\(\int \csc ^2(c+d x) \sin ^3(a+b x) \, dx\) [127]

Optimal result

Integrand size = 17, antiderivative size = 273 \[ \int \csc ^2(c+d x) \sin ^3(a+b x) \, dx=-\frac {e^{-3 i a-3 i b x+2 i (c+d x)} \operatorname {Hypergeometric2F1}\left (2,1-\frac {3 b}{2 d},2-\frac {3 b}{2 d},e^{2 i (c+d x)}\right )}{2 (3 b-2 d)}+\frac {3 e^{-i a-i b x+2 i (c+d x)} \operatorname {Hypergeometric2F1}\left (2,1-\frac {b}{2 d},2-\frac {b}{2 d},e^{2 i (c+d x)}\right )}{2 (b-2 d)}+\frac {3 e^{i a+i b x+2 i (c+d x)} \operatorname {Hypergeometric2F1}\left (2,1+\frac {b}{2 d},2+\frac {b}{2 d},e^{2 i (c+d x)}\right )}{2 (b+2 d)}-\frac {e^{3 i a+3 i b x+2 i (c+d x)} \operatorname {Hypergeometric2F1}\left (2,1+\frac {3 b}{2 d},2+\frac {3 b}{2 d},e^{2 i (c+d x)}\right )}{2 (3 b+2 d)} \] Output:

-1/2*exp(-3*I*a-3*I*b*x+2*I*(d*x+c))*hypergeom([2, 1-3/2*b/d],[2-3/2*b/d], 
exp(2*I*(d*x+c)))/(3*b-2*d)+3*exp(-I*a-I*b*x+2*I*(d*x+c))*hypergeom([2, 1- 
1/2*b/d],[2-1/2*b/d],exp(2*I*(d*x+c)))/(2*b-4*d)+3*exp(I*a+I*b*x+2*I*(d*x+ 
c))*hypergeom([2, 1+1/2*b/d],[2+1/2*b/d],exp(2*I*(d*x+c)))/(2*b+4*d)-exp(3 
*I*a+3*I*b*x+2*I*(d*x+c))*hypergeom([2, 1+3/2*b/d],[2+3/2*b/d],exp(2*I*(d* 
x+c)))/(6*b+4*d)
 

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(570\) vs. \(2(273)=546\).

Time = 4.03 (sec) , antiderivative size = 570, normalized size of antiderivative = 2.09 \[ \int \csc ^2(c+d x) \sin ^3(a+b x) \, dx=\frac {\frac {e^{-i (3 a-2 c+3 b x)} \left (3 b e^{2 i d x} \operatorname {Hypergeometric2F1}\left (1,1-\frac {3 b}{2 d},2-\frac {3 b}{2 d},e^{2 i (c+d x)}\right )+(-3 b+2 d) \operatorname {Hypergeometric2F1}\left (1,-\frac {3 b}{2 d},1-\frac {3 b}{2 d},e^{2 i (c+d x)}\right )\right )}{(3 b-2 d) \left (-1+e^{2 i c}\right )}+\frac {e^{-i (a-2 c+b x)} \left (-3 b e^{2 i d x} \operatorname {Hypergeometric2F1}\left (1,1-\frac {b}{2 d},2-\frac {b}{2 d},e^{2 i (c+d x)}\right )+3 (b-2 d) \operatorname {Hypergeometric2F1}\left (1,-\frac {b}{2 d},1-\frac {b}{2 d},e^{2 i (c+d x)}\right )\right )}{(b-2 d) \left (-1+e^{2 i c}\right )}+\frac {3 e^{i (a+2 c+b x)} \left (b e^{2 i d x} \operatorname {Hypergeometric2F1}\left (1,1+\frac {b}{2 d},2+\frac {b}{2 d},e^{2 i (c+d x)}\right )-(b+2 d) \operatorname {Hypergeometric2F1}\left (1,\frac {b}{2 d},1+\frac {b}{2 d},e^{2 i (c+d x)}\right )\right )}{(b+2 d) \left (-1+e^{2 i c}\right )}+\frac {e^{i (3 a+2 c+3 b x)} \left (-3 b e^{2 i d x} \operatorname {Hypergeometric2F1}\left (1,1+\frac {3 b}{2 d},2+\frac {3 b}{2 d},e^{2 i (c+d x)}\right )+(3 b+2 d) \operatorname {Hypergeometric2F1}\left (1,\frac {3 b}{2 d},1+\frac {3 b}{2 d},e^{2 i (c+d x)}\right )\right )}{(3 b+2 d) \left (-1+e^{2 i c}\right )}+3 \cos (b x) \csc (c) \csc (c+d x) \sin (a) \sin (d x)-\cos (3 b x) \csc (c) \csc (c+d x) \sin (3 a) \sin (d x)+3 \cos (a) \csc (c) \csc (c+d x) \sin (b x) \sin (d x)-\cos (3 a) \csc (c) \csc (c+d x) \sin (3 b x) \sin (d x)}{4 d} \] Input:

