\(\int \frac {\cos (e+f x) (a+a \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx\) [921]

Optimal result

Integrand size = 31, antiderivative size = 59 \[ \int \frac {\cos (e+f x) (a+a \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d (1+\sin (e+f x))}{c-d}\right ) (a+a \sin (e+f x))^{1+m}}{a (c-d) f (1+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (e+f x) (a+a \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d (1+\sin (e+f x))}{c-d}\right ) (a+a \sin (e+f x))^{1+m}}{a (c-d) f (1+m)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {3042, 3312, 27, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3312

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 78

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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