\(\int \cos (a+b x) \sec ^3(c+d x) \, dx\) [332]

Optimal result

Integrand size = 15, antiderivative size = 145 \[ \int \cos (a+b x) \sec ^3(c+d x) \, dx=\frac {4 i e^{-i a-i b x+3 i (c+d x)} \operatorname {Hypergeometric2F1}\left (3,\frac {1}{2} \left (3-\frac {b}{d}\right ),\frac {1}{2} \left (5-\frac {b}{d}\right ),-e^{2 i (c+d x)}\right )}{b-3 d}-\frac {4 i e^{i a+i b x+3 i (c+d x)} \operatorname {Hypergeometric2F1}\left (3,\frac {1}{2} \left (3+\frac {b}{d}\right ),\frac {1}{2} \left (5+\frac {b}{d}\right ),-e^{2 i (c+d x)}\right )}{b+3 d} \] Output:

4*I*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)-4*I*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)
 

Mathematica [A] (verified)

Time = 2.03 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.19 \[ \int \cos (a+b x) \sec ^3(c+d x) \, dx=\frac {-2 i (b+d) e^{-i (a-c+(b-d) x)} \operatorname {Hypergeometric2F1}\left (1,\frac {-b+d}{2 d},\frac {3}{2}-\frac {b}{2 d},-e^{2 i (c+d x)}\right )+2 i (b-d) e^{i (a+c+(b+d) x)} \operatorname {Hypergeometric2F1}\left (1,\frac {b+d}{2 d},\frac {1}{2} \left (3+\frac {b}{d}\right ),-e^{2 i (c+d x)}\right )+\sec ^2(c+d x) ((b-d) \sin (a-c+b x-d x)+(b+d) \sin (a+c+(b+d) x))}{4 d^2} \] Input:

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

Output:

(((-2*I)*(b + d)*Hypergeometric2F1[1, (-b + d)/(2*d), 3/2 - b/(2*d), -E^(( 
2*I)*(c + d*x))])/E^(I*(a - c + (b - d)*x)) + (2*I)*(b - d)*E^(I*(a + c + 
(b + d)*x))*Hypergeometric2F1[1, (b + d)/(2*d), (3 + b/d)/2, -E^((2*I)*(c 
+ d*x))] + Sec[c + d*x]^2*((b - d)*Sin[a - c + b*x - d*x] + (b + d)*Sin[a 
+ c + (b + d)*x]))/(4*d^2)
 

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.

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

1/2*((b - d)*cos(b*x + a)*sin((2*b + d)*x + 2*a + c) - (b - d)*cos((2*b + 
d)*x + 2*a + c)*sin(b*x + a) + (b - d)*cos(3*d*x + 3*c)*sin(b*x + a) + (b 
+ d)*cos(d*x + c)*sin(b*x + a) - (b - d)*cos(b*x + a)*sin(3*d*x + 3*c) - ( 
b + d)*cos(b*x + a)*sin(d*x + c) - (2*(b + d)*sin((b + 2*d)*x + a + 2*c) + 
 (b + d)*sin(b*x + a))*cos((2*b + 3*d)*x + 2*a + 3*c) + ((b + d)*sin((2*b 
+ 3*d)*x + 2*a + 3*c) + (b - d)*sin((2*b + d)*x + 2*a + c) - (b - d)*sin(3 
*d*x + 3*c) - (b + d)*sin(d*x + c))*cos((b + 4*d)*x + a + 4*c) + 2*((b - d 
)*sin((2*b + d)*x + 2*a + c) - (b - d)*sin(3*d*x + 3*c) - (b + d)*sin(d*x 
+ c))*cos((b + 2*d)*x + a + 2*c) + 2*(d^2*cos((b + 4*d)*x + a + 4*c)^2 + 4 
*d^2*cos((b + 2*d)*x + a + 2*c)^2 + 4*d^2*cos((b + 2*d)*x + a + 2*c)*cos(b 
*x + a) + d^2*cos(b*x + a)^2 + d^2*sin((b + 4*d)*x + a + 4*c)^2 + 4*d^2*si 
n((b + 2*d)*x + a + 2*c)^2 + 4*d^2*sin((b + 2*d)*x + a + 2*c)*sin(b*x + a) 
 + d^2*sin(b*x + a)^2 + 2*(2*d^2*cos((b + 2*d)*x + a + 2*c) + d^2*cos(b*x 
+ a))*cos((b + 4*d)*x + a + 4*c) + 2*(2*d^2*sin((b + 2*d)*x + a + 2*c) + d 
^2*sin(b*x + a))*sin((b + 4*d)*x + a + 4*c))*integrate(-1/2*((b^2 - d^2)*c 
os((2*b + d)*x + 2*a + c)*cos(b*x + a) + (b^2 - d^2)*cos(b*x + a)*cos(d*x 
+ c) + (b^2 - d^2)*sin((2*b + d)*x + 2*a + c)*sin(b*x + a) + (b^2 - d^2)*s 
in(b*x + a)*sin(d*x + c) + ((b^2 - d^2)*cos((2*b + d)*x + 2*a + c) + (b^2 
- d^2)*cos(d*x + c))*cos((b + 2*d)*x + a + 2*c) + ((b^2 - d^2)*sin((2*b + 
d)*x + 2*a + c) + (b^2 - d^2)*sin(d*x + c))*sin((b + 2*d)*x + a + 2*c))...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \cos (a+b x) \sec ^3(c+d x) \, dx=\text {too large to display} \] Input:

