\(\int \sec ^2(c+d x) \sin (a+b x) \, dx\) [189]

Optimal result

Integrand size = 15, antiderivative size = 135 \[ \int \sec ^2(c+d x) \sin (a+b x) \, dx=-\frac {2 e^{-i a-i b x+2 i (c+d x)} \operatorname {Hypergeometric2F1}\left (2,1-\frac {b}{2 d},2-\frac {b}{2 d},-e^{2 i (c+d x)}\right )}{b-2 d}-\frac {2 e^{i a+i b x+2 i (c+d x)} \operatorname {Hypergeometric2F1}\left (2,1+\frac {b}{2 d},2+\frac {b}{2 d},-e^{2 i (c+d x)}\right )}{b+2 d} \] Output:

-2*exp(-I*a-I*b*x+2*I*(d*x+c))*hypergeom([2, 1-1/2*b/d],[2-1/2*b/d],-exp(2 
*I*(d*x+c)))/(b-2*d)-2*exp(I*a+I*b*x+2*I*(d*x+c))*hypergeom([2, 1+1/2*b/d] 
,[2+1/2*b/d],-exp(2*I*(d*x+c)))/(b+2*d)
 

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(289\) vs. \(2(135)=270\).

Time = 3.98 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.14 \[ \int \sec ^2(c+d x) \sin (a+b x) \, dx=\frac {-\frac {b e^{-i (a-2 c)} \left (\frac {e^{-i (b-2 d) x} \operatorname {Hypergeometric2F1}\left (1,1-\frac {b}{2 d},2-\frac {b}{2 d},-e^{2 i (c+d x)}\right )}{b-2 d}-\frac {e^{-i b x} \operatorname {Hypergeometric2F1}\left (1,-\frac {b}{2 d},1-\frac {b}{2 d},-e^{2 i (c+d x)}\right )}{b}\right )}{1+e^{2 i c}}+\frac {e^{i (a+2 c+b x)} \left (b e^{2 i d x} \operatorname {Hypergeometric2F1}\left (1,1+\frac {b}{2 d},2+\frac {b}{2 d},-e^{2 i (c+d x)}\right )-(b+2 d) \operatorname {Hypergeometric2F1}\left (1,\frac {b}{2 d},1+\frac {b}{2 d},-e^{2 i (c+d x)}\right )\right )}{(b+2 d) \left (1+e^{2 i c}\right )}+\cos (b x) \sec (c) \sec (c+d x) \sin (a) \sin (d x)+\cos (a) \sec (c) \sec (c+d x) \sin (b x) \sin (d x)}{d} \] Input:

Integrate[Sec[c + d*x]^2*Sin[a + b*x],x]
 

Output:

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

\(\displaystyle \int \sin (a+b x) \sec ^2(c+d x) \, dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \sin (a+b x) \sec ^2(c+d x)dx\)

Input:

Int[Sec[c + d*x]^2*Sin[a + b*x],x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

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

Maple [F]

\[\int \sec \left (d x +c \right )^{2} \sin \left (b x +a \right )d x\]

Input:

int(sec(d*x+c)^2*sin(b*x+a),x)
 

Output:

int(sec(d*x+c)^2*sin(b*x+a),x)
 

Fricas [F]

\[ \int \sec ^2(c+d x) \sin (a+b x) \, dx=\int { \sec \left (d x + c\right )^{2} \sin \left (b x + a\right ) \,d x } \] Input:

integrate(sec(d*x+c)^2*sin(b*x+a),x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \sec ^2(c+d x) \sin (a+b x) \, dx=\text {Timed out} \] Input:

integrate(sec(d*x+c)**2*sin(b*x+a),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \sec ^2(c+d x) \sin (a+b x) \, dx=\int { \sec \left (d x + c\right )^{2} \sin \left (b x + a\right ) \,d x } \] Input:

integrate(sec(d*x+c)^2*sin(b*x+a),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \sec ^2(c+d x) \sin (a+b x) \, dx=\int { \sec \left (d x + c\right )^{2} \sin \left (b x + a\right ) \,d x } \] Input:

integrate(sec(d*x+c)^2*sin(b*x+a),x, algorithm="giac")
 