Integrate[Csc[c + d*x]^2*Sin[a + b*x]^3,x]
 

Output:

((3*b*E^((2*I)*d*x)*Hypergeometric2F1[1, 1 - (3*b)/(2*d), 2 - (3*b)/(2*d), 
 E^((2*I)*(c + d*x))] + (-3*b + 2*d)*Hypergeometric2F1[1, (-3*b)/(2*d), 1 
- (3*b)/(2*d), E^((2*I)*(c + d*x))])/((3*b - 2*d)*E^(I*(3*a - 2*c + 3*b*x) 
)*(-1 + E^((2*I)*c))) + (-3*b*E^((2*I)*d*x)*Hypergeometric2F1[1, 1 - b/(2* 
d), 2 - b/(2*d), E^((2*I)*(c + d*x))] + 3*(b - 2*d)*Hypergeometric2F1[1, - 
1/2*b/d, 1 - b/(2*d), E^((2*I)*(c + d*x))])/((b - 2*d)*E^(I*(a - 2*c + b*x 
))*(-1 + E^((2*I)*c))) + (3*E^(I*(a + 2*c + b*x))*(b*E^((2*I)*d*x)*Hyperge 
ometric2F1[1, 1 + b/(2*d), 2 + b/(2*d), E^((2*I)*(c + d*x))] - (b + 2*d)*H 
ypergeometric2F1[1, b/(2*d), 1 + b/(2*d), E^((2*I)*(c + d*x))]))/((b + 2*d 
)*(-1 + E^((2*I)*c))) + (E^(I*(3*a + 2*c + 3*b*x))*(-3*b*E^((2*I)*d*x)*Hyp 
ergeometric2F1[1, 1 + (3*b)/(2*d), 2 + (3*b)/(2*d), E^((2*I)*(c + d*x))] + 
 (3*b + 2*d)*Hypergeometric2F1[1, (3*b)/(2*d), 1 + (3*b)/(2*d), E^((2*I)*( 
c + d*x))]))/((3*b + 2*d)*(-1 + E^((2*I)*c))) + 3*Cos[b*x]*Csc[c]*Csc[c + 
d*x]*Sin[a]*Sin[d*x] - Cos[3*b*x]*Csc[c]*Csc[c + d*x]*Sin[3*a]*Sin[d*x] + 
3*Cos[a]*Csc[c]*Csc[c + d*x]*Sin[b*x]*Sin[d*x] - Cos[3*a]*Csc[c]*Csc[c + d 
*x]*Sin[3*b*x]*Sin[d*x])/(4*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.

\(\displaystyle \int \sin ^3(a+b x) \csc ^2(c+d x) \, dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \sin ^3(a+b x) \csc ^2(c+d x)dx\)

Input:

Int[Csc[c + d*x]^2*Sin[a + b*x]^3,x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

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

Maple [F]

\[\int \csc \left (d x +c \right )^{2} \sin \left (b x +a \right )^{3}d x\]

Input:

int(csc(d*x+c)^2*sin(b*x+a)^3,x)
 

Output:

int(csc(d*x+c)^2*sin(b*x+a)^3,x)
 

Fricas [F]

\[ \int \csc ^2(c+d x) \sin ^3(a+b x) \, dx=\int { \csc \left (d x + c\right )^{2} \sin \left (b x + a\right )^{3} \,d x } \] Input:

integrate(csc(d*x+c)^2*sin(b*x+a)^3,x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \csc ^2(c+d x) \sin ^3(a+b x) \, dx=\text {Timed out} \] Input:

integrate(csc(d*x+c)**2*sin(b*x+a)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \csc ^2(c+d x) \sin ^3(a+b x) \, dx=\int { \csc \left (d x + c\right )^{2} \sin \left (b x + a\right )^{3} \,d x } \] Input:

integrate(csc(d*x+c)^2*sin(b*x+a)^3,x, algorithm="maxima")
 

