Integrand size = 13, antiderivative size = 138 \[ \int \sec (c+d x) \sin (a+b x) \, dx=-\frac {e^{-i a-i b x+i (c+d x)} \operatorname {Hypergeometric2F1}\left (1,-\frac {b-d}{2 d},\frac {1}{2} \left (3-\frac {b}{d}\right ),-e^{2 i (c+d x)}\right )}{b-d}-\frac {e^{i a+i b x+i (c+d x)} \operatorname {Hypergeometric2F1}\left (1,\frac {b+d}{2 d},\frac {1}{2} \left (3+\frac {b}{d}\right ),-e^{2 i (c+d x)}\right )}{b+d} \] Output:
-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)-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)
Time = 0.38 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.90 \[ \int \sec (c+d x) \sin (a+b x) \, dx=-\frac {e^{-i (a-c+(b-d) x)} \operatorname {Hypergeometric2F1}\left (1,\frac {-b+d}{2 d},\frac {3}{2}-\frac {b}{2 d},-e^{2 i (c+d x)}\right )}{b-d}-\frac {e^{i (a+c+(b+d) x)} \operatorname {Hypergeometric2F1}\left (1,\frac {b+d}{2 d},\frac {1}{2} \left (3+\frac {b}{d}\right ),-e^{2 i (c+d x)}\right )}{b+d} \] Input:
Integrate[Sec[c + d*x]*Sin[a + b*x],x]
Output:
-(Hypergeometric2F1[1, (-b + d)/(2*d), 3/2 - b/(2*d), -E^((2*I)*(c + d*x)) ]/((b - d)*E^(I*(a - c + (b - d)*x)))) - (E^(I*(a + c + (b + d)*x))*Hyperg eometric2F1[1, (b + d)/(2*d), (3 + b/d)/2, -E^((2*I)*(c + d*x))])/(b + 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.
\(\displaystyle \int \sin (a+b x) \sec (c+d x) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \sin (a+b x) \sec (c+d x)dx\) |
Input:
Int[Sec[c + d*x]*Sin[a + b*x],x]
Output:
$Aborted
\[\int \sec \left (d x +c \right ) \sin \left (b x +a \right )d x\]
Input:
int(sec(d*x+c)*sin(b*x+a),x)
Output:
int(sec(d*x+c)*sin(b*x+a),x)
\[ \int \sec (c+d x) \sin (a+b x) \, dx=\int { \sec \left (d x + c\right ) \sin \left (b x + a\right ) \,d x } \] Input:
integrate(sec(d*x+c)*sin(b*x+a),x, algorithm="fricas")
Output:
integral(sec(d*x + c)*sin(b*x + a), x)
\[ \int \sec (c+d x) \sin (a+b x) \, dx=\int \sin {\left (a + b x \right )} \sec {\left (c + d x \right )}\, dx \] Input:
integrate(sec(d*x+c)*sin(b*x+a),x)
Output:
Integral(sin(a + b*x)*sec(c + d*x), x)
\[ \int \sec (c+d x) \sin (a+b x) \, dx=\int { \sec \left (d x + c\right ) \sin \left (b x + a\right ) \,d x } \] Input:
integrate(sec(d*x+c)*sin(b*x+a),x, algorithm="maxima")
Output:
integrate(sec(d*x + c)*sin(b*x + a), x)
\[ \int \sec (c+d x) \sin (a+b x) \, dx=\int { \sec \left (d x + c\right ) \sin \left (b x + a\right ) \,d x } \] Input:
integrate(sec(d*x+c)*sin(b*x+a),x, algorithm="giac")
Output:
integrate(sec(d*x + c)*sin(b*x + a), x)
Timed out. \[ \int \sec (c+d x) \sin (a+b x) \, dx=\int \frac {\sin \left (a+b\,x\right )}{\cos \left (c+d\,x\right )} \,d x \] Input:
int(sin(a + b*x)/cos(c + d*x),x)
Output:
int(sin(a + b*x)/cos(c + d*x), x)
\[ \int \sec (c+d x) \sin (a+b x) \, dx=\frac {\left (\int \frac {\sin \left (b x +a \right )}{\cos \left (d x +c \right )}d x \right ) b -1}{b} \] Input:
int(sec(d*x+c)*sin(b*x+a),x)
Output:
(int(sin(a + b*x)/cos(c + d*x),x)*b - 1)/b