\(\int \cos (a+b x) \cos ^q(c+d x) \, dx\) [360]

Optimal result

Integrand size = 15, antiderivative size = 234 \[ \int \cos (a+b x) \cos ^q(c+d x) \, dx=-\frac {i e^{i (a-c q)+i (b-d q) x+i q (c+d x)} \left (1+e^{2 i (c+d x)}\right )^{-q} \cos ^q(c+d x) \operatorname {Hypergeometric2F1}\left (-q,\frac {b-d q}{2 d},\frac {1}{2} \left (2+\frac {b}{d}-q\right ),-e^{2 i (c+d x)}\right )}{2 (b-d q)}+\frac {i e^{-i (a+c q)-i (b+d q) x+i q (c+d x)} \left (1+e^{2 i (c+d x)}\right )^{-q} \cos ^q(c+d x) \operatorname {Hypergeometric2F1}\left (-q,-\frac {b+d q}{2 d},-\frac {b-d (2-q)}{2 d},-e^{2 i (c+d x)}\right )}{2 (b+d q)} \] Output:

-1/2*I*exp(I*(-c*q+a)+I*(-d*q+b)*x+I*q*(d*x+c))*cos(d*x+c)^q*hypergeom([-q 
, 1/2*(-d*q+b)/d],[1+1/2*b/d-1/2*q],-exp(2*I*(d*x+c)))/((1+exp(2*I*(d*x+c) 
))^q)/(-d*q+b)+1/2*I*exp(-I*(c*q+a)-I*(d*q+b)*x+I*q*(d*x+c))*cos(d*x+c)^q* 
hypergeom([-q, -1/2*(d*q+b)/d],[-1/2*(b-d*(2-q))/d],-exp(2*I*(d*x+c)))/((1 
+exp(2*I*(d*x+c)))^q)/(d*q+b)
 

Mathematica [A] (warning: unable to verify)

Time = 1.43 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.79 \[ \int \cos (a+b x) \cos ^q(c+d x) \, dx=-\frac {i 2^{-1-q} e^{-i (a-c+(b-d) x)} \left (e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )\right )^{1+q} \left ((-b+d q) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (2-\frac {b}{d}+q\right ),-\frac {b+d (-2+q)}{2 d},-e^{2 i (c+d x)}\right )+e^{2 i (a+b x)} (b+d q) \operatorname {Hypergeometric2F1}\left (1,\frac {b+d (2+q)}{2 d},\frac {1}{2} \left (2+\frac {b}{d}-q\right ),-e^{2 i (c+d x)}\right )\right )}{(b-d q) (b+d q)} \] Input:

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

Output:

((-I)*2^(-1 - q)*((1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x)))^(1 + q)*((-b 
+ d*q)*Hypergeometric2F1[1, (2 - b/d + q)/2, -1/2*(b + d*(-2 + q))/d, -E^( 
(2*I)*(c + d*x))] + E^((2*I)*(a + b*x))*(b + d*q)*Hypergeometric2F1[1, (b 
+ d*(2 + q))/(2*d), (2 + b/d - q)/2, -E^((2*I)*(c + d*x))]))/(E^(I*(a - c 
+ (b - d)*x))*(b - d*q)*(b + d*q))
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {5065, 2009}

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.

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

\(\Big \downarrow \) 5065

\(\displaystyle 2^{-q-1} \int \left (e^{-i a-i b x} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^q+e^{i a+i b x} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^q\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 2^{-q-1} \left (\frac {i \left (1+e^{2 i c+2 i d x}\right )^{-q} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^q \operatorname {Hypergeometric2F1}\left (-q,-\frac {b+d q}{2 d},1-\frac {b+d q}{2 d},-e^{2 i (c+d x)}\right ) \exp (-i (a+c q)-i x (b+d q)+i q (c+d x))}{b+d q}-\frac {i \left (1+e^{2 i c+2 i d x}\right )^{-q} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^q \operatorname {Hypergeometric2F1}\left (-q,\frac {b-d q}{2 d},\frac {1}{2} \left (\frac {b}{d}-q+2\right ),-e^{2 i (c+d x)}\right ) \exp (i (a-c q)+i x (b-d q)+i q (c+d x))}{b-d q}\right )\)

Input:

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

Output:

2^(-1 - q)*(((-I)*E^(I*(a - c*q) + I*(b - d*q)*x + I*q*(c + d*x))*(E^((-I) 
*(c + d*x)) + E^(I*(c + d*x)))^q*Hypergeometric2F1[-q, (b - d*q)/(2*d), (2 
 + b/d - q)/2, -E^((2*I)*(c + d*x))])/((1 + E^((2*I)*c + (2*I)*d*x))^q*(b 
- d*q)) + (I*E^((-I)*(a + c*q) - I*(b + d*q)*x + I*q*(c + d*x))*(E^((-I)*( 
c + d*x)) + E^(I*(c + d*x)))^q*Hypergeometric2F1[-q, -1/2*(b + d*q)/d, 1 - 
 (b + d*q)/(2*d), -E^((2*I)*(c + d*x))])/((1 + E^((2*I)*c + (2*I)*d*x))^q* 
(b + d*q)))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5065

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*Cos[(c_.) + (d_.)*(x_)]^(q_.), x_Symbol] 
:> Simp[1/2^(p + q)   Int[ExpandIntegrand[(E^((-I)*(c + d*x)) + E^(I*(c + d 
*x)))^q, (E^((-I)*(a + b*x)) + E^(I*(a + b*x)))^p, x], x], x] /; FreeQ[{a, 
b, c, d, q}, x] && IGtQ[p, 0] &&  !IntegerQ[q]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(cos(c + d*x)**q*cos(a + b*x),x)