,d x \] Input:

int(sin(a + b*x)^2/sin(c + d*x)^4,x)
 

Output:

int(sin(a + b*x)^2/sin(c + d*x)^4, x)
 

Reduce [F]

\[ \int \csc ^4(c+d x) \sin ^2(a+b x) \, dx=\int \csc \left (d x +c \right )^{4} \sin \left (b x +a \right )^{2}d x \] Input:

int(csc(d*x+c)^4*sin(b*x+a)^2,x)
 

Output:

int(csc(c + d*x)**4*sin(a + b*x)**2,x)