\(\int \cos (a+b x) \sec ^2(c+d x) \, dx\) [331]

Optimal result

Integrand size = 15, antiderivative size = 139 \[ \int \cos (a+b x) \sec ^2(c+d x) \, dx=\frac {2 i e^{-i a-i b x+2 i (c+d x)} \operatorname {Hypergeometric2F1}\left (2,1-\frac {b}{2 d},2-\frac {b}{2 d},-e^{2 i (c+d x)}\right )}{b-2 d}-\frac {2 i e^{i a+i b x+2 i (c+d x)} \operatorname {Hypergeometric2F1}\left (2,1+\frac {b}{2 d},2+\frac {b}{2 d},-e^{2 i (c+d x)}\right )}{b+2 d} \] Output:

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

Mathematica [A] (verified)

Time = 1.01 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.19 \[ \int \cos (a+b x) \sec ^2(c+d x) \, dx=\frac {e^{-i (a+b x)} \left (-i \left (1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (1,-\frac {b}{2 d},1-\frac {b}{2 d},-e^{2 i (c+d x)}\right )-i e^{2 i (a+b x)} \left (1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {b}{2 d},1+\frac {b}{2 d},-e^{2 i (c+d x)}\right )+\left (1+e^{2 i (a+b x)}\right ) \cos (c) \sec (c+d x) (i \cos (d x)+\sin (d x))\right )}{d \left (1+e^{2 i c}\right )} \] Input:

Integrate[Cos[a + b*x]*Sec[c + d*x]^2,x]
 

Output:

((-I)*(1 + E^((2*I)*c))*Hypergeometric2F1[1, -1/2*b/d, 1 - b/(2*d), -E^((2 
*I)*(c + d*x))] - I*E^((2*I)*(a + b*x))*(1 + E^((2*I)*c))*Hypergeometric2F 
1[1, b/(2*d), 1 + b/(2*d), -E^((2*I)*(c + d*x))] + (1 + E^((2*I)*(a + b*x) 
))*Cos[c]*Sec[c + d*x]*(I*Cos[d*x] + Sin[d*x]))/(d*E^(I*(a + b*x))*(1 + E^ 
((2*I)*c)))
 

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.

\(\displaystyle \int \cos (a+b x) \sec ^2(c+d x) \, dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \cos (a+b x) \sec ^2(c+d x)dx\)

Input:

Int[Cos[a + b*x]*Sec[c + d*x]^2,x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

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

Maple [F]

\[\int \cos \left (b x +a \right ) \sec \left (d x +c \right )^{2}d x\]

Input:

int(cos(b*x+a)*sec(d*x+c)^2,x)
 

Output:

int(cos(b*x+a)*sec(d*x+c)^2,x)
 

Fricas [F]

\[ \int \cos (a+b x) \sec ^2(c+d x) \, dx=\int { \cos \left (b x + a\right ) \sec \left (d x + c\right )^{2} \,d x } \] Input:

integrate(cos(b*x+a)*sec(d*x+c)^2,x, algorithm="fricas")
 

Output:

integral(cos(b*x + a)*sec(d*x + c)^2, x)
 

Sympy [F(-1)]

Timed out. \[ \int \cos (a+b x) \sec ^2(c+d x) \, dx=\text {Timed out} \] Input:

integrate(cos(b*x+a)*sec(d*x+c)**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \cos (a+b x) \sec ^2(c+d x) \, dx=\int { \cos \left (b x + a\right ) \sec \left (d x + c\right )^{2} \,d x } \] Input:

integrate(cos(b*x+a)*sec(d*x+c)^2,x, algorithm="maxima")
 

Output:

((d*cos((b + 2*d)*x + a + 2*c)^2 + 2*d*cos((b + 2*d)*x + a + 2*c)*cos(b*x 
+ a) + d*cos(b*x + a)^2 + d*sin((b + 2*d)*x + a + 2*c)^2 + 2*d*sin((b + 2* 
d)*x + a + 2*c)*sin(b*x + a) + d*sin(b*x + a)^2)*integrate((b*cos(2*b*x + 
2*a)*cos(b*x + a) + b*sin((b + 2*d)*x + a + 2*c)*sin(2*b*x + 2*a) + b*sin( 
2*b*x + 2*a)*sin(b*x + a) + (b*cos(2*b*x + 2*a) - b)*cos((b + 2*d)*x + a + 
 2*c) - b*cos(b*x + a))/(d*cos((b + 2*d)*x + a + 2*c)^2 + 2*d*cos((b + 2*d 
)*x + a + 2*c)*cos(b*x + a) + d*cos(b*x + a)^2 + d*sin((b + 2*d)*x + a + 2 
*c)^2 + 2*d*sin((b + 2*d)*x + a + 2*c)*sin(b*x + a) + d*sin(b*x + a)^2), x 
) + (cos(2*b*x + 2*a) + 1)*sin((b + 2*d)*x + a + 2*c) - cos((b + 2*d)*x + 
a + 2*c)*sin(2*b*x + 2*a) - cos(b*x + a)*sin(2*b*x + 2*a) + cos(2*b*x + 2* 
a)*sin(b*x + a) + sin(b*x + a))/(d*cos((b + 2*d)*x + a + 2*c)^2 + 2*d*cos( 
(b + 2*d)*x + a + 2*c)*cos(b*x + a) + d*cos(b*x + a)^2 + d*sin((b + 2*d)*x 
 + a + 2*c)^2 + 2*d*sin((b + 2*d)*x + a + 2*c)*sin(b*x + a) + d*sin(b*x + 
a)^2)
 

