\(\int (a \cot (e+f x))^m (b \cot (e+f x))^n \, dx\) [41]

Optimal result

Integrand size = 21, antiderivative size = 64 \[ \int (a \cot (e+f x))^m (b \cot (e+f x))^n \, dx=-\frac {(a \cot (e+f x))^{1+m} (b \cot (e+f x))^n \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1+m+n),\frac {1}{2} (3+m+n),-\cot ^2(e+f x)\right )}{a f (1+m+n)} \] Output:

-(a*cot(f*x+e))^(1+m)*(b*cot(f*x+e))^n*hypergeom([1, 1/2+1/2*m+1/2*n],[3/2 
+1/2*m+1/2*n],-cot(f*x+e)^2)/a/f/(1+m+n)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.02 \[ \int (a \cot (e+f x))^m (b \cot (e+f x))^n \, dx=-\frac {\cot (e+f x) (a \cot (e+f x))^m (b \cot (e+f x))^n \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1+m+n),\frac {1}{2} (3+m+n),-\cot ^2(e+f x)\right )}{f (1+m+n)} \] Input:

Integrate[(a*Cot[e + f*x])^m*(b*Cot[e + f*x])^n,x]
 

Output:

-((Cot[e + f*x]*(a*Cot[e + f*x])^m*(b*Cot[e + f*x])^n*Hypergeometric2F1[1, 
 (1 + m + n)/2, (3 + m + n)/2, -Cot[e + f*x]^2])/(f*(1 + m + n)))
 

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2034, 3042, 3957, 278}

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.

\(\displaystyle \int (a \cot (e+f x))^m (b \cot (e+f x))^n \, dx\)

\(\Big \downarrow \) 2034

\(\displaystyle (a \cot (e+f x))^{-n} (b \cot (e+f x))^n \int (a \cot (e+f x))^{m+n}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle (a \cot (e+f x))^{-n} (b \cot (e+f x))^n \int \left (-a \tan \left (e+f x+\frac {\pi }{2}\right )\right )^{m+n}dx\)

\(\Big \downarrow \) 3957

\(\displaystyle -\frac {a (a \cot (e+f x))^{-n} (b \cot (e+f x))^n \int \frac {(a \cot (e+f x))^{m+n}}{\cot ^2(e+f x) a^2+a^2}d(a \cot (e+f x))}{f}\)

\(\Big \downarrow \) 278

\(\displaystyle -\frac {(a \cot (e+f x))^{m+1} (b \cot (e+f x))^n \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (m+n+1),\frac {1}{2} (m+n+3),-\cot ^2(e+f x)\right )}{a f (m+n+1)}\)

Input:

Int[(a*Cot[e + f*x])^m*(b*Cot[e + f*x])^n,x]
 

Output:

-(((a*Cot[e + f*x])^(1 + m)*(b*Cot[e + f*x])^n*Hypergeometric2F1[1, (1 + m 
 + n)/2, (3 + m + n)/2, -Cot[e + f*x]^2])/(a*f*(1 + m + n)))
 

Defintions of rubi rules used

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 2034

Int[(Fx_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Simp[b^IntPart 
[n]*((b*v)^FracPart[n]/(a^IntPart[n]*(a*v)^FracPart[n]))   Int[(a*v)^(m + n 
)*Fx, x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] && 
  !IntegerQ[m + n]
 

rule 3042

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

rule 3957

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d   Subst[Int 
[x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && 
!IntegerQ[n]
 

Maple [F]

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

Input:

int((a*cot(f*x+e))^m*(b*cot(f*x+e))^n,x)
 

Output:

int((a*cot(f*x+e))^m*(b*cot(f*x+e))^n,x)
 

Fricas [F]

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

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

Output:

integral((a*cot(f*x + e))^m*(b*cot(f*x + e))^n, x)
 

Sympy [F]

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

integrate((a*cot(f*x+e))**m*(b*cot(f*x+e))**n,x)
 

Output:

Integral((a*cot(e + f*x))**m*(b*cot(e + f*x))**n, x)
 

Maxima [F]

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

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

Output:

integrate((a*cot(f*x + e))^m*(b*cot(f*x + e))^n, x)
 

Giac [F]

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

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

Output:

integrate((a*cot(f*x + e))^m*(b*cot(f*x + e))^n, x)
 

Mupad [F(-1)]

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

int((a*cot(e + f*x))^m*(b*cot(e + f*x))^n,x)
 

Output:

int((a*cot(e + f*x))^m*(b*cot(e + f*x))^n, x)
 

Reduce [F]

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

int((a*cot(f*x+e))^m*(b*cot(f*x+e))^n,x)
 

Output:

b**n*a**m*int(cot(e + f*x)**(m + n),x)