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

Output:

( - 12*cos(a + b*x)*cos(c + d*x)*sin(c + d*x)*b**3*d**2 + 12*cos(a + b*x)* 
cos(c + d*x)*sin(c + d*x)*b*d**4 - 18*cos(a + b*x)*sin(c + d*x)*b**3*d**2 
- 12*cos(a + b*x)*sin(c + d*x)*b*d**4 + 6*cos(c + d*x)*sin(a + b*x)*b**4*d 
 - 36*cos(c + d*x)*sin(a + b*x)*b**2*d**3 + 6*cos(c + d*x)*sin(c + d*x)*b* 
*5 - 18*cos(c + d*x)*sin(c + d*x)*b**3*d**2 + 12*cos(c + d*x)*sin(c + d*x) 
*b*d**4 - 192*int((tan((a + b*x)/2)*tan((c + d*x)/2))/(tan((a + b*x)/2)**2 
*tan((c + d*x)/2)**6*b**4 + 15*tan((a + b*x)/2)**2*tan((c + d*x)/2)**6*b** 
2*d**2 + 14*tan((a + b*x)/2)**2*tan((c + d*x)/2)**6*d**4 - 3*tan((a + b*x) 
/2)**2*tan((c + d*x)/2)**4*b**4 - 45*tan((a + b*x)/2)**2*tan((c + d*x)/2)* 
*4*b**2*d**2 - 42*tan((a + b*x)/2)**2*tan((c + d*x)/2)**4*d**4 + 3*tan((a 
+ b*x)/2)**2*tan((c + d*x)/2)**2*b**4 + 45*tan((a + b*x)/2)**2*tan((c + d* 
x)/2)**2*b**2*d**2 + 42*tan((a + b*x)/2)**2*tan((c + d*x)/2)**2*d**4 - tan 
((a + b*x)/2)**2*b**4 - 15*tan((a + b*x)/2)**2*b**2*d**2 - 14*tan((a + b*x 
)/2)**2*d**4 + tan((c + d*x)/2)**6*b**4 + 15*tan((c + d*x)/2)**6*b**2*d**2 
 + 14*tan((c + d*x)/2)**6*d**4 - 3*tan((c + d*x)/2)**4*b**4 - 45*tan((c + 
d*x)/2)**4*b**2*d**2 - 42*tan((c + d*x)/2)**4*d**4 + 3*tan((c + d*x)/2)**2 
*b**4 + 45*tan((c + d*x)/2)**2*b**2*d**2 + 42*tan((c + d*x)/2)**2*d**4 - b 
**4 - 15*b**2*d**2 - 14*d**4),x)*sin(c + d*x)**2*b**8*d**2 - 2688*int((tan 
((a + b*x)/2)*tan((c + d*x)/2))/(tan((a + b*x)/2)**2*tan((c + d*x)/2)**6*b 
**4 + 15*tan((a + b*x)/2)**2*tan((c + d*x)/2)**6*b**2*d**2 + 14*tan((a ...