\(\int \sec ^3(c+d x) \sin (a+b x) \, dx\) [190]

Optimal result

Integrand size = 15, antiderivative size = 141 \[ \int \sec ^3(c+d x) \sin (a+b x) \, dx=-\frac {4 e^{-i a-i b x+3 i (c+d x)} \operatorname {Hypergeometric2F1}\left (3,\frac {1}{2} \left (3-\frac {b}{d}\right ),\frac {1}{2} \left (5-\frac {b}{d}\right ),-e^{2 i (c+d x)}\right )}{b-3 d}-\frac {4 e^{i a+i b x+3 i (c+d x)} \operatorname {Hypergeometric2F1}\left (3,\frac {1}{2} \left (3+\frac {b}{d}\right ),\frac {1}{2} \left (5+\frac {b}{d}\right ),-e^{2 i (c+d x)}\right )}{b+3 d} \] Output:

-4*exp(-I*a-I*b*x+3*I*(d*x+c))*hypergeom([3, 3/2-1/2*b/d],[5/2-1/2*b/d],-e 
xp(2*I*(d*x+c)))/(b-3*d)-4*exp(I*a+I*b*x+3*I*(d*x+c))*hypergeom([3, 3/2+1/ 
2*b/d],[5/2+1/2*b/d],-exp(2*I*(d*x+c)))/(b+3*d)
 

Mathematica [A] (verified)

Time = 1.75 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.21 \[ \int \sec ^3(c+d x) \sin (a+b x) \, dx=\frac {2 (b+d) e^{-i (a-c+(b-d) x)} \operatorname {Hypergeometric2F1}\left (1,\frac {-b+d}{2 d},\frac {3}{2}-\frac {b}{2 d},-e^{2 i (c+d x)}\right )+2 (b-d) e^{i (a+c+(b+d) x)} \operatorname {Hypergeometric2F1}\left (1,\frac {b+d}{2 d},\frac {1}{2} \left (3+\frac {b}{d}\right ),-e^{2 i (c+d x)}\right )-((b-d) \cos (a-c+b x-d x)+(b+d) \cos (a+c+(b+d) x)) \sec ^2(c+d x)}{4 d^2} \] Input:

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

Output:

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

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.

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-1/2*((b - d)*cos((2*b + d)*x + 2*a + c)*cos(b*x + a) + (b - d)*cos(b*x + 
a)*cos(3*d*x + 3*c) + (b + d)*cos(b*x + a)*cos(d*x + c) + (b - d)*sin((2*b 
 + d)*x + 2*a + c)*sin(b*x + a) + (b - d)*sin(b*x + a)*sin(3*d*x + 3*c) + 
(b + d)*sin(b*x + a)*sin(d*x + c) + (2*(b + d)*cos((b + 2*d)*x + a + 2*c) 
+ (b + d)*cos(b*x + a))*cos((2*b + 3*d)*x + 2*a + 3*c) + ((b + d)*cos((2*b 
 + 3*d)*x + 2*a + 3*c) + (b - d)*cos((2*b + d)*x + 2*a + c) + (b - d)*cos( 
3*d*x + 3*c) + (b + d)*cos(d*x + c))*cos((b + 4*d)*x + a + 4*c) + 2*((b - 
d)*cos((2*b + d)*x + 2*a + c) + (b - d)*cos(3*d*x + 3*c) + (b + d)*cos(d*x 
 + c))*cos((b + 2*d)*x + a + 2*c) + 2*(d^2*cos((b + 4*d)*x + a + 4*c)^2 + 
4*d^2*cos((b + 2*d)*x + a + 2*c)^2 + 4*d^2*cos((b + 2*d)*x + a + 2*c)*cos( 
b*x + a) + d^2*cos(b*x + a)^2 + d^2*sin((b + 4*d)*x + a + 4*c)^2 + 4*d^2*s 
in((b + 2*d)*x + a + 2*c)^2 + 4*d^2*sin((b + 2*d)*x + a + 2*c)*sin(b*x + a 
) + d^2*sin(b*x + a)^2 + 2*(2*d^2*cos((b + 2*d)*x + a + 2*c) + d^2*cos(b*x 
 + a))*cos((b + 4*d)*x + a + 4*c) + 2*(2*d^2*sin((b + 2*d)*x + a + 2*c) + 
d^2*sin(b*x + a))*sin((b + 4*d)*x + a + 4*c))*integrate(1/2*((b^2 - d^2)*c 
os(b*x + a)*sin((2*b + d)*x + 2*a + c) - (b^2 - d^2)*cos((2*b + d)*x + 2*a 
 + c)*sin(b*x + a) + (b^2 - d^2)*cos(d*x + c)*sin(b*x + a) - (b^2 - d^2)*c 
os(b*x + a)*sin(d*x + c) + ((b^2 - d^2)*sin((2*b + d)*x + 2*a + c) - (b^2 
- d^2)*sin(d*x + c))*cos((b + 2*d)*x + a + 2*c) - ((b^2 - d^2)*cos((2*b + 
d)*x + 2*a + c) - (b^2 - d^2)*cos(d*x + c))*sin((b + 2*d)*x + a + 2*c))...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \sec ^3(c+d x) \sin (a+b x) \, dx=\text {too large to display} \] Input:

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

Output:

( - 6*cos(a + b*x)*cos(c + d*x)*tan((a + b*x)/2)**2*tan((c + d*x)/2)**4*b* 
*6*d**2 + 24*cos(a + b*x)*cos(c + d*x)*tan((a + b*x)/2)**2*tan((c + d*x)/2 
)**4*b**4*d**4 + 72*cos(a + b*x)*cos(c + d*x)*tan((a + b*x)/2)**2*tan((c + 
 d*x)/2)**4*b**2*d**6 + 12*cos(a + b*x)*cos(c + d*x)*tan((a + b*x)/2)**2*t 
an((c + d*x)/2)**2*b**6*d**2 - 48*cos(a + b*x)*cos(c + d*x)*tan((a + b*x)/ 
2)**2*tan((c + d*x)/2)**2*b**4*d**4 - 144*cos(a + b*x)*cos(c + d*x)*tan((a 
 + b*x)/2)**2*tan((c + d*x)/2)**2*b**2*d**6 - 6*cos(a + b*x)*cos(c + d*x)* 
tan((a + b*x)/2)**2*b**6*d**2 + 24*cos(a + b*x)*cos(c + d*x)*tan((a + b*x) 
/2)**2*b**4*d**4 + 72*cos(a + b*x)*cos(c + d*x)*tan((a + b*x)/2)**2*b**2*d 
**6 - 6*cos(a + b*x)*cos(c + d*x)*tan((c + d*x)/2)**4*b**6*d**2 + 24*cos(a 
 + b*x)*cos(c + d*x)*tan((c + d*x)/2)**4*b**4*d**4 + 72*cos(a + b*x)*cos(c 
 + d*x)*tan((c + d*x)/2)**4*b**2*d**6 + 12*cos(a + b*x)*cos(c + d*x)*tan(( 
c + d*x)/2)**2*b**6*d**2 - 48*cos(a + b*x)*cos(c + d*x)*tan((c + d*x)/2)** 
2*b**4*d**4 - 144*cos(a + b*x)*cos(c + d*x)*tan((c + d*x)/2)**2*b**2*d**6 
- 6*cos(a + b*x)*cos(c + d*x)*b**6*d**2 + 24*cos(a + b*x)*cos(c + d*x)*b** 
4*d**4 + 72*cos(a + b*x)*cos(c + d*x)*b**2*d**6 + 2*cos(a + b*x)*sin(c + d 
*x)**2*tan((a + b*x)/2)**2*tan((c + d*x)/2)**4*b**6*d**2 - 2*cos(a + b*x)* 
sin(c + d*x)**2*tan((a + b*x)/2)**2*tan((c + d*x)/2)**4*b**4*d**4 - 8*cos( 
a + b*x)*sin(c + d*x)**2*tan((a + b*x)/2)**2*tan((c + d*x)/2)**4*b**2*d**6 
 + 8*cos(a + b*x)*sin(c + d*x)**2*tan((a + b*x)/2)**2*tan((c + d*x)/2)*...