\(\int \csc ^4(c+d x) \sin ^3(a+b x) \, dx\) [129]

Optimal result

Integrand size = 17, antiderivative size = 265 \[ \int \csc ^4(c+d x) \sin ^3(a+b x) \, dx=\frac {2 e^{-3 i a-3 i b x+4 i (c+d x)} \operatorname {Hypergeometric2F1}\left (4,2-\frac {3 b}{2 d},3-\frac {3 b}{2 d},e^{2 i (c+d x)}\right )}{3 b-4 d}-\frac {6 e^{-i a-i b x+4 i (c+d x)} \operatorname {Hypergeometric2F1}\left (4,2-\frac {b}{2 d},3-\frac {b}{2 d},e^{2 i (c+d x)}\right )}{b-4 d}-\frac {6 e^{i a+i b x+4 i (c+d x)} \operatorname {Hypergeometric2F1}\left (4,2+\frac {b}{2 d},3+\frac {b}{2 d},e^{2 i (c+d x)}\right )}{b+4 d}+\frac {2 e^{3 i a+3 i b x+4 i (c+d x)} \operatorname {Hypergeometric2F1}\left (4,2+\frac {3 b}{2 d},3+\frac {3 b}{2 d},e^{2 i (c+d x)}\right )}{3 b+4 d} \] Output:

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

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(913\) vs. \(2(265)=530\).

Time = 5.99 (sec) , antiderivative size = 913, normalized size of antiderivative = 3.45 \[ \int \csc ^4(c+d x) \sin ^3(a+b x) \, dx =\text {Too large to display} \] Input:

Integrate[Csc[c + d*x]^4*Sin[a + b*x]^3,x]
 

Output:

((4*(3*b + 2*d)*(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))]))/E^(I*(3*a - 2*c + 3*b 
*x)) - (12*(b + 2*d)*(b*E^((2*I)*d*x)*Hypergeometric2F1[1, 1 - b/(2*d), 2 
- b/(2*d), E^((2*I)*(c + d*x))] - (b - 2*d)*Hypergeometric2F1[1, -1/2*b/d, 
 1 - b/(2*d), E^((2*I)*(c + d*x))]))/E^(I*(a - 2*c + b*x)) + 12*(b - 2*d)* 
E^(I*(a + 2*c + b*x))*(b*E^((2*I)*d*x)*Hypergeometric2F1[1, 1 + b/(2*d), 2 
 + b/(2*d), E^((2*I)*(c + d*x))] - (b + 2*d)*Hypergeometric2F1[1, b/(2*d), 
 1 + b/(2*d), E^((2*I)*(c + d*x))]) - 4*(3*b - 2*d)*E^(I*(3*a + 2*c + 3*b* 
x))*(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))]) + (6*I)*(-1 + E^((2*I)*c))*Csc[c + d 
*x]*(Cos[a] - I*Sin[a])*(Cos[b*x] - I*Sin[b*x])*(d*((-I)*b + 2*d*Cot[c])*C 
sc[c + d*x] + (b^2 - 4*d^2)*Csc[c]*Sin[d*x] - 2*d^2*Csc[c]*Csc[c + d*x]^2* 
Sin[d*x]) + 2*(-1 + E^((2*I)*c))*Csc[c + d*x]*(I*Cos[3*a] + Sin[3*a])*(Cos 
[3*b*x] - I*Sin[3*b*x])*(d*((3*I)*b - 2*d*Cot[c])*Csc[c + d*x] + (-9*b^2 + 
 4*d^2)*Csc[c]*Sin[d*x] + 2*d^2*Csc[c]*Csc[c + d*x]^2*Sin[d*x]) + (I/2)*(- 
1 + E^((2*I)*c))*Csc[c]*Csc[c + d*x]^3*(Cos[3*a] + I*Sin[3*a])*(Cos[3*b*x] 
 + I*Sin[3*b*x])*((6*I)*b*d*Cos[d*x] - (6*I)*b*d*Cos[2*c + d*x] + 18*b^2*S 
in[d*x] - 12*d^2*Sin[d*x] + 9*b^2*Sin[2*c + d*x] - 9*b^2*Sin[2*c + 3*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.

\(\displaystyle \int \sin ^3(a+b x) \csc ^4(c+d x) \, dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \sin ^3(a+b x) \csc ^4(c+d x)dx\)

Input:

Int[Csc[c + d*x]^4*Sin[a + b*x]^3,x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

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

Maple [F]

\[\int \csc \left (d x +c \right )^{4} \sin \left (b x +a \right )^{3}d x\]

Input:

int(csc(d*x+c)^4*sin(b*x+a)^3,x)
 

Output:

int(csc(d*x+c)^4*sin(b*x+a)^3,x)
 

