\(\int \cot ^3(e+f x) (a+a \sin (e+f x))^m \, dx\) [133]

Optimal result

Integrand size = 21, antiderivative size = 83 \[ \int \cot ^3(e+f x) (a+a \sin (e+f x))^m \, dx=-\frac {\csc ^2(e+f x) (a+a \sin (e+f x))^{2+m}}{2 a^2 f}-\frac {(2-m) \operatorname {Hypergeometric2F1}(2,2+m,3+m,1+\sin (e+f x)) (a+a \sin (e+f x))^{2+m}}{2 a^2 f (2+m)} \] Output:

-1/2*csc(f*x+e)^2*(a+a*sin(f*x+e))^(2+m)/a^2/f-1/2*(2-m)*hypergeom([2, 2+m 
],[3+m],1+sin(f*x+e))*(a+a*sin(f*x+e))^(2+m)/a^2/f/(2+m)
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.82 \[ \int \cot ^3(e+f x) (a+a \sin (e+f x))^m \, dx=-\frac {\left ((2+m) \csc ^2(e+f x)-(-2+m) \operatorname {Hypergeometric2F1}(2,2+m,3+m,1+\sin (e+f x))\right ) (1+\sin (e+f x))^2 (a (1+\sin (e+f x)))^m}{2 f (2+m)} \] Input:

Integrate[Cot[e + f*x]^3*(a + a*Sin[e + f*x])^m,x]
 

Output:

-1/2*(((2 + m)*Csc[e + f*x]^2 - (-2 + m)*Hypergeometric2F1[2, 2 + m, 3 + m 
, 1 + Sin[e + f*x]])*(1 + Sin[e + f*x])^2*(a*(1 + Sin[e + f*x]))^m)/(f*(2 
+ m))
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3186, 87, 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.

\(\displaystyle \int \cot ^3(e+f x) (a \sin (e+f x)+a)^m \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a \sin (e+f x)+a)^m}{\tan (e+f x)^3}dx\)

\(\Big \downarrow \) 3186

\(\displaystyle \frac {\int \frac {\csc ^3(e+f x) (a-a \sin (e+f x)) (\sin (e+f x) a+a)^{m+1}}{a^3}d(a \sin (e+f x))}{f}\)

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 75

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

Input:

Int[Cot[e + f*x]^3*(a + a*Sin[e + f*x])^m,x]
 

Output:

(-1/2*(Csc[e + f*x]^2*(a + a*Sin[e + f*x])^(2 + m))/a^2 - ((2 - m)*Hyperge 
ometric2F1[2, 2 + m, 3 + m, 1 + Sin[e + f*x]]*(a + a*Sin[e + f*x])^(2 + m) 
)/(2*a^2*(2 + m)))/f
 

Defintions of rubi rules used

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 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 3042

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

rule 3186

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p 
_.), x_Symbol] :> Simp[1/f   Subst[Int[x^p*((a + x)^(m - (p + 1)/2)/(a - x) 
^((p + 1)/2)), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && E 
qQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]
 

Maple [F]

\[\int \cot \left (f x +e \right )^{3} \left (a +\sin \left (f x +e \right ) a \right )^{m}d x\]

Input:

int(cot(f*x+e)^3*(a+sin(f*x+e)*a)^m,x)
 

Output:

int(cot(f*x+e)^3*(a+sin(f*x+e)*a)^m,x)
 

Fricas [F]

\[ \int \cot ^3(e+f x) (a+a \sin (e+f x))^m \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cot \left (f x + e\right )^{3} \,d x } \] Input:

integrate(cot(f*x+e)^3*(a+a*sin(f*x+e))^m,x, algorithm="fricas")
 

Output:

integral((a*sin(f*x + e) + a)^m*cot(f*x + e)^3, x)
 

Sympy [F]

\[ \int \cot ^3(e+f x) (a+a \sin (e+f x))^m \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \cot ^{3}{\left (e + f x \right )}\, dx \] Input:

integrate(cot(f*x+e)**3*(a+a*sin(f*x+e))**m,x)
 

Output:

Integral((a*(sin(e + f*x) + 1))**m*cot(e + f*x)**3, x)
 

Maxima [F]

\[ \int \cot ^3(e+f x) (a+a \sin (e+f x))^m \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cot \left (f x + e\right )^{3} \,d x } \] Input:

integrate(cot(f*x+e)^3*(a+a*sin(f*x+e))^m,x, algorithm="maxima")
 

Output:

integrate((a*sin(f*x + e) + a)^m*cot(f*x + e)^3, x)
 

Giac [F]

\[ \int \cot ^3(e+f x) (a+a \sin (e+f x))^m \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cot \left (f x + e\right )^{3} \,d x } \] Input:

integrate(cot(f*x+e)^3*(a+a*sin(f*x+e))^m,x, algorithm="giac")
 

Output:

integrate((a*sin(f*x + e) + a)^m*cot(f*x + e)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \cot ^3(e+f x) (a+a \sin (e+f x))^m \, dx=\int {\mathrm {cot}\left (e+f\,x\right )}^3\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m \,d x \] Input:

int(cot(e + f*x)^3*(a + a*sin(e + f*x))^m,x)
 

Output:

int(cot(e + f*x)^3*(a + a*sin(e + f*x))^m, x)
 

Reduce [F]

\[ \int \cot ^3(e+f x) (a+a \sin (e+f x))^m \, dx=\int \left (a +a \sin \left (f x +e \right )\right )^{m} \cot \left (f x +e \right )^{3}d x \] Input:

int(cot(f*x+e)^3*(a+a*sin(f*x+e))^m,x)
 

Output:

int((sin(e + f*x)*a + a)**m*cot(e + f*x)**3,x)