\(\int (a (b \cot (c+d x))^p)^n \, dx\) [38]

Optimal result

Integrand size = 14, antiderivative size = 62 \[ \int \left (a (b \cot (c+d x))^p\right )^n \, dx=-\frac {\cot (c+d x) \left (a (b \cot (c+d x))^p\right )^n \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),-\cot ^2(c+d x)\right )}{d (1+n p)} \] Output:

-cot(d*x+c)*(a*(b*cot(d*x+c))^p)^n*hypergeom([1, 1/2*n*p+1/2],[1/2*n*p+3/2 
],-cot(d*x+c)^2)/d/(n*p+1)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00 \[ \int \left (a (b \cot (c+d x))^p\right )^n \, dx=-\frac {\cot (c+d x) \left (a (b \cot (c+d x))^p\right )^n \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),-\cot ^2(c+d x)\right )}{d (1+n p)} \] Input:

Integrate[(a*(b*Cot[c + d*x])^p)^n,x]
 

Output:

-((Cot[c + d*x]*(a*(b*Cot[c + d*x])^p)^n*Hypergeometric2F1[1, (1 + n*p)/2, 
 (3 + n*p)/2, -Cot[c + d*x]^2])/(d*(1 + n*p)))
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3042, 4142, 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.

\(\displaystyle \int \left (a (b \cot (c+d x))^p\right )^n \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (a \left (-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )^p\right )^ndx\)

\(\Big \downarrow \) 4142

\(\displaystyle (b \cot (c+d x))^{-n p} \left (a (b \cot (c+d x))^p\right )^n \int (b \cot (c+d x))^{n p}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle (b \cot (c+d x))^{-n p} \left (a (b \cot (c+d x))^p\right )^n \int \left (-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{n p}dx\)

\(\Big \downarrow \) 3957

\(\displaystyle -\frac {b (b \cot (c+d x))^{-n p} \left (a (b \cot (c+d x))^p\right )^n \int \frac {(b \cot (c+d x))^{n p}}{\cot ^2(c+d x) b^2+b^2}d(b \cot (c+d x))}{d}\)

\(\Big \downarrow \) 278

\(\displaystyle -\frac {\cot (c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (n p+1),\frac {1}{2} (n p+3),-\cot ^2(c+d x)\right ) \left (a (b \cot (c+d x))^p\right )^n}{d (n p+1)}\)

Input:

Int[(a*(b*Cot[c + d*x])^p)^n,x]
 

Output:

-((Cot[c + d*x]*(a*(b*Cot[c + d*x])^p)^n*Hypergeometric2F1[1, (1 + n*p)/2, 
 (3 + n*p)/2, -Cot[c + d*x]^2])/(d*(1 + n*p)))
 

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 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]
 

rule 4142

Int[(u_.)*((b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> S 
imp[b^IntPart[p]*((b*(c*Tan[e + f*x])^n)^FracPart[p]/(c*Tan[e + f*x])^(n*Fr 
acPart[p]))   Int[ActivateTrig[u]*(c*Tan[e + f*x])^(n*p), x], x] /; FreeQ[{ 
b, c, e, f, n, p}, x] &&  !IntegerQ[p] &&  !IntegerQ[n] && (EqQ[u, 1] || Ma 
tchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, 
 cos, tan, cot, sec, csc}