Integrand size = 17, antiderivative size = 144 \[ \int \cos ^2(a+b x) \sec ^2(c+d x) \, dx=\frac {i e^{-2 i a-2 i b x+2 i (c+d x)} \operatorname {Hypergeometric2F1}\left (2,1-\frac {b}{d},2-\frac {b}{d},-e^{2 i (c+d x)}\right )}{2 (b-d)}-\frac {i e^{2 i a+2 i b x+2 i (c+d x)} \operatorname {Hypergeometric2F1}\left (2,\frac {b+d}{d},2+\frac {b}{d},-e^{2 i (c+d x)}\right )}{2 (b+d)}+\frac {\tan (c+d x)}{2 d} \] Output:
1/2*I*exp(-2*I*a-2*I*b*x+2*I*(d*x+c))*hypergeom([2, 1-b/d],[2-b/d],-exp(2* I*(d*x+c)))/(b-d)-1/2*I*exp(2*I*a+2*I*b*x+2*I*(d*x+c))*hypergeom([2, (b+d) /d],[2+b/d],-exp(2*I*(d*x+c)))/(b+d)+1/2*tan(d*x+c)/d
Time = 1.46 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.10 \[ \int \cos ^2(a+b x) \sec ^2(c+d x) \, dx=-\frac {i e^{-2 i (a+b x)} \left (-1-e^{4 i (a+b x)}+\left (1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (1,-\frac {b}{d},1-\frac {b}{d},-e^{2 i (c+d x)}\right )+e^{4 i (a+b x)} \left (1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {b}{d},\frac {b+d}{d},-e^{2 i (c+d x)}\right )\right )}{2 d \left (1+e^{2 i c}\right )}+\frac {\cos ^2(a+b x) \sec (c) \sec (c+d x) \sin (d x)}{d} \] Input:
Integrate[Cos[a + b*x]^2*Sec[c + d*x]^2,x]
Output:
((-1/2*I)*(-1 - E^((4*I)*(a + b*x)) + (1 + E^((2*I)*c))*Hypergeometric2F1[ 1, -(b/d), 1 - b/d, -E^((2*I)*(c + d*x))] + E^((4*I)*(a + b*x))*(1 + E^((2 *I)*c))*Hypergeometric2F1[1, b/d, (b + d)/d, -E^((2*I)*(c + d*x))]))/(d*E^ ((2*I)*(a + b*x))*(1 + E^((2*I)*c))) + (Cos[a + b*x]^2*Sec[c]*Sec[c + d*x] *Sin[d*x])/d
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.
\(\displaystyle \int \cos ^2(a+b x) \sec ^2(c+d x) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \cos ^2(a+b x) \sec ^2(c+d x)dx\) |
Input:
Int[Cos[a + b*x]^2*Sec[c + d*x]^2,x]
Output:
$Aborted
\[\int \cos \left (b x +a \right )^{2} \sec \left (d x +c \right )^{2}d x\]
Input:
int(cos(b*x+a)^2*sec(d*x+c)^2,x)
Output:
int(cos(b*x+a)^2*sec(d*x+c)^2,x)
\[ \int \cos ^2(a+b x) \sec ^2(c+d x) \, dx=\int { \cos \left (b x + a\right )^{2} \sec \left (d x + c\right )^{2} \,d x } \] Input:
integrate(cos(b*x+a)^2*sec(d*x+c)^2,x, algorithm="fricas")
Output:
integral(cos(b*x + a)^2*sec(d*x + c)^2, x)
Timed out. \[ \int \cos ^2(a+b x) \sec ^2(c+d x) \, dx=\text {Timed out} \] Input:
integrate(cos(b*x+a)**2*sec(d*x+c)**2,x)
Output:
Timed out
\[ \int \cos ^2(a+b x) \sec ^2(c+d x) \, dx=\int { \cos \left (b x + a\right )^{2} \sec \left (d x + c\right )^{2} \,d x } \] Input:
integrate(cos(b*x+a)^2*sec(d*x+c)^2,x, algorithm="maxima")
Output:
-1/2*((sin(4*b*x + 4*a) + 2*sin(2*b*x + 2*a))*cos(2*(b + d)*x + 2*a + 2*c) - 2*(d*cos(2*(b + d)*x + 2*a + 2*c)^2 + 2*d*cos(2*(b + d)*x + 2*a + 2*c)* cos(2*b*x + 2*a) + d*cos(2*b*x + 2*a)^2 + d*sin(2*(b + d)*x + 2*a + 2*c)^2 + 2*d*sin(2*(b + d)*x + 2*a + 2*c)*sin(2*b*x + 2*a) + d*sin(2*b*x + 2*a)^ 2)*integrate((b*cos(4*b*x + 4*a)*cos(2*b*x + 2*a) + b*sin(2*(b + d)*x + 2* a + 2*c)*sin(4*b*x + 4*a) + b*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + (b*cos(4 *b*x + 4*a) - b)*cos(2*(b + d)*x + 2*a + 2*c) - b*cos(2*b*x + 2*a))/(d*cos (2*(b + d)*x + 2*a + 2*c)^2 + 2*d*cos(2*(b + d)*x + 2*a + 2*c)*cos(2*b*x + 2*a) + d*cos(2*b*x + 2*a)^2 + d*sin(2*(b + d)*x + 2*a + 2*c)^2 + 2*d*sin( 2*(b + d)*x + 2*a + 2*c)*sin(2*b*x + 2*a) + d*sin(2*b*x + 2*a)^2), x) - (c os(4*b*x + 4*a) + 2*cos(2*b*x + 2*a) + 1)*sin(2*(b + d)*x + 2*a + 2*c) + c os(2*b*x + 2*a)*sin(4*b*x + 4*a) - cos(4*b*x + 4*a)*sin(2*b*x + 2*a) - sin (2*b*x + 2*a))/(d*cos(2*(b + d)*x + 2*a + 2*c)^2 + 2*d*cos(2*(b + d)*x + 2 *a + 2*c)*cos(2*b*x + 2*a) + d*cos(2*b*x + 2*a)^2 + d*sin(2*(b + d)*x + 2* a + 2*c)^2 + 2*d*sin(2*(b + d)*x + 2*a + 2*c)*sin(2*b*x + 2*a) + d*sin(2*b *x + 2*a)^2)
\[ \int \cos ^2(a+b x) \sec ^2(c+d x) \, dx=\int { \cos \left (b x + a\right )^{2} \sec \left (d x + c\right )^{2} \,d x } \] Input:
integrate(cos(b*x+a)^2*sec(d*x+c)^2,x, algorithm="giac")
Output:
integrate(cos(b*x + a)^2*sec(d*x + c)^2, x)
Timed out. \[ \int \cos ^2(a+b x) \sec ^2(c+d x) \, dx=\int \frac {{\cos \left (a+b\,x\right )}^2}{{\cos \left (c+d\,x\right )}^2} \,d x \] Input:
int(cos(a + b*x)^2/cos(c + d*x)^2,x)
Output:
int(cos(a + b*x)^2/cos(c + d*x)^2, x)
\[ \int \cos ^2(a+b x) \sec ^2(c+d x) \, dx=\int \cos \left (b x +a \right )^{2} \sec \left (d x +c \right )^{2}d x \] Input:
int(cos(b*x+a)^2*sec(d*x+c)^2,x)
Output:
int(cos(a + b*x)**2*sec(c + d*x)**2,x)