\(\int \frac {\cos (c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^2} \, dx\) [263]

Optimal result

Integrand size = 27, antiderivative size = 38 \[ \int \frac {\cos (c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\operatorname {Hypergeometric2F1}(2,1+n,2+n,-\sin (c+d x)) \sin ^{1+n}(c+d x)}{a^2 d (1+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\operatorname {Hypergeometric2F1}(2,1+n,2+n,-\sin (c+d x)) \sin ^{1+n}(c+d x)}{a^2 d (1+n)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3042, 3312, 74}

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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (c+d x) \sin (c+d x)^n}{(a \sin (c+d x)+a)^2}dx\)

\(\Big \downarrow \) 3312

\(\displaystyle \frac {\int \frac {\sin ^n(c+d x)}{(\sin (c+d x) a+a)^2}d(a \sin (c+d x))}{a d}\)

\(\Big \downarrow \) 74

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 74

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x 
)^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] 
/; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] 
 &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
 

rule 3042

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

rule 3312

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*(( 
c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b*f)   Su 
bst[Int[(a + x)^m*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, 
b, c, d, e, f, m, n}, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\cos (c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\sin \left (d x +c \right )^{n}-\left (\int \frac {\sin \left (d x +c \right )^{n} \cos \left (d x +c \right )}{\sin \left (d x +c \right )^{3} n -\sin \left (d x +c \right )^{3}+2 \sin \left (d x +c \right )^{2} n -2 \sin \left (d x +c \right )^{2}+\sin \left (d x +c \right ) n -\sin \left (d x +c \right )}d x \right ) \sin \left (d x +c \right ) d \,n^{2}+\left (\int \frac {\sin \left (d x +c \right )^{n} \cos \left (d x +c \right )}{\sin \left (d x +c \right )^{3} n -\sin \left (d x +c \right )^{3}+2 \sin \left (d x +c \right )^{2} n -2 \sin \left (d x +c \right )^{2}+\sin \left (d x +c \right ) n -\sin \left (d x +c \right )}d x \right ) \sin \left (d x +c \right ) d n -\left (\int \frac {\sin \left (d x +c \right )^{n} \cos \left (d x +c \right )}{\sin \left (d x +c \right )^{3} n -\sin \left (d x +c \right )^{3}+2 \sin \left (d x +c \right )^{2} n -2 \sin \left (d x +c \right )^{2}+\sin \left (d x +c \right ) n -\sin \left (d x +c \right )}d x \right ) d \,n^{2}+\left (\int \frac {\sin \left (d x +c \right )^{n} \cos \left (d x +c \right )}{\sin \left (d x +c \right )^{3} n -\sin \left (d x +c \right )^{3}+2 \sin \left (d x +c \right )^{2} n -2 \sin \left (d x +c \right )^{2}+\sin \left (d x +c \right ) n -\sin \left (d x +c \right )}d x \right ) d n}{a^{2} d \left (\sin \left (d x +c \right ) n -\sin \left (d x +c \right )+n -1\right )} \] Input:

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

Output:

(sin(c + d*x)**n - int((sin(c + d*x)**n*cos(c + d*x))/(sin(c + d*x)**3*n - 
 sin(c + d*x)**3 + 2*sin(c + d*x)**2*n - 2*sin(c + d*x)**2 + sin(c + d*x)* 
n - sin(c + d*x)),x)*sin(c + d*x)*d*n**2 + int((sin(c + d*x)**n*cos(c + d* 
x))/(sin(c + d*x)**3*n - sin(c + d*x)**3 + 2*sin(c + d*x)**2*n - 2*sin(c + 
 d*x)**2 + sin(c + d*x)*n - sin(c + d*x)),x)*sin(c + d*x)*d*n - int((sin(c 
 + d*x)**n*cos(c + d*x))/(sin(c + d*x)**3*n - sin(c + d*x)**3 + 2*sin(c + 
d*x)**2*n - 2*sin(c + d*x)**2 + sin(c + d*x)*n - sin(c + d*x)),x)*d*n**2 + 
 int((sin(c + d*x)**n*cos(c + d*x))/(sin(c + d*x)**3*n - sin(c + d*x)**3 + 
 2*sin(c + d*x)**2*n - 2*sin(c + d*x)**2 + sin(c + d*x)*n - sin(c + d*x)), 
x)*d*n)/(a**2*d*(sin(c + d*x)*n - sin(c + d*x) + n - 1))