\(\int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx\) [909]

Optimal result

Integrand size = 31, antiderivative size = 88 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,-\frac {d (1+\sin (e+f x))}{c-d}\right ) (a+a \sin (e+f x))^{1+m} (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n}}{a f (1+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,-\frac {d (1+\sin (e+f x))}{c-d}\right ) (a+a \sin (e+f x))^{1+m} (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n}}{a f (1+m)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.12, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {3042, 3312, 80, 79}

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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^ndx\)

\(\Big \downarrow \) 3312

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

\(\Big \downarrow \) 80

\(\displaystyle \frac {(c+d \sin (e+f x))^n \left (\frac {a c+a d \sin (e+f x)}{a (c-d)}\right )^{-n} \int (\sin (e+f x) a+a)^m \left (\frac {c}{c-d}+\frac {d \sin (e+f x)}{c-d}\right )^nd(a \sin (e+f x))}{a f}\)

\(\Big \downarrow \) 79

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 79

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

rule 80

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

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

\[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \left (c + d \sin {\left (e + f x \right )}\right )^{n} \cos {\left (e + f x \right )}\, dx \] Input:

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

Output:

Integral((a*(sin(e + f*x) + 1))**m*(c + d*sin(e + f*x))**n*cos(e + f*x), x 
)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

integrate(cos(f*x+e)*(a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^n,x, algorithm="g 
iac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\text {too large to display} \] Input:

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

Output:

((sin(e + f*x)*d + c)**n*(sin(e + f*x)*a + a)**m*sin(e + f*x)*d*m + (sin(e 
 + f*x)*d + c)**n*(sin(e + f*x)*a + a)**m*sin(e + f*x)*d*n + (sin(e + f*x) 
*d + c)**n*(sin(e + f*x)*a + a)**m*c*n + (sin(e + f*x)*d + c)**n*(sin(e + 
f*x)*a + a)**m*d*m - int(((sin(e + f*x)*d + c)**n*(sin(e + f*x)*a + a)**m* 
cos(e + f*x))/(sin(e + f*x)**2*d*m**2 + 2*sin(e + f*x)**2*d*m*n + sin(e + 
f*x)**2*d*m + sin(e + f*x)**2*d*n**2 + sin(e + f*x)**2*d*n + sin(e + f*x)* 
c*m**2 + 2*sin(e + f*x)*c*m*n + sin(e + f*x)*c*m + sin(e + f*x)*c*n**2 + s 
in(e + f*x)*c*n + sin(e + f*x)*d*m**2 + 2*sin(e + f*x)*d*m*n + sin(e + f*x 
)*d*m + sin(e + f*x)*d*n**2 + sin(e + f*x)*d*n + c*m**2 + 2*c*m*n + c*m + 
c*n**2 + c*n),x)*c**2*f*m**3*n - 2*int(((sin(e + f*x)*d + c)**n*(sin(e + f 
*x)*a + a)**m*cos(e + f*x))/(sin(e + f*x)**2*d*m**2 + 2*sin(e + f*x)**2*d* 
m*n + sin(e + f*x)**2*d*m + sin(e + f*x)**2*d*n**2 + sin(e + f*x)**2*d*n + 
 sin(e + f*x)*c*m**2 + 2*sin(e + f*x)*c*m*n + sin(e + f*x)*c*m + sin(e + f 
*x)*c*n**2 + sin(e + f*x)*c*n + sin(e + f*x)*d*m**2 + 2*sin(e + f*x)*d*m*n 
 + sin(e + f*x)*d*m + sin(e + f*x)*d*n**2 + sin(e + f*x)*d*n + c*m**2 + 2* 
c*m*n + c*m + c*n**2 + c*n),x)*c**2*f*m**2*n**2 - int(((sin(e + f*x)*d + c 
)**n*(sin(e + f*x)*a + a)**m*cos(e + f*x))/(sin(e + f*x)**2*d*m**2 + 2*sin 
(e + f*x)**2*d*m*n + sin(e + f*x)**2*d*m + sin(e + f*x)**2*d*n**2 + sin(e 
+ f*x)**2*d*n + sin(e + f*x)*c*m**2 + 2*sin(e + f*x)*c*m*n + sin(e + f*x)* 
c*m + sin(e + f*x)*c*n**2 + sin(e + f*x)*c*n + sin(e + f*x)*d*m**2 + 2*...