\(\int (a \cot (e+f x))^m (b \tan (e+f x))^n \, dx\) [225]

Optimal result

Integrand size = 21, antiderivative size = 69 \[ \int (a \cot (e+f x))^m (b \tan (e+f x))^n \, dx=\frac {(a \cot (e+f x))^m \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1-m+n),\frac {1}{2} (3-m+n),-\tan ^2(e+f x)\right ) (b \tan (e+f x))^{1+n}}{b f (1-m+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.97 \[ \int (a \cot (e+f x))^m (b \tan (e+f x))^n \, dx=\frac {a (a \cot (e+f x))^{-1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1-m+n),\frac {1}{2} (3-m+n),-\tan ^2(e+f x)\right ) (b \tan (e+f x))^n}{f (1-m+n)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 3084, 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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int (a \cot (e+f x))^m (b \tan (e+f x))^ndx\)

\(\Big \downarrow \) 3084

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3957

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

\(\Big \downarrow \) 278

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

Input:

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

Output:

((a*Cot[e + f*x])^m*Hypergeometric2F1[1, (1 - m + n)/2, (3 - m + n)/2, -Ta 
n[e + f*x]^2]*(b*Tan[e + f*x])^(1 + n))/(b*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 3042

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

rule 3084

Int[(cot[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n 
_), x_Symbol] :> Simp[(a*Cot[e + f*x])^m*(b*Tan[e + f*x])^m   Int[(b*Tan[e 
+ f*x])^(n - m), x], x] /; FreeQ[{a, b, e, f, m, n}, x] &&  !IntegerQ[m] && 
  !IntegerQ[n]
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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