\(\int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx\) [654]

Optimal result

Integrand size = 29, antiderivative size = 200 \[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 \cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(c+d x)\right ) \sin ^{1+n}(c+d x)}{d (1+n) \sqrt {\cos ^2(c+d x)}}+\frac {2 a^2 \cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {2+n}{2},\frac {4+n}{2},\sin ^2(c+d x)\right ) \sin ^{2+n}(c+d x)}{d (2+n) \sqrt {\cos ^2(c+d x)}}+\frac {a^2 \cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {3+n}{2},\frac {5+n}{2},\sin ^2(c+d x)\right ) \sin ^{3+n}(c+d x)}{d (3+n) \sqrt {\cos ^2(c+d x)}} \] Output:

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

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.82 \[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 \sqrt {\cos ^2(c+d x)} \sec (c+d x) \sin ^{1+n}(c+d x) \left (\left (6+5 n+n^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(c+d x)\right )+(1+n) \sin (c+d x) \left (2 (3+n) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {2+n}{2},\frac {4+n}{2},\sin ^2(c+d x)\right )+(2+n) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {3+n}{2},\frac {5+n}{2},\sin ^2(c+d x)\right ) \sin (c+d x)\right )\right )}{d (1+n) (2+n) (3+n)} \] Input:

Integrate[Cos[c + d*x]^6*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^2,x]
 

Output:

(a^2*Sqrt[Cos[c + d*x]^2]*Sec[c + d*x]*Sin[c + d*x]^(1 + n)*((6 + 5*n + n^ 
2)*Hypergeometric2F1[-5/2, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2] + (1 + n) 
*Sin[c + d*x]*(2*(3 + n)*Hypergeometric2F1[-5/2, (2 + n)/2, (4 + n)/2, Sin 
[c + d*x]^2] + (2 + n)*Hypergeometric2F1[-5/2, (3 + n)/2, (5 + n)/2, Sin[c 
 + d*x]^2]*Sin[c + d*x])))/(d*(1 + n)*(2 + n)*(3 + n))
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3042, 3352, 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.

\(\displaystyle \int \cos ^6(c+d x) (a \sin (c+d x)+a)^2 \sin ^n(c+d x) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cos (c+d x)^6 (a \sin (c+d x)+a)^2 \sin (c+d x)^ndx\)

\(\Big \downarrow \) 3352

\(\displaystyle \int \left (2 a^2 \cos ^6(c+d x) \sin ^{n+1}(c+d x)+a^2 \cos ^6(c+d x) \sin ^{n+2}(c+d x)+a^2 \cos ^6(c+d x) \sin ^n(c+d x)\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

Int[Cos[c + d*x]^6*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^2,x]
 

Output:

(a^2*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]) + (2*a^2*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]) + (a^2*Cos[c + d*x]*Hy 
pergeometric2F1[-5/2, (3 + n)/2, (5 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^( 
3 + n))/(d*(3 + n)*Sqrt[Cos[c + d*x]^2])
 

Defintions of rubi rules used

rule 2009

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

rule 3042

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

rule 3352

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n 
_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig 
[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; F 
reeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
 

Maple [F]

\[\int \cos \left (d x +c \right )^{6} \sin \left (d x +c \right )^{n} \left (a +a \sin \left (d x +c \right )\right )^{2}d x\]

Input:

int(cos(d*x+c)^6*sin(d*x+c)^n*(a+a*sin(d*x+c))^2,x)
 

Output:

int(cos(d*x+c)^6*sin(d*x+c)^n*(a+a*sin(d*x+c))^2,x)
 

Fricas [F]

\[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{2} \sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{6} \,d x } \] Input:

integrate(cos(d*x+c)^6*sin(d*x+c)^n*(a+a*sin(d*x+c))^2,x, algorithm="frica 
s")
 

Output:

integral(-(a^2*cos(d*x + c)^8 - 2*a^2*cos(d*x + c)^6*sin(d*x + c) - 2*a^2* 
cos(d*x + c)^6)*sin(d*x + c)^n, x)
 

Sympy [F(-1)]

Timed out. \[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**6*sin(d*x+c)**n*(a+a*sin(d*x+c))**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{2} \sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{6} \,d x } \] Input:

integrate(cos(d*x+c)^6*sin(d*x+c)^n*(a+a*sin(d*x+c))^2,x, algorithm="maxim 
a")
 

Output:

integrate((a*sin(d*x + c) + a)^2*sin(d*x + c)^n*cos(d*x + c)^6, x)
 

Giac [F]

\[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{2} \sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{6} \,d x } \] Input:

integrate(cos(d*x+c)^6*sin(d*x+c)^n*(a+a*sin(d*x+c))^2,x, algorithm="giac" 
)
 

Output:

integrate((a*sin(d*x + c) + a)^2*sin(d*x + c)^n*cos(d*x + c)^6, x)
 

Mupad [F(-1)]

Timed out. \[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=\int {\cos \left (c+d\,x\right )}^6\,{\sin \left (c+d\,x\right )}^n\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^2 \,d x \] Input:

int(cos(c + d*x)^6*sin(c + d*x)^n*(a + a*sin(c + d*x))^2,x)
 

Output:

int(cos(c + d*x)^6*sin(c + d*x)^n*(a + a*sin(c + d*x))^2, x)
 

Reduce [F]

\[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=a^{2} \left (\int \sin \left (d x +c \right )^{n} \cos \left (d x +c \right )^{6} \sin \left (d x +c \right )^{2}d x +2 \left (\int \sin \left (d x +c \right )^{n} \cos \left (d x +c \right )^{6} \sin \left (d x +c \right )d x \right )+\int \sin \left (d x +c \right )^{n} \cos \left (d x +c \right )^{6}d x \right ) \] Input:

int(cos(d*x+c)^6*sin(d*x+c)^n*(a+a*sin(d*x+c))^2,x)
 

Output:

a**2*(int(sin(c + d*x)**n*cos(c + d*x)**6*sin(c + d*x)**2,x) + 2*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))