3.3.23 \(\int \cos ^n(c+d x) \sin ^3(a+b x) \, dx\) [223]

3.3.23.1 Optimal result

Integrand size = 17, antiderivative size = 568 \[ \int \cos ^n(c+d x) \sin ^3(a+b x) \, dx=\frac {2^{-3-n} e^{i (3 a-c n)+i (3 b-d n) x+i n (c+d x)} \left (1+e^{2 i c+2 i d x}\right )^{-n} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (\frac {3 b}{d}-n\right ),-n,\frac {1}{2} \left (2+\frac {3 b}{d}-n\right ),-e^{2 i (c+d x)}\right )}{3 b-d n}-\frac {3\ 2^{-3-n} e^{i (a-c n)+i (b-d n) x+i n (c+d x)} \left (1+e^{2 i c+2 i d x}\right )^{-n} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \operatorname {Hypergeometric2F1}\left (-n,\frac {b-d n}{2 d},\frac {1}{2} \left (2+\frac {b}{d}-n\right ),-e^{2 i (c+d x)}\right )}{b-d n}-\frac {3\ 2^{-3-n} e^{-i (a+c n)-i (b+d n) x+i n (c+d x)} \left (1+e^{2 i c+2 i d x}\right )^{-n} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \operatorname {Hypergeometric2F1}\left (-n,-\frac {b+d n}{2 d},1-\frac {b+d n}{2 d},-e^{2 i (c+d x)}\right )}{b+d n}+\frac {2^{-3-n} e^{-i (3 a+c n)-i (3 b+d n) x+i n (c+d x)} \left (1+e^{2 i c+2 i d x}\right )^{-n} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \operatorname {Hypergeometric2F1}\left (-n,-\frac {3 b+d n}{2 d},\frac {1}{2} \left (2-\frac {3 b}{d}-n\right ),-e^{2 i (c+d x)}\right )}{3 b+d n} \]

output

2^(-3-n)*exp(I*(-c*n+3*a)+I*(-d*n+3*b)*x+I*n*(d*x+c))*(exp(-I*(d*x+c))+exp 
(I*(d*x+c)))^n*hypergeom([-n, 3/2*b/d-1/2*n],[1+3/2*b/d-1/2*n],-exp(2*I*(d 
*x+c)))/((1+exp(2*I*c+2*I*d*x))^n)/(-d*n+3*b)-3*2^(-3-n)*exp(I*(-c*n+a)+I* 
(-d*n+b)*x+I*n*(d*x+c))*(exp(-I*(d*x+c))+exp(I*(d*x+c)))^n*hypergeom([-n, 
1/2*(-d*n+b)/d],[1+1/2*b/d-1/2*n],-exp(2*I*(d*x+c)))/((1+exp(2*I*c+2*I*d*x 
))^n)/(-d*n+b)-3*2^(-3-n)*exp(-I*(c*n+a)-I*(d*n+b)*x+I*n*(d*x+c))*(exp(-I* 
(d*x+c))+exp(I*(d*x+c)))^n*hypergeom([-n, 1/2*(-d*n-b)/d],[1+1/2*(-d*n-b)/ 
d],-exp(2*I*(d*x+c)))/((1+exp(2*I*c+2*I*d*x))^n)/(d*n+b)+2^(-3-n)*exp(-I*( 
c*n+3*a)-I*(d*n+3*b)*x+I*n*(d*x+c))*(exp(-I*(d*x+c))+exp(I*(d*x+c)))^n*hyp 
ergeom([-n, 1/2*(-d*n-3*b)/d],[1-3/2*b/d-1/2*n],-exp(2*I*(d*x+c)))/((1+exp 
(2*I*c+2*I*d*x))^n)/(d*n+3*b)
 

