\(\int \cos ^3(a+b x) \sec (c+d x) \, dx\) [343]

Optimal result

Integrand size = 15, antiderivative size = 298 \[ \int \cos ^3(a+b x) \sec (c+d x) \, dx=\frac {3 i e^{-i a-i b x+i (c+d x)} \operatorname {Hypergeometric2F1}\left (1,-\frac {b-d}{2 d},\frac {1}{2} \left (3-\frac {b}{d}\right ),-e^{2 i (c+d x)}\right )}{4 (b-d)}+\frac {i e^{-3 i a-3 i b x+i (c+d x)} \operatorname {Hypergeometric2F1}\left (1,-\frac {3 b-d}{2 d},\frac {3}{2} \left (1-\frac {b}{d}\right ),-e^{2 i (c+d x)}\right )}{4 (3 b-d)}-\frac {3 i e^{i a+i b x+i (c+d x)} \operatorname {Hypergeometric2F1}\left (1,\frac {b+d}{2 d},\frac {1}{2} \left (3+\frac {b}{d}\right ),-e^{2 i (c+d x)}\right )}{4 (b+d)}-\frac {i e^{3 i a+3 i b x+i (c+d x)} \operatorname {Hypergeometric2F1}\left (1,\frac {3 b+d}{2 d},\frac {3 (b+d)}{2 d},-e^{2 i (c+d x)}\right )}{4 (3 b+d)} \] Output:

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

Mathematica [A] (verified)

Time = 7.60 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.05 \[ \int \cos ^3(a+b x) \sec (c+d x) \, dx=\frac {e^{-3 i (a+b x)} \left (-\operatorname {Hypergeometric2F1}\left (1,-\frac {3 b}{d},1-\frac {3 b}{d},-i e^{i (c+d x)}\right )+\operatorname {Hypergeometric2F1}\left (1,-\frac {3 b}{d},1-\frac {3 b}{d},i e^{i (c+d x)}\right )+e^{2 i (a+b x)} \left (-9 \operatorname {Hypergeometric2F1}\left (1,-\frac {b}{d},1-\frac {b}{d},-i e^{i (c+d x)}\right )+9 \operatorname {Hypergeometric2F1}\left (1,-\frac {b}{d},1-\frac {b}{d},i e^{i (c+d x)}\right )+e^{2 i (a+b x)} \left (9 \operatorname {Hypergeometric2F1}\left (1,\frac {b}{d},\frac {b+d}{d},-i e^{i (c+d x)}\right )-9 \operatorname {Hypergeometric2F1}\left (1,\frac {b}{d},\frac {b+d}{d},i e^{i (c+d x)}\right )+e^{2 i (a+b x)} \left (\operatorname {Hypergeometric2F1}\left (1,\frac {3 b}{d},1+\frac {3 b}{d},-i e^{i (c+d x)}\right )-\operatorname {Hypergeometric2F1}\left (1,\frac {3 b}{d},1+\frac {3 b}{d},i e^{i (c+d x)}\right )\right )\right )\right )\right )}{24 b} \] Input:

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

Output:

(-Hypergeometric2F1[1, (-3*b)/d, 1 - (3*b)/d, (-I)*E^(I*(c + d*x))] + Hype 
rgeometric2F1[1, (-3*b)/d, 1 - (3*b)/d, I*E^(I*(c + d*x))] + E^((2*I)*(a + 
 b*x))*(-9*Hypergeometric2F1[1, -(b/d), 1 - b/d, (-I)*E^(I*(c + d*x))] + 9 
*Hypergeometric2F1[1, -(b/d), 1 - b/d, I*E^(I*(c + d*x))] + E^((2*I)*(a + 
b*x))*(9*Hypergeometric2F1[1, b/d, (b + d)/d, (-I)*E^(I*(c + d*x))] - 9*Hy 
pergeometric2F1[1, b/d, (b + d)/d, I*E^(I*(c + d*x))] + E^((2*I)*(a + b*x) 
)*(Hypergeometric2F1[1, (3*b)/d, 1 + (3*b)/d, (-I)*E^(I*(c + d*x))] - Hype 
rgeometric2F1[1, (3*b)/d, 1 + (3*b)/d, I*E^(I*(c + d*x))]))))/(24*b*E^((3* 
I)*(a + b*x)))
 

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.

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral(cos(a + b*x)**3*sec(c + d*x), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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