Integrand size = 13, antiderivative size = 111 \[ \int \cos (a+b x) \tan (c+d x) \, dx=-\frac {e^{-i (a+b x)} \operatorname {Hypergeometric2F1}\left (1,-\frac {b}{2 d},1-\frac {b}{2 d},-e^{2 i (c+d x)}\right )}{b}+\frac {e^{i (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {b}{2 d},1+\frac {b}{2 d},-e^{2 i (c+d x)}\right )}{b}-\frac {i \sin (a+b x)}{b} \] Output:
-hypergeom([1, -1/2*b/d],[1-1/2*b/d],-exp(2*I*(d*x+c)))/b/exp(I*(b*x+a))+e xp(I*(b*x+a))*hypergeom([1, 1/2*b/d],[1+1/2*b/d],-exp(2*I*(d*x+c)))/b-I*si n(b*x+a)/b
Time = 1.06 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.03 \[ \int \cos (a+b x) \tan (c+d x) \, dx=\frac {e^{-i (a+b x)} \left (1-e^{2 i (a+b x)}-2 \operatorname {Hypergeometric2F1}\left (1,-\frac {b}{2 d},1-\frac {b}{2 d},-e^{2 i (c+d x)}\right )+2 e^{2 i (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {b}{2 d},1+\frac {b}{2 d},-e^{2 i (c+d x)}\right )\right )}{2 b} \] Input:
Integrate[Cos[a + b*x]*Tan[c + d*x],x]
Output:
(1 - E^((2*I)*(a + b*x)) - 2*Hypergeometric2F1[1, -1/2*b/d, 1 - b/(2*d), - E^((2*I)*(c + d*x))] + 2*E^((2*I)*(a + b*x))*Hypergeometric2F1[1, b/(2*d), 1 + b/(2*d), -E^((2*I)*(c + d*x))])/(2*b*E^(I*(a + b*x)))
Time = 0.32 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.21, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5071, 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.
\(\displaystyle \int \cos (a+b x) \tan (c+d x) \, dx\) |
\(\Big \downarrow \) 5071 |
\(\displaystyle \int \left (\frac {i e^{-i (a+b x)}}{1+e^{2 i (c+d x)}}+\frac {i e^{i (a+b x)}}{1+e^{2 i (c+d x)}}-\frac {1}{2} i e^{-i (a+b x)}-\frac {1}{2} i e^{i (a+b x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {e^{-i (a+b x)} \operatorname {Hypergeometric2F1}\left (1,-\frac {b}{2 d},1-\frac {b}{2 d},-e^{2 i (c+d x)}\right )}{b}+\frac {e^{i (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {b}{2 d},\frac {b}{2 d}+1,-e^{2 i (c+d x)}\right )}{b}+\frac {e^{-i (a+b x)}}{2 b}-\frac {e^{i (a+b x)}}{2 b}\) |
Input:
Int[Cos[a + b*x]*Tan[c + d*x],x]
Output:
1/(2*b*E^(I*(a + b*x))) - E^(I*(a + b*x))/(2*b) - Hypergeometric2F1[1, -1/ 2*b/d, 1 - b/(2*d), -E^((2*I)*(c + d*x))]/(b*E^(I*(a + b*x))) + (E^(I*(a + b*x))*Hypergeometric2F1[1, b/(2*d), 1 + b/(2*d), -E^((2*I)*(c + d*x))])/b
Int[Cos[(a_.) + (b_.)*(x_)]*Tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[(-I)* (1/(E^(I*(a + b*x))*2)) - I*(E^(I*(a + b*x))/2) + I*(1/(E^(I*(a + b*x))*(1 + E^(2*I*(c + d*x))))) + I*(E^(I*(a + b*x))/(1 + E^(2*I*(c + d*x)))), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]
\[\int \cos \left (b x +a \right ) \tan \left (d x +c \right )d x\]
Input:
int(cos(b*x+a)*tan(d*x+c),x)
Output:
int(cos(b*x+a)*tan(d*x+c),x)
\[ \int \cos (a+b x) \tan (c+d x) \, dx=\int { \cos \left (b x + a\right ) \tan \left (d x + c\right ) \,d x } \] Input:
integrate(cos(b*x+a)*tan(d*x+c),x, algorithm="fricas")
Output:
integral(cos(b*x + a)*tan(d*x + c), x)
\[ \int \cos (a+b x) \tan (c+d x) \, dx=\int \cos {\left (a + b x \right )} \tan {\left (c + d x \right )}\, dx \] Input:
integrate(cos(b*x+a)*tan(d*x+c),x)
Output:
Integral(cos(a + b*x)*tan(c + d*x), x)
\[ \int \cos (a+b x) \tan (c+d x) \, dx=\int { \cos \left (b x + a\right ) \tan \left (d x + c\right ) \,d x } \] Input:
integrate(cos(b*x+a)*tan(d*x+c),x, algorithm="maxima")
Output:
integrate(cos(b*x + a)*tan(d*x + c), x)
\[ \int \cos (a+b x) \tan (c+d x) \, dx=\int { \cos \left (b x + a\right ) \tan \left (d x + c\right ) \,d x } \] Input:
integrate(cos(b*x+a)*tan(d*x+c),x, algorithm="giac")
Output:
integrate(cos(b*x + a)*tan(d*x + c), x)
Timed out. \[ \int \cos (a+b x) \tan (c+d x) \, dx=\int \cos \left (a+b\,x\right )\,\mathrm {tan}\left (c+d\,x\right ) \,d x \] Input:
int(cos(a + b*x)*tan(c + d*x),x)
Output:
int(cos(a + b*x)*tan(c + d*x), x)
\[ \int \cos (a+b x) \tan (c+d x) \, dx=\int \cos \left (b x +a \right ) \tan \left (d x +c \right )d x \] Input:
int(cos(b*x+a)*tan(d*x+c),x)
Output:
int(cos(a + b*x)*tan(c + d*x),x)