Output:

integrate(sec(d*x + c)^2*sin(b*x + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \sec ^2(c+d x) \sin (a+b x) \, dx=\int \frac {\sin \left (a+b\,x\right )}{{\cos \left (c+d\,x\right )}^2} \,d x \] Input:

int(sin(a + b*x)/cos(c + d*x)^2,x)
 

Output:

int(sin(a + b*x)/cos(c + d*x)^2, x)
 

Reduce [F]

\[ \int \sec ^2(c+d x) \sin (a+b x) \, dx =\text {Too large to display} \] Input:

int(sec(d*x+c)^2*sin(b*x+a),x)
 

Output:

(cos(a + b*x)*cos(c + d*x)*b + 2*cos(a + b*x)*b + 8*cos(c + d*x)*int(tan(( 
a + b*x)/2)/(tan((a + b*x)/2)**2*tan((c + d*x)/2)**4*b**2 + 2*tan((a + b*x 
)/2)**2*tan((c + d*x)/2)**4*d**2 - 2*tan((a + b*x)/2)**2*tan((c + d*x)/2)* 
*2*b**2 - 4*tan((a + b*x)/2)**2*tan((c + d*x)/2)**2*d**2 + tan((a + b*x)/2 
)**2*b**2 + 2*tan((a + b*x)/2)**2*d**2 + tan((c + d*x)/2)**4*b**2 + 2*tan( 
(c + d*x)/2)**4*d**2 - 2*tan((c + d*x)/2)**2*b**2 - 4*tan((c + d*x)/2)**2* 
d**2 + b**2 + 2*d**2),x)*b**4 + 16*cos(c + d*x)*int(tan((a + b*x)/2)/(tan( 
(a + b*x)/2)**2*tan((c + d*x)/2)**4*b**2 + 2*tan((a + b*x)/2)**2*tan((c + 
d*x)/2)**4*d**2 - 2*tan((a + b*x)/2)**2*tan((c + d*x)/2)**2*b**2 - 4*tan(( 
a + b*x)/2)**2*tan((c + d*x)/2)**2*d**2 + tan((a + b*x)/2)**2*b**2 + 2*tan 
((a + b*x)/2)**2*d**2 + tan((c + d*x)/2)**4*b**2 + 2*tan((c + d*x)/2)**4*d 
**2 - 2*tan((c + d*x)/2)**2*b**2 - 4*tan((c + d*x)/2)**2*d**2 + b**2 + 2*d 
**2),x)*b**2*d**2 - 16*cos(c + d*x)*int(tan((c + d*x)/2)/(tan((a + b*x)/2) 
**2*tan((c + d*x)/2)**4*b**2 + 2*tan((a + b*x)/2)**2*tan((c + d*x)/2)**4*d 
**2 - 2*tan((a + b*x)/2)**2*tan((c + d*x)/2)**2*b**2 - 4*tan((a + b*x)/2)* 
*2*tan((c + d*x)/2)**2*d**2 + tan((a + b*x)/2)**2*b**2 + 2*tan((a + b*x)/2 
)**2*d**2 + tan((c + d*x)/2)**4*b**2 + 2*tan((c + d*x)/2)**4*d**2 - 2*tan( 
(c + d*x)/2)**2*b**2 - 4*tan((c + d*x)/2)**2*d**2 + b**2 + 2*d**2),x)*b**3 
*d - 32*cos(c + d*x)*int(tan((c + d*x)/2)/(tan((a + b*x)/2)**2*tan((c + d* 
x)/2)**4*b**2 + 2*tan((a + b*x)/2)**2*tan((c + d*x)/2)**4*d**2 - 2*tan(...