\(\int \sec (c+d x) \sin ^3(a+b x) \, dx\) [200]

Optimal result

Integrand size = 15, antiderivative size = 290 \[ \int \sec (c+d x) \sin ^3(a+b x) \, dx=-\frac {3 e^{-i a-i b x+i (c+d x)} \operatorname {Hypergeometric2F1}\left (1,-\frac {b-d}{2 d},\frac {1}{2} \left (3-\frac {b}{d}\right ),-e^{2 i (c+d x)}\right )}{4 (b-d)}+\frac {e^{-3 i a-3 i b x+i (c+d x)} \operatorname {Hypergeometric2F1}\left (1,-\frac {3 b-d}{2 d},\frac {3}{2} \left (1-\frac {b}{d}\right ),-e^{2 i (c+d x)}\right )}{4 (3 b-d)}-\frac {3 e^{i a+i b x+i (c+d x)} \operatorname {Hypergeometric2F1}\left (1,\frac {b+d}{2 d},\frac {1}{2} \left (3+\frac {b}{d}\right ),-e^{2 i (c+d x)}\right )}{4 (b+d)}+\frac {e^{3 i a+3 i b x+i (c+d x)} \operatorname {Hypergeometric2F1}\left (1,\frac {3 b+d}{2 d},\frac {3 (b+d)}{2 d},-e^{2 i (c+d x)}\right )}{4 (3 b+d)} \] Output:

-3*exp(-I*a-I*b*x+I*(d*x+c))*hypergeom([1, -1/2*(b-d)/d],[3/2-1/2*b/d],-ex 
p(2*I*(d*x+c)))/(4*b-4*d)+exp(-3*I*a-3*I*b*x+I*(d*x+c))*hypergeom([1, -1/2 
*(3*b-d)/d],[3/2-3/2*b/d],-exp(2*I*(d*x+c)))/(12*b-4*d)-3*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)))/(4*b+ 
4*d)+exp(3*I*a+3*I*b*x+I*(d*x+c))*hypergeom([1, 1/2*(3*b+d)/d],[3/2*(b+d)/ 
d],-exp(2*I*(d*x+c)))/(12*b+4*d)
 

Mathematica [A] (verified)

Time = 0.70 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.92 \[ \int \sec (c+d x) \sin ^3(a+b x) \, dx=\frac {e^{-i (3 a-c+3 b x-d x)} \operatorname {Hypergeometric2F1}\left (1,\frac {-3 b+d}{2 d},\frac {3}{2}-\frac {3 b}{2 d},-e^{2 i (c+d x)}\right )}{4 (3 b-d)}-\frac {3 e^{-i (a-c+(b-d) x)} \operatorname {Hypergeometric2F1}\left (1,\frac {-b+d}{2 d},\frac {3}{2}-\frac {b}{2 d},-e^{2 i (c+d x)}\right )}{4 (b-d)}-\frac {3 e^{i (a+c+(b+d) x)} \operatorname {Hypergeometric2F1}\left (1,\frac {b+d}{2 d},\frac {1}{2} \left (3+\frac {b}{d}\right ),-e^{2 i (c+d x)}\right )}{4 (b+d)}+\frac {e^{i (3 a+c+(3 b+d) x)} \operatorname {Hypergeometric2F1}\left (1,\frac {3 b+d}{2 d},\frac {3 (b+d)}{2 d},-e^{2 i (c+d x)}\right )}{4 (3 b+d)} \] Input:

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

Output:

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

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.

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral(sin(a + b*x)**3*sec(c + d*x), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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