\(\int \frac {\cos (e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx\) [914]

Optimal result

Integrand size = 31, antiderivative size = 60 \[ \int \frac {\cos (e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {c+d \sin (e+f x)}{c-d}\right ) (c+d \sin (e+f x))^{1+n}}{a (c-d) f (1+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {c+d \sin (e+f x)}{c-d}\right ) (c+d \sin (e+f x))^{1+n}}{a (c-d) f (1+n)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {3042, 3312, 78}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3312

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

\(\Big \downarrow \) 78

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 78

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

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]

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

Input:

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

Output:

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

Fricas [F]

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

integrate(cos(f*x+e)*(c+d*sin(f*x+e))^n/(a+a*sin(f*x+e)),x, algorithm="fri 
cas")
 

Output:

integral((d*sin(f*x + e) + c)^n*cos(f*x + e)/(a*sin(f*x + e) + a), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate(cos(f*x+e)*(c+d*sin(f*x+e))^n/(a+a*sin(f*x+e)),x, algorithm="max 
ima")
 

Output:

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

Giac [F]

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

integrate(cos(f*x+e)*(c+d*sin(f*x+e))^n/(a+a*sin(f*x+e)),x, algorithm="gia 
c")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

((sin(e + f*x)*d + c)**n + int(((sin(e + f*x)*d + c)**n*cos(e + f*x))/(sin 
(e + f*x)**2*d + sin(e + f*x)*c + sin(e + f*x)*d + c),x)*c*f*n - int(((sin 
(e + f*x)*d + c)**n*cos(e + f*x))/(sin(e + f*x)**2*d + sin(e + f*x)*c + si 
n(e + f*x)*d + c),x)*d*f*n)/(a*f*n)