\(\int (d \csc (e+f x))^m (a+a \sin (e+f x))^n \, dx\) [227]

Optimal result

Integrand size = 23, antiderivative size = 103 \[ \int (d \csc (e+f x))^m (a+a \sin (e+f x))^n \, dx=-\frac {2^{\frac {1}{2}+n} \operatorname {AppellF1}\left (\frac {1}{2},m,\frac {1}{2}-n,\frac {3}{2},1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right ) \cos (e+f x) (d \csc (e+f x))^m \sin ^m(e+f x) (1+\sin (e+f x))^{-\frac {1}{2}-n} (a+a \sin (e+f x))^n}{f} \] Output:

-2^(1/2+n)*AppellF1(1/2,m,1/2-n,3/2,1-sin(f*x+e),1/2-1/2*sin(f*x+e))*cos(f 
*x+e)*(d*csc(f*x+e))^m*sin(f*x+e)^m*(1+sin(f*x+e))^(-1/2-n)*(a+a*sin(f*x+e 
))^n/f
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3042, 4710, 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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int (a \sin (e+f x)+a)^n (d \csc (e+f x))^mdx\)

\(\Big \downarrow \) 4710

\(\displaystyle (d \sin (e+f x))^m (d \csc (e+f x))^m \int (d \sin (e+f x))^{-m} (\sin (e+f x) a+a)^ndx\)

\(\Big \downarrow \) 3042

\(\displaystyle (d \sin (e+f x))^m (d \csc (e+f x))^m \int (d \sin (e+f x))^{-m} (\sin (e+f x) a+a)^ndx\)

\(\Big \downarrow \) 3266

\(\displaystyle (\sin (e+f x)+1)^{-n} (a \sin (e+f x)+a)^n (d \sin (e+f x))^m (d \csc (e+f x))^m \int (d \sin (e+f x))^{-m} (\sin (e+f x)+1)^ndx\)

\(\Big \downarrow \) 3042

\(\displaystyle (\sin (e+f x)+1)^{-n} (a \sin (e+f x)+a)^n (d \sin (e+f x))^m (d \csc (e+f x))^m \int (d \sin (e+f x))^{-m} (\sin (e+f x)+1)^ndx\)

\(\Big \downarrow \) 3265

\(\displaystyle \sin ^m(e+f x) (\sin (e+f x)+1)^{-n} (a \sin (e+f x)+a)^n (d \csc (e+f x))^m \int \sin ^{-m}(e+f x) (\sin (e+f x)+1)^ndx\)

\(\Big \downarrow \) 3042

\(\displaystyle \sin ^m(e+f x) (\sin (e+f x)+1)^{-n} (a \sin (e+f x)+a)^n (d \csc (e+f x))^m \int \sin (e+f x)^{-m} (\sin (e+f x)+1)^ndx\)

\(\Big \downarrow \) 3264

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

\(\Big \downarrow \) 150

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

Input:

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

Output:

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

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 4710

Int[(csc[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Simp[(c*Csc[a 
+ b*x])^m*(c*Sin[a + b*x])^m   Int[ActivateTrig[u]/(c*Sin[a + b*x])^m, x], 
x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownSineIntegrandQ[u, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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