\(\int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^m \, dx\) [931]

Optimal result

Integrand size = 27, antiderivative size = 43 \[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^m \, dx=-\frac {\operatorname {Hypergeometric2F1}(3,1+m,2+m,1+\sin (c+d x)) (a+a \sin (c+d x))^{1+m}}{a d (1+m)} \] Output:

-hypergeom([3, 1+m],[2+m],1+sin(d*x+c))*(a+a*sin(d*x+c))^(1+m)/a/d/(1+m)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^m \, dx=-\frac {\operatorname {Hypergeometric2F1}(3,1+m,2+m,1+\sin (c+d x)) (a+a \sin (c+d x))^{1+m}}{a d (1+m)} \] Input:

Integrate[Cot[c + d*x]*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^m,x]
 

Output:

-((Hypergeometric2F1[3, 1 + m, 2 + m, 1 + Sin[c + d*x]]*(a + a*Sin[c + d*x 
])^(1 + m))/(a*d*(1 + m)))
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3042, 3312, 27, 75}

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.

\(\displaystyle \int \cot (c+d x) \csc ^2(c+d x) (a \sin (c+d x)+a)^m \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (c+d x) (a \sin (c+d x)+a)^m}{\sin (c+d x)^3}dx\)

\(\Big \downarrow \) 3312

\(\displaystyle \frac {\int \csc ^3(c+d x) (\sin (c+d x) a+a)^md(a \sin (c+d x))}{a d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a^2 \int \frac {\csc ^3(c+d x) (\sin (c+d x) a+a)^m}{a^3}d(a \sin (c+d x))}{d}\)

\(\Big \downarrow \) 75

\(\displaystyle -\frac {(a \sin (c+d x)+a)^{m+1} \operatorname {Hypergeometric2F1}(3,m+1,m+2,\sin (c+d x)+1)}{a d (m+1)}\)

Input:

Int[Cot[c + d*x]*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^m,x]
 

Output:

-((Hypergeometric2F1[3, 1 + m, 2 + m, 1 + Sin[c + d*x]]*(a + a*Sin[c + d*x 
])^(1 + m))/(a*d*(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 75

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

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]

\[\int \cot \left (d x +c \right ) \csc \left (d x +c \right )^{2} \left (a +a \sin \left (d x +c \right )\right )^{m}d x\]

Input:

int(cot(d*x+c)*csc(d*x+c)^2*(a+a*sin(d*x+c))^m,x)
 

Output:

int(cot(d*x+c)*csc(d*x+c)^2*(a+a*sin(d*x+c))^m,x)
 

Fricas [F]

\[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^m \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \cot \left (d x + c\right ) \csc \left (d x + c\right )^{2} \,d x } \] Input:

integrate(cot(d*x+c)*csc(d*x+c)^2*(a+a*sin(d*x+c))^m,x, algorithm="fricas" 
)
 

Output:

integral((a*sin(d*x + c) + a)^m*cot(d*x + c)*csc(d*x + c)^2, x)
 

Sympy [F]

\[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^m \, dx=\int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{m} \cot {\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx \] Input:

integrate(cot(d*x+c)*csc(d*x+c)**2*(a+a*sin(d*x+c))**m,x)
 

Output:

Integral((a*(sin(c + d*x) + 1))**m*cot(c + d*x)*csc(c + d*x)**2, x)
 

Maxima [F]

\[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^m \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \cot \left (d x + c\right ) \csc \left (d x + c\right )^{2} \,d x } \] Input:

integrate(cot(d*x+c)*csc(d*x+c)^2*(a+a*sin(d*x+c))^m,x, algorithm="maxima" 
)
 

Output:

integrate((a*sin(d*x + c) + a)^m*cot(d*x + c)*csc(d*x + c)^2, x)
 

Giac [F]

\[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^m \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \cot \left (d x + c\right ) \csc \left (d x + c\right )^{2} \,d x } \] Input:

integrate(cot(d*x+c)*csc(d*x+c)^2*(a+a*sin(d*x+c))^m,x, algorithm="giac")
 

Output:

integrate((a*sin(d*x + c) + a)^m*cot(d*x + c)*csc(d*x + c)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^m \, dx=\int \frac {\mathrm {cot}\left (c+d\,x\right )\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^m}{{\sin \left (c+d\,x\right )}^2} \,d x \] Input:

int((cot(c + d*x)*(a + a*sin(c + d*x))^m)/sin(c + d*x)^2,x)
 

Output:

int((cot(c + d*x)*(a + a*sin(c + d*x))^m)/sin(c + d*x)^2, x)
 

Reduce [F]

\[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^m \, dx=\frac {-\left (\sin \left (d x +c \right ) a +a \right )^{m} \csc \left (d x +c \right )^{2}+\left (\int \frac {\left (\sin \left (d x +c \right ) a +a \right )^{m} \cos \left (d x +c \right ) \csc \left (d x +c \right )^{2}}{\sin \left (d x +c \right )+1}d x \right ) d m}{2 d} \] Input:

int(cot(d*x+c)*csc(d*x+c)^2*(a+a*sin(d*x+c))^m,x)
 

Output:

( - (sin(c + d*x)*a + a)**m*csc(c + d*x)**2 + int(((sin(c + d*x)*a + a)**m 
*cos(c + d*x)*csc(c + d*x)**2)/(sin(c + d*x) + 1),x)*d*m)/(2*d)