Giac [F]

\[ \int \cos (a+b x) \sec ^2(c+d x) \, dx=\int { \cos \left (b x + a\right ) \sec \left (d x + c\right )^{2} \,d x } \] Input:

integrate(cos(b*x+a)*sec(d*x+c)^2,x, algorithm="giac")
 

Output:

integrate(cos(b*x + a)*sec(d*x + c)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \cos (a+b x) \sec ^2(c+d x) \, dx=\int \frac {\cos \left (a+b\,x\right )}{{\cos \left (c+d\,x\right )}^2} \,d x \] Input:

int(cos(a + b*x)/cos(c + d*x)^2,x)
 

Output:

int(cos(a + b*x)/cos(c + d*x)^2, x)
 

Reduce [F]

\[ \int \cos (a+b x) \sec ^2(c+d x) \, dx=\text {too large to display} \] Input:

int(cos(b*x+a)*sec(d*x+c)^2,x)
 

Output:

(2*cos(a + b*x)*sin(c + d*x)*tan((a + b*x)/2)**2*tan((c + d*x)/2)**2*b*d** 
2 - 2*cos(a + b*x)*sin(c + d*x)*tan((a + b*x)/2)**2*b*d**2 + 2*cos(a + b*x 
)*sin(c + d*x)*tan((c + d*x)/2)**2*b*d**2 - 2*cos(a + b*x)*sin(c + d*x)*b* 
d**2 + 8*cos(c + d*x)*int(tan((c + d*x)/2)**2/(tan((a + b*x)/2)**2*tan((c 
+ d*x)/2)**4 - 2*tan((a + b*x)/2)**2*tan((c + d*x)/2)**2 + tan((a + b*x)/2 
)**2 + tan((c + d*x)/2)**4 - 2*tan((c + d*x)/2)**2 + 1),x)*tan((a + b*x)/2 
)**2*tan((c + d*x)/2)**2*b**3*d + 16*cos(c + d*x)*int(tan((c + d*x)/2)**2/ 
(tan((a + b*x)/2)**2*tan((c + d*x)/2)**4 - 2*tan((a + b*x)/2)**2*tan((c + 
d*x)/2)**2 + tan((a + b*x)/2)**2 + tan((c + d*x)/2)**4 - 2*tan((c + d*x)/2 
)**2 + 1),x)*tan((a + b*x)/2)**2*tan((c + d*x)/2)**2*b*d**3 - 8*cos(c + d* 
x)*int(tan((c + d*x)/2)**2/(tan((a + b*x)/2)**2*tan((c + d*x)/2)**4 - 2*ta 
n((a + b*x)/2)**2*tan((c + d*x)/2)**2 + tan((a + b*x)/2)**2 + tan((c + d*x 
)/2)**4 - 2*tan((c + d*x)/2)**2 + 1),x)*tan((a + b*x)/2)**2*b**3*d - 16*co 
s(c + d*x)*int(tan((c + d*x)/2)**2/(tan((a + b*x)/2)**2*tan((c + d*x)/2)** 
4 - 2*tan((a + b*x)/2)**2*tan((c + d*x)/2)**2 + tan((a + b*x)/2)**2 + tan( 
(c + d*x)/2)**4 - 2*tan((c + d*x)/2)**2 + 1),x)*tan((a + b*x)/2)**2*b*d**3 
 + 8*cos(c + d*x)*int(tan((c + d*x)/2)**2/(tan((a + b*x)/2)**2*tan((c + d* 
x)/2)**4 - 2*tan((a + b*x)/2)**2*tan((c + d*x)/2)**2 + tan((a + b*x)/2)**2 
 + tan((c + d*x)/2)**4 - 2*tan((c + d*x)/2)**2 + 1),x)*tan((c + d*x)/2)**2 
*b**3*d + 16*cos(c + d*x)*int(tan((c + d*x)/2)**2/(tan((a + b*x)/2)**2*...