\(\int (c (d \sin (e+f x))^p)^n (a+a \sin (e+f x))^m \, dx\) [826]

Optimal result

Integrand size = 27, antiderivative size = 113 \[ \int \left (c (d \sin (e+f x))^p\right )^n (a+a \sin (e+f x))^m \, dx=-\frac {2^{\frac {1}{2}+m} \operatorname {AppellF1}\left (\frac {1}{2},-n p,\frac {1}{2}-m,\frac {3}{2},1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right ) \cos (e+f x) \sin ^{-n p}(e+f x) \left (c (d \sin (e+f x))^p\right )^n (1+\sin (e+f x))^{-\frac {1}{2}-m} (a+a \sin (e+f x))^m}{f} \] Output:

-2^(1/2+m)*AppellF1(1/2,-n*p,1/2-m,3/2,1-sin(f*x+e),1/2-1/2*sin(f*x+e))*co 
s(f*x+e)*(c*(d*sin(f*x+e))^p)^n*(1+sin(f*x+e))^(-1/2-m)*(a+a*sin(f*x+e))^m 
/f/(sin(f*x+e)^(n*p))
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3305, 3042, 3266, 3042, 3265, 3042, 3264, 150}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3305

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3266

\(\displaystyle (\sin (e+f x)+1)^{-m} (a \sin (e+f x)+a)^m (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n \int (d \sin (e+f x))^{n p} (\sin (e+f x)+1)^mdx\)

\(\Big \downarrow \) 3042

\(\displaystyle (\sin (e+f x)+1)^{-m} (a \sin (e+f x)+a)^m (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n \int (d \sin (e+f x))^{n p} (\sin (e+f x)+1)^mdx\)

\(\Big \downarrow \) 3265

\(\displaystyle (\sin (e+f x)+1)^{-m} (a \sin (e+f x)+a)^m \sin ^{-n p}(e+f x) \left (c (d \sin (e+f x))^p\right )^n \int \sin ^{n p}(e+f x) (\sin (e+f x)+1)^mdx\)

\(\Big \downarrow \) 3042

\(\displaystyle (\sin (e+f x)+1)^{-m} (a \sin (e+f x)+a)^m \sin ^{-n p}(e+f x) \left (c (d \sin (e+f x))^p\right )^n \int \sin (e+f x)^{n p} (\sin (e+f x)+1)^mdx\)

\(\Big \downarrow \) 3264

\(\displaystyle -\frac {\cos (e+f x) (\sin (e+f x)+1)^{-m-\frac {1}{2}} (a \sin (e+f x)+a)^m \sin ^{-n p}(e+f x) \left (c (d \sin (e+f x))^p\right )^n \int \frac {\sin ^{n p}(e+f x) (\sin (e+f x)+1)^{m-\frac {1}{2}}}{\sqrt {1-\sin (e+f x)}}d(1-\sin (e+f x))}{f \sqrt {1-\sin (e+f x)}}\)

\(\Big \downarrow \) 150

\(\displaystyle -\frac {2^{m+\frac {1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac {1}{2}} (a \sin (e+f x)+a)^m \sin ^{-n p}(e+f x) \operatorname {AppellF1}\left (\frac {1}{2},-n p,\frac {1}{2}-m,\frac {3}{2},1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right ) \left (c (d \sin (e+f x))^p\right )^n}{f}\)

Input:

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

Output:

-((2^(1/2 + m)*AppellF1[1/2, -(n*p), 1/2 - m, 3/2, 1 - Sin[e + f*x], (1 - 
Sin[e + f*x])/2]*Cos[e + f*x]*(c*(d*Sin[e + f*x])^p)^n*(1 + Sin[e + f*x])^ 
(-1/2 - m)*(a + a*Sin[e + f*x])^m)/(f*Sin[e + f*x]^(n*p)))
 

Defintions of rubi rules used

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 3042

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

rule 3264

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_), x_Symbol] :> Simp[(-b)*(d/b)^n*(Cos[e + f*x]/(f*Sqrt[a + b*Sin[e 
 + f*x]]*Sqrt[a - b*Sin[e + f*x]]))   Subst[Int[(a - x)^n*((2*a - x)^(m - 1 
/2)/Sqrt[x]), x], x, a - b*Sin[e + f*x]], x] /; FreeQ[{a, b, d, e, f, m, n} 
, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[m] && GtQ[a, 0] && GtQ[d/b, 0]
 

rule 3265

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*( 
x_)])^(m_), x_Symbol] :> Simp[(d/b)^IntPart[n]*((d*Sin[e + f*x])^FracPart[n 
]/(b*Sin[e + f*x])^FracPart[n])   Int[(a + b*Sin[e + f*x])^m*(b*Sin[e + f*x 
])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] &&  !I 
ntegerQ[m] && GtQ[a, 0] &&  !GtQ[d/b, 0]
 

rule 3266

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*( 
x_)])^(m_), x_Symbol] :> Simp[a^IntPart[m]*((a + b*Sin[e + f*x])^FracPart[m 
]/(1 + (b/a)*Sin[e + f*x])^FracPart[m])   Int[(1 + (b/a)*Sin[e + f*x])^m*(d 
*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^ 
2, 0] &&  !IntegerQ[m] &&  !GtQ[a, 0]
 

rule 3305

Int[((c_.)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(p_))^(n_)*((a_.) + (b_.)*sin[(e 
_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Simp[c^IntPart[n]*((c*(d*Sin[e + f*x 
])^p)^FracPart[n]/(d*Sin[e + f*x])^(p*FracPart[n]))   Int[(a + b*Sin[e + f* 
x])^m*(d*Sin[e + f*x])^(n*p), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, 
x] &&  !IntegerQ[n]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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