Output:

1/4*((cos(6*b*x + 6*a) - 3*cos(4*b*x + 4*a) + 3*cos(2*b*x + 2*a) - 1)*cos( 
(3*b + 2*d)*x + 3*a + 2*c) - (3*cos(2*b*x + 2*a) - 1)*cos(3*b*x + 3*a) - c 
os(6*b*x + 6*a)*cos(3*b*x + 3*a) + 3*cos(4*b*x + 4*a)*cos(3*b*x + 3*a) + 4 
*(d*cos((3*b + 2*d)*x + 3*a + 2*c)^2 - 2*d*cos((3*b + 2*d)*x + 3*a + 2*c)* 
cos(3*b*x + 3*a) + d*cos(3*b*x + 3*a)^2 + d*sin((3*b + 2*d)*x + 3*a + 2*c) 
^2 - 2*d*sin((3*b + 2*d)*x + 3*a + 2*c)*sin(3*b*x + 3*a) + d*sin(3*b*x + 3 
*a)^2)*integrate(-3/8*(b*cos(3*b*x + 3*a)*sin(6*b*x + 6*a) - b*cos(3*b*x + 
 3*a)*sin(4*b*x + 4*a) - b*cos(6*b*x + 6*a)*sin(3*b*x + 3*a) + b*cos(4*b*x 
 + 4*a)*sin(3*b*x + 3*a) - b*cos(3*b*x + 3*a)*sin(2*b*x + 2*a) + (b*sin(6* 
b*x + 6*a) - b*sin(4*b*x + 4*a) - b*sin(2*b*x + 2*a))*cos((3*b + d)*x + 3* 
a + c) - (b*cos(6*b*x + 6*a) - b*cos(4*b*x + 4*a) - b*cos(2*b*x + 2*a) + b 
)*sin((3*b + d)*x + 3*a + c) + (b*cos(2*b*x + 2*a) - b)*sin(3*b*x + 3*a))/ 
(d*cos((3*b + d)*x + 3*a + c)^2 + 2*d*cos((3*b + d)*x + 3*a + c)*cos(3*b*x 
 + 3*a) + d*cos(3*b*x + 3*a)^2 + d*sin((3*b + d)*x + 3*a + c)^2 + 2*d*sin( 
(3*b + d)*x + 3*a + c)*sin(3*b*x + 3*a) + d*sin(3*b*x + 3*a)^2), x) - 4*(d 
*cos((3*b + 2*d)*x + 3*a + 2*c)^2 - 2*d*cos((3*b + 2*d)*x + 3*a + 2*c)*cos 
(3*b*x + 3*a) + d*cos(3*b*x + 3*a)^2 + d*sin((3*b + 2*d)*x + 3*a + 2*c)^2 
- 2*d*sin((3*b + 2*d)*x + 3*a + 2*c)*sin(3*b*x + 3*a) + d*sin(3*b*x + 3*a) 
^2)*integrate(3/8*(b*cos(3*b*x + 3*a)*sin(6*b*x + 6*a) - b*cos(3*b*x + 3*a 
)*sin(4*b*x + 4*a) - b*cos(6*b*x + 6*a)*sin(3*b*x + 3*a) + b*cos(4*b*x ...
 

Giac [F]

\[ \int \csc ^2(c+d x) \sin ^3(a+b x) \, dx=\int { \csc \left (d x + c\right )^{2} \sin \left (b x + a\right )^{3} \,d x } \] Input:

integrate(csc(d*x+c)^2*sin(b*x+a)^3,x, algorithm="giac")
 

Output:

integrate(csc(d*x + c)^2*sin(b*x + a)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \csc ^2(c+d x) \sin ^3(a+b x) \, dx=\int \frac {{\sin \left (a+b\,x\right )}^3}{{\sin \left (c+d\,x\right )}^2} \,d x \] Input:

int(sin(a + b*x)^3/sin(c + d*x)^2,x)
 

Output:

int(sin(a + b*x)^3/sin(c + d*x)^2, x)
 

Reduce [F]

\[ \int \csc ^2(c+d x) \sin ^3(a+b x) \, dx=\int \csc \left (d x +c \right )^{2} \sin \left (b x +a \right )^{3}d x \] Input:

int(csc(d*x+c)^2*sin(b*x+a)^3,x)
 

Output:

int(csc(c + d*x)**2*sin(a + b*x)**3,x)