\(\int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx\) [655]

Optimal result

Integrand size = 27, antiderivative size = 129 \[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(c+d x)\right ) \sin ^{1+n}(c+d x)}{d (1+n) \sqrt {\cos ^2(c+d x)}}+\frac {a \cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {2+n}{2},\frac {4+n}{2},\sin ^2(c+d x)\right ) \sin ^{2+n}(c+d x)}{d (2+n) \sqrt {\cos ^2(c+d x)}} \] Output:

a*cos(d*x+c)*hypergeom([-5/2, 1/2+1/2*n],[3/2+1/2*n],sin(d*x+c)^2)*sin(d*x 
+c)^(1+n)/d/(1+n)/(cos(d*x+c)^2)^(1/2)+a*cos(d*x+c)*hypergeom([-5/2, 1+1/2 
*n],[2+1/2*n],sin(d*x+c)^2)*sin(d*x+c)^(2+n)/d/(2+n)/(cos(d*x+c)^2)^(1/2)
 

Mathematica [A] (verified)

Time = 1.32 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.09 \[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \sqrt {\cos ^2(c+d x)} \sec (c+d x) \sin ^{1+n}(c+d x) (1+\sin (c+d x)) \left ((2+n) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(c+d x)\right )+(1+n) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {2+n}{2},\frac {4+n}{2},\sin ^2(c+d x)\right ) \sin (c+d x)\right )}{d (1+n) (2+n) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2} \] Input:

Integrate[Cos[c + d*x]^6*Sin[c + d*x]^n*(a + a*Sin[c + d*x]),x]
 

Output:

(a*Sqrt[Cos[c + d*x]^2]*Sec[c + d*x]*Sin[c + d*x]^(1 + n)*(1 + Sin[c + d*x 
])*((2 + n)*Hypergeometric2F1[-5/2, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2] 
+ (1 + n)*Hypergeometric2F1[-5/2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Si 
n[c + d*x]))/(d*(1 + n)*(2 + n)*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3042, 3317, 3042, 3057}

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.

\(\displaystyle \int \cos ^6(c+d x) (a \sin (c+d x)+a) \sin ^n(c+d x) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cos (c+d x)^6 (a \sin (c+d x)+a) \sin (c+d x)^ndx\)

\(\Big \downarrow \) 3317

\(\displaystyle a \int \cos ^6(c+d x) \sin ^{n+1}(c+d x)dx+a \int \cos ^6(c+d x) \sin ^n(c+d x)dx\)

\(\Big \downarrow \) 3042

\(\displaystyle a \int \cos (c+d x)^6 \sin (c+d x)^ndx+a \int \cos (c+d x)^6 \sin (c+d x)^{n+1}dx\)

\(\Big \downarrow \) 3057

\(\displaystyle \frac {a \cos (c+d x) \sin ^{n+1}(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {n+1}{2},\frac {n+3}{2},\sin ^2(c+d x)\right )}{d (n+1) \sqrt {\cos ^2(c+d x)}}+\frac {a \cos (c+d x) \sin ^{n+2}(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {n+2}{2},\frac {n+4}{2},\sin ^2(c+d x)\right )}{d (n+2) \sqrt {\cos ^2(c+d x)}}\)

Input:

Int[Cos[c + d*x]^6*Sin[c + d*x]^n*(a + a*Sin[c + d*x]),x]
 

Output:

(a*Cos[c + d*x]*Hypergeometric2F1[-5/2, (1 + n)/2, (3 + n)/2, Sin[c + d*x] 
^2]*Sin[c + d*x]^(1 + n))/(d*(1 + n)*Sqrt[Cos[c + d*x]^2]) + (a*Cos[c + d* 
x]*Hypergeometric2F1[-5/2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d 
*x]^(2 + n))/(d*(2 + n)*Sqrt[Cos[c + d*x]^2])
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3057

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_), x_Symbol] :> Simp[b^(2*IntPart[(n - 1)/2] + 1)*(b*Cos[e + f*x])^(2*Frac 
Part[(n - 1)/2])*((a*Sin[e + f*x])^(m + 1)/(a*f*(m + 1)*(Cos[e + f*x]^2)^Fr 
acPart[(n - 1)/2]))*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, Sin[ 
e + f*x]^2], x] /; FreeQ[{a, b, e, f, m, n}, x]
 

rule 3317

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n 
_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a   Int[(g*Co 
s[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d   Int[(g*Cos[e + f*x])^ 
p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]
 

Maple [F]

\[\int \cos \left (d x +c \right )^{6} \sin \left (d x +c \right )^{n} \left (a +a \sin \left (d x +c \right )\right )d x\]

Input:

int(cos(d*x+c)^6*sin(d*x+c)^n*(a+a*sin(d*x+c)),x)
 

Output:

int(cos(d*x+c)^6*sin(d*x+c)^n*(a+a*sin(d*x+c)),x)
 

Fricas [F]

\[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )} \sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{6} \,d x } \] Input:

integrate(cos(d*x+c)^6*sin(d*x+c)^n*(a+a*sin(d*x+c)),x, algorithm="fricas" 
)
 

Output:

integral((a*cos(d*x + c)^6*sin(d*x + c) + a*cos(d*x + c)^6)*sin(d*x + c)^n 
, x)
 

Sympy [F(-1)]

Timed out. \[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**6*sin(d*x+c)**n*(a+a*sin(d*x+c)),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )} \sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{6} \,d x } \] Input:

integrate(cos(d*x+c)^6*sin(d*x+c)^n*(a+a*sin(d*x+c)),x, algorithm="maxima" 
)
 

Output:

integrate((a*sin(d*x + c) + a)*sin(d*x + c)^n*cos(d*x + c)^6, x)
 

Giac [F]

\[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )} \sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{6} \,d x } \] Input:

integrate(cos(d*x+c)^6*sin(d*x+c)^n*(a+a*sin(d*x+c)),x, algorithm="giac")
 

Output:

integrate((a*sin(d*x + c) + a)*sin(d*x + c)^n*cos(d*x + c)^6, x)
 

Mupad [F(-1)]

Timed out. \[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=\int {\cos \left (c+d\,x\right )}^6\,{\sin \left (c+d\,x\right )}^n\,\left (a+a\,\sin \left (c+d\,x\right )\right ) \,d x \] Input:

int(cos(c + d*x)^6*sin(c + d*x)^n*(a + a*sin(c + d*x)),x)
 

Output:

int(cos(c + d*x)^6*sin(c + d*x)^n*(a + a*sin(c + d*x)), x)
 

Reduce [F]

\[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=a \left (\int \sin \left (d x +c \right )^{n} \cos \left (d x +c \right )^{6} \sin \left (d x +c \right )d x +\int \sin \left (d x +c \right )^{n} \cos \left (d x +c \right )^{6}d x \right ) \] Input:

int(cos(d*x+c)^6*sin(d*x+c)^n*(a+a*sin(d*x+c)),x)
 

Output:

a*(int(sin(c + d*x)**n*cos(c + d*x)**6*sin(c + d*x),x) + int(sin(c + d*x)* 
*n*cos(c + d*x)**6,x))