\(\int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^m \, dx\) [924]

Optimal result

Integrand size = 27, antiderivative size = 54 \[ \int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^m \, dx=-\frac {\operatorname {Hypergeometric2F1}(1,2+m+n,2+m,1+\sin (c+d x)) \sin ^{1+n}(c+d x) (a+a \sin (c+d x))^{1+m}}{a d (1+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.13 \[ \int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^m \, dx=\frac {\operatorname {Hypergeometric2F1}(-m,1+n,2+n,-\sin (c+d x)) \sin ^{1+n}(c+d x) (1+\sin (c+d x))^{-m} (a+a \sin (c+d x))^m}{d (1+n)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.13, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3042, 3312, 76, 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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \cos (c+d x) \sin (c+d x)^n (a \sin (c+d x)+a)^mdx\)

\(\Big \downarrow \) 3312

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

\(\Big \downarrow \) 76

\(\displaystyle \frac {(\sin (c+d x)+1)^{-m} (a \sin (c+d x)+a)^m \int \sin ^n(c+d x) (\sin (c+d x)+1)^md(a \sin (c+d x))}{a d}\)

\(\Big \downarrow \) 74

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

Input:

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

Output:

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

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 76

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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