3.3.23.2 Mathematica [A] (warning: unable to verify)

Time = 2.60 (sec) , antiderivative size = 329, normalized size of antiderivative = 0.58 \[ \int \cos ^n(c+d x) \sin ^3(a+b x) \, dx=2^{-3-n} e^{i (-3 a+c+d (1+n) x)} \left (e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )\right )^{1+n} \left (\frac {e^{-i (3 b+d n) x} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (2-\frac {3 b}{d}+n\right ),1-\frac {3 b}{2 d}-\frac {n}{2},-e^{2 i (c+d x)}\right )}{3 b+d n}-\frac {3 e^{2 i a-i (b+d n) x} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (2-\frac {b}{d}+n\right ),-\frac {b+d (-2+n)}{2 d},-e^{2 i (c+d x)}\right )}{b+d n}+e^{i (4 a+b x-d n x)} \left (\frac {e^{2 i (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (2+\frac {3 b}{d}+n\right ),1+\frac {3 b}{2 d}-\frac {n}{2},-e^{2 i (c+d x)}\right )}{3 b-d n}-\frac {3 \operatorname {Hypergeometric2F1}\left (1,\frac {b+d (2+n)}{2 d},\frac {1}{2} \left (2+\frac {b}{d}-n\right ),-e^{2 i (c+d x)}\right )}{b-d n}\right )\right ) \]

input

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

output

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

3.3.23.3 Rubi [A] (verified)

Time = 1.37 (sec) , antiderivative size = 548, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {5066, 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.

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

\(\Big \downarrow \) 5066

\(\displaystyle 2^{-n-3} \int \left (3 i e^{-i a-i b x} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n-3 i e^{i a+i b x} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n-i e^{-3 i a-3 i b x} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n+i e^{3 i a+3 i b x} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 2^{-n-3} \left (\frac {\left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \left (1+e^{2 i c+2 i d x}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (\frac {3 b}{d}-n\right ),-n,\frac {1}{2} \left (\frac {3 b}{d}-n+2\right ),-e^{2 i (c+d x)}\right ) \exp (i (3 a-c n)+i x (3 b-d n)+i n (c+d x))}{3 b-d n}-\frac {3 \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \left (1+e^{2 i c+2 i d x}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,\frac {b-d n}{2 d},\frac {1}{2} \left (\frac {b}{d}-n+2\right ),-e^{2 i (c+d x)}\right ) \exp (i (a-c n)+i x (b-d n)+i n (c+d x))}{b-d n}-\frac {3 \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \left (1+e^{2 i c+2 i d x}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,-\frac {b+d n}{2 d},1-\frac {b+d n}{2 d},-e^{2 i (c+d x)}\right ) \exp (-i (a+c n)-i x (b+d n)+i n (c+d x))}{b+d n}+\frac {\left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \left (1+e^{2 i c+2 i d x}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,-\frac {3 b+d n}{2 d},\frac {1}{2} \left (-\frac {3 b}{d}-n+2\right ),-e^{2 i (c+d x)}\right ) \exp (-i (3 a+c n)-i x (3 b+d n)+i n (c+d x))}{3 b+d n}\right )\)

input

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

output

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

3.3.23.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5066

Int[Cos[(c_.) + (d_.)*(x_)]^(q_.)*Sin[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] 
:> Simp[1/2^(p + q)   Int[ExpandIntegrand[(E^((-I)*(c + d*x)) + E^(I*(c + d 
*x)))^q, (I/E^(I*(a + b*x)) - I*E^(I*(a + b*x)))^p, x], x], x] /; FreeQ[{a, 
 b, c, d, q}, x] && IGtQ[p, 0] &&  !IntegerQ[q]
 

3.3.23.4 Maple [F]

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

input

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

output

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

3.3.23.5 Fricas [F]

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

input

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

output

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

3.3.23.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.3.23.7 Maxima [F]

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

input

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

output

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

3.3.23.8 Giac [F]

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

input

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

output

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

3.3.23.9 Mupad [F(-1)]

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

input

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

output

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