\(\int \cos ^3(a+b x) \csc ^3(c+d x) \, dx\) [215]

Optimal result

Integrand size = 17, antiderivative size = 275 \[ \int \cos ^3(a+b x) \csc ^3(c+d x) \, dx=-\frac {e^{-3 i a-3 i b x+3 i (c+d x)} \operatorname {Hypergeometric2F1}\left (3,\frac {3}{2} \left (1-\frac {b}{d}\right ),\frac {1}{2} \left (5-\frac {3 b}{d}\right ),e^{2 i (c+d x)}\right )}{3 (b-d)}-\frac {3 e^{-i a-i b x+3 i (c+d x)} \operatorname {Hypergeometric2F1}\left (3,\frac {1}{2} \left (3-\frac {b}{d}\right ),\frac {1}{2} \left (5-\frac {b}{d}\right ),e^{2 i (c+d x)}\right )}{b-3 d}+\frac {3 e^{i a+i b x+3 i (c+d x)} \operatorname {Hypergeometric2F1}\left (3,\frac {1}{2} \left (3+\frac {b}{d}\right ),\frac {1}{2} \left (5+\frac {b}{d}\right ),e^{2 i (c+d x)}\right )}{b+3 d}+\frac {e^{3 i a+3 i b x+3 i (c+d x)} \operatorname {Hypergeometric2F1}\left (3,\frac {3 (b+d)}{2 d},\frac {1}{2} \left (5+\frac {3 b}{d}\right ),e^{2 i (c+d x)}\right )}{3 (b+d)} \] Output:

-1/3*exp(-3*I*a-3*I*b*x+3*I*(d*x+c))*hypergeom([3, 3/2-3/2*b/d],[5/2-3/2*b 
/d],exp(2*I*(d*x+c)))/(b-d)-3*exp(-I*a-I*b*x+3*I*(d*x+c))*hypergeom([3, 3/ 
2-1/2*b/d],[5/2-1/2*b/d],exp(2*I*(d*x+c)))/(b-3*d)+3*exp(I*a+I*b*x+3*I*(d* 
x+c))*hypergeom([3, 3/2+1/2*b/d],[5/2+1/2*b/d],exp(2*I*(d*x+c)))/(b+3*d)+e 
xp(3*I*a+3*I*b*x+3*I*(d*x+c))*hypergeom([3, 3/2*(b+d)/d],[5/2+3/2*b/d],exp 
(2*I*(d*x+c)))/(3*b+3*d)
 

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1795\) vs. \(2(275)=550\).

Time = 12.27 (sec) , antiderivative size = 1795, normalized size of antiderivative = 6.53 \[ \int \cos ^3(a+b x) \csc ^3(c+d x) \, dx =\text {Too large to display} \] Input:

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

Output:

-1/4*(Cos[a + b*x]^2*((-3*b + d)*Cos[a - c + b*x - d*x] + (3*b + d)*Cos[a 
+ c + (b + d)*x])*Csc[c + d*x]^2)/d^2 + ((-4*E^(((3*I)/2)*c)*(-1 + E^((2*I 
)*(a + b*x)))*(9*b^2*(1 + E^((2*I)*(a + b*x)))^2 - d^2*(1 + 10*E^((2*I)*(a 
 + b*x)) + E^((4*I)*(a + b*x)))))/(-1 + E^((2*I)*c)) + ((9*I)*b^2*E^(I*c)* 
Csc[c/2]*(3*b*E^(I*(6*a + (6*b + d)*x))*Hypergeometric2F1[1, 1 + (3*b)/d, 
2 + (3*b)/d, E^(I*(c + d*x))] + (3*b + d)*Hypergeometric2F1[1, (-3*b)/d, 1 
 - (3*b)/d, E^(I*(c + d*x))]))/(3*b + d) - (I*d^2*E^(I*c)*Csc[c/2]*(3*b*E^ 
(I*(6*a + (6*b + d)*x))*Hypergeometric2F1[1, 1 + (3*b)/d, 2 + (3*b)/d, E^( 
I*(c + d*x))] + (3*b + d)*Hypergeometric2F1[1, (-3*b)/d, 1 - (3*b)/d, E^(I 
*(c + d*x))]))/(3*b + d) - (9*I)*b^2*E^((2*I)*(a + b*x))*Csc[c/2]*(-1 + Hy 
pergeometric2F1[1, -(b/d), 1 - b/d, E^(I*(c + d*x))] + E^(I*(2*a + c + 2*b 
*x))*Hypergeometric2F1[1, b/d, (b + d)/d, E^(I*(c + d*x))]) + (9*I)*d^2*E^ 
((2*I)*(a + b*x))*Csc[c/2]*(-1 + Hypergeometric2F1[1, -(b/d), 1 - b/d, E^( 
I*(c + d*x))] + E^(I*(2*a + c + 2*b*x))*Hypergeometric2F1[1, b/d, (b + d)/ 
d, E^(I*(c + d*x))]) - (9*I)*b^2*Csc[c/2]*(-1 + Hypergeometric2F1[1, (-3*b 
)/d, 1 - (3*b)/d, E^(I*(c + d*x))] + E^(I*(6*a + c + 6*b*x))*Hypergeometri 
c2F1[1, (3*b)/d, 1 + (3*b)/d, E^(I*(c + d*x))]) + I*d^2*Csc[c/2]*(-1 + Hyp 
ergeometric2F1[1, (-3*b)/d, 1 - (3*b)/d, E^(I*(c + d*x))] + E^(I*(6*a + c 
+ 6*b*x))*Hypergeometric2F1[1, (3*b)/d, 1 + (3*b)/d, E^(I*(c + d*x))]) + ( 
(9*I)*b^2*E^(I*(2*a + c + 2*b*x))*Csc[c/2]*((b + d)*Hypergeometric2F1[1...
 

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.

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(cos(b*x + a)^3*csc(d*x + c)^3, x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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