Fricas [F]

\[ \int \csc ^4(c+d x) \sin ^3(a+b x) \, dx=\int { \csc \left (d x + c\right )^{4} \sin \left (b x + a\right )^{3} \,d x } \] Input:

integrate(csc(d*x+c)^4*sin(b*x+a)^3,x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \csc ^4(c+d x) \sin ^3(a+b x) \, dx=\text {Timed out} \] Input:

integrate(csc(d*x+c)**4*sin(b*x+a)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \csc ^4(c+d x) \sin ^3(a+b x) \, dx=\int { \csc \left (d x + c\right )^{4} \sin \left (b x + a\right )^{3} \,d x } \] Input:

integrate(csc(d*x+c)^4*sin(b*x+a)^3,x, algorithm="maxima")
 

Output:

-1/24*(6*(b^2 - b*d - 6*d^2)*cos(2*(b + d)*x + 2*a + 2*c)*cos(3*b*x + 3*a) 
 - (9*b^2 - 4*d^2)*cos(6*b*x + 6*a)*cos(3*b*x + 3*a) + 3*(b^2 - 4*d^2)*cos 
(4*b*x + 4*a)*cos(3*b*x + 3*a) + 3*(3*b^2 - 2*b*d)*cos(3*b*x + 3*a)*cos(4* 
d*x + 4*c) - 6*(3*b^2 - b*d - 2*d^2)*cos(3*b*x + 3*a)*cos(2*d*x + 2*c) + 6 
*(b^2 - b*d - 6*d^2)*sin(2*(b + d)*x + 2*a + 2*c)*sin(3*b*x + 3*a) - (9*b^ 
2 - 4*d^2)*sin(6*b*x + 6*a)*sin(3*b*x + 3*a) + 3*(b^2 - 4*d^2)*sin(4*b*x + 
 4*a)*sin(3*b*x + 3*a) - 3*(b^2 - 4*d^2)*sin(3*b*x + 3*a)*sin(2*b*x + 2*a) 
 + 3*(3*b^2 - 2*b*d)*sin(3*b*x + 3*a)*sin(4*d*x + 4*c) - 6*(3*b^2 - b*d - 
2*d^2)*sin(3*b*x + 3*a)*sin(2*d*x + 2*c) + 3*(9*b^2 - 4*d^2 + 6*(3*b^2 + b 
*d - 2*d^2)*cos(2*(3*b + d)*x + 6*a + 2*c) - 6*(b^2 + b*d - 6*d^2)*cos(2*( 
2*b + d)*x + 4*a + 2*c) - 3*(b^2 - 2*b*d)*cos(2*(b + 2*d)*x + 2*a + 4*c) + 
 6*(b^2 - b*d - 6*d^2)*cos(2*(b + d)*x + 2*a + 2*c) - (9*b^2 - 4*d^2)*cos( 
6*b*x + 6*a) + 3*(b^2 - 4*d^2)*cos(4*b*x + 4*a) - 3*(b^2 - 4*d^2)*cos(2*b* 
x + 2*a) + 3*(3*b^2 - 2*b*d)*cos(4*d*x + 4*c) - 6*(3*b^2 - b*d - 2*d^2)*co 
s(2*d*x + 2*c))*cos((3*b + 4*d)*x + 3*a + 4*c) - 3*(3*(3*b^2 + 2*b*d)*cos( 
(3*b + 4*d)*x + 3*a + 4*c) - 3*(3*b^2 + 2*b*d)*cos((3*b + 2*d)*x + 3*a + 2 
*c) + (3*b^2 + 2*b*d)*cos(3*b*x + 3*a))*cos(2*(3*b + 2*d)*x + 6*a + 4*c) - 
 3*(9*b^2 - 4*d^2 + 6*(b^2 - b*d - 6*d^2)*cos(2*(b + d)*x + 2*a + 2*c) - ( 
9*b^2 - 4*d^2)*cos(6*b*x + 6*a) + 3*(b^2 - 4*d^2)*cos(4*b*x + 4*a) - 3*(b^ 
2 - 4*d^2)*cos(2*b*x + 2*a) + 3*(3*b^2 - 2*b*d)*cos(4*d*x + 4*c) - 6*(3...
 

Giac [F]

\[ \int \csc ^4(c+d x) \sin ^3(a+b x) \, dx=\int { \csc \left (d x + c\right )^{4} \sin \left (b x + a\right )^{3} \,d x } \] Input:

integrate(csc(d*x+c)^4*sin(b*x+a)^3,x, algorithm="giac")
 

Output:

integrate(csc(d*x + c)^4*sin(b*x + a)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \csc ^4(c+d x) \sin ^3(a+b x) \, dx=\int \frac {{\sin \left (a+b\,x\right )}^3}{{\sin \left (c+d\,x\right )}^4} \,d x \] Input:

int(sin(a + b*x)^3/sin(c + d*x)^4,x)
 

Output:

int(sin(a + b*x)^3/sin(c + d*x)^4, x)
 

Reduce [F]

\[ \int \csc ^4(c+d x) \sin ^3(a+b x) \, dx=\int \csc \left (d x +c \right )^{4} \sin \left (b x +a \right )^{3}d x \] Input:

int(csc(d*x+c)^4*sin(b*x+a)^3,x)
 

Output:

int(csc(d*x+c)^4*sin(b*x+a)^3,x)