\(\int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^4} \, dx\) [703]

Optimal result

Integrand size = 29, antiderivative size = 109 \[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^4} \, dx=-\frac {7 \sin ^{1+n}(c+d x)}{a^4 d (1+n)}+\frac {8 \operatorname {Hypergeometric2F1}(1,1+n,2+n,-\sin (c+d x)) \sin ^{1+n}(c+d x)}{a^4 d (1+n)}+\frac {4 \sin ^{2+n}(c+d x)}{a^4 d (2+n)}-\frac {\sin ^{3+n}(c+d x)}{a^4 d (3+n)} \] Output:

-7*sin(d*x+c)^(1+n)/a^4/d/(1+n)+8*hypergeom([1, 1+n],[2+n],-sin(d*x+c))*si 
n(d*x+c)^(1+n)/a^4/d/(1+n)+4*sin(d*x+c)^(2+n)/a^4/d/(2+n)-sin(d*x+c)^(3+n) 
/a^4/d/(3+n)
 

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.95 \[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {-\frac {7 a^3 \sin ^{1+n}(c+d x)}{1+n}+\frac {8 a^3 \operatorname {Hypergeometric2F1}(1,1+n,2+n,-\sin (c+d x)) \sin ^{1+n}(c+d x)}{1+n}+\frac {4 a^3 \sin ^{2+n}(c+d x)}{2+n}-\frac {a^3 \sin ^{3+n}(c+d x)}{3+n}}{a^7 d} \] Input:

Integrate[(Cos[c + d*x]^7*Sin[c + d*x]^n)/(a + a*Sin[c + d*x])^4,x]
 

Output:

((-7*a^3*Sin[c + d*x]^(1 + n))/(1 + n) + (8*a^3*Hypergeometric2F1[1, 1 + n 
, 2 + n, -Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(1 + n) + (4*a^3*Sin[c + d*x 
]^(2 + n))/(2 + n) - (a^3*Sin[c + d*x]^(3 + n))/(3 + n))/(a^7*d)
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {3042, 3315, 99, 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.

\(\displaystyle \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a \sin (c+d x)+a)^4} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (c+d x)^7 \sin (c+d x)^n}{(a \sin (c+d x)+a)^4}dx\)

\(\Big \downarrow \) 3315

\(\displaystyle \frac {\int \frac {\sin ^n(c+d x) (a-a \sin (c+d x))^3}{\sin (c+d x) a+a}d(a \sin (c+d x))}{a^7 d}\)

\(\Big \downarrow \) 99

\(\displaystyle \frac {\int \left (-4 a^2 \sin ^n(c+d x)-(a-a \sin (c+d x))^2 \sin ^n(c+d x)-2 a (a-a \sin (c+d x)) \sin ^n(c+d x)+\frac {8 a^3 \sin ^n(c+d x)}{\sin (c+d x) a+a}\right )d(a \sin (c+d x))}{a^7 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {8 a^3 \sin ^{n+1}(c+d x) \operatorname {Hypergeometric2F1}(1,n+1,n+2,-\sin (c+d x))}{n+1}-\frac {7 a^3 \sin ^{n+1}(c+d x)}{n+1}+\frac {4 a^3 \sin ^{n+2}(c+d x)}{n+2}-\frac {a^3 \sin ^{n+3}(c+d x)}{n+3}}{a^7 d}\)

Input:

Int[(Cos[c + d*x]^7*Sin[c + d*x]^n)/(a + a*Sin[c + d*x])^4,x]
 

Output:

((-7*a^3*Sin[c + d*x]^(1 + n))/(1 + n) + (8*a^3*Hypergeometric2F1[1, 1 + n 
, 2 + n, -Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(1 + n) + (4*a^3*Sin[c + d*x 
]^(2 + n))/(2 + n) - (a^3*Sin[c + d*x]^(3 + n))/(3 + n))/(a^7*d)
 

Defintions of rubi rules used

rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | 
| (GtQ[m, 0] && GeQ[n, -1]))
 

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 3315

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ 
.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* 
f)   Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x)^n, 
 x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && Intege 
rQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]
 

Maple [F]

\[\int \frac {\cos \left (d x +c \right )^{7} \sin \left (d x +c \right )^{n}}{\left (a +a \sin \left (d x +c \right )\right )^{4}}d x\]

Input:

int(cos(d*x+c)^7*sin(d*x+c)^n/(a+a*sin(d*x+c))^4,x)
 

Output:

int(cos(d*x+c)^7*sin(d*x+c)^n/(a+a*sin(d*x+c))^4,x)
 

Fricas [F]

\[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\int { \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{7}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{4}} \,d x } \] Input:

integrate(cos(d*x+c)^7*sin(d*x+c)^n/(a+a*sin(d*x+c))^4,x, algorithm="frica 
s")
 

Output:

integral(sin(d*x + c)^n*cos(d*x + c)^7/(a^4*cos(d*x + c)^4 - 8*a^4*cos(d*x 
 + c)^2 + 8*a^4 - 4*(a^4*cos(d*x + c)^2 - 2*a^4)*sin(d*x + c)), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**7*sin(d*x+c)**n/(a+a*sin(d*x+c))**4,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\int { \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{7}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{4}} \,d x } \] Input:

integrate(cos(d*x+c)^7*sin(d*x+c)^n/(a+a*sin(d*x+c))^4,x, algorithm="maxim 
a")
 

Output:

integrate(sin(d*x + c)^n*cos(d*x + c)^7/(a*sin(d*x + c) + a)^4, x)
 

Giac [F]

\[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\int { \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{7}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{4}} \,d x } \] Input:

integrate(cos(d*x+c)^7*sin(d*x+c)^n/(a+a*sin(d*x+c))^4,x, algorithm="giac" 
)
 

Output:

integrate(sin(d*x + c)^n*cos(d*x + c)^7/(a*sin(d*x + c) + a)^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^7\,{\sin \left (c+d\,x\right )}^n}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^4} \,d x \] Input:

int((cos(c + d*x)^7*sin(c + d*x)^n)/(a + a*sin(c + d*x))^4,x)
 

Output:

int((cos(c + d*x)^7*sin(c + d*x)^n)/(a + a*sin(c + d*x))^4, x)
 

Reduce [F]

\[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\int \frac {\cos \left (d x +c \right )^{7} \sin \left (d x +c \right )^{n}}{\left (\sin \left (d x +c \right ) a +a \right )^{4}}d x \] Input:

int(cos(d*x+c)^7*sin(d*x+c)^n/(a+a*sin(d*x+c))^4,x)
 

Output:

int(cos(d*x+c)^7*sin(d*x+c)^n/(a+a*sin(d*x+c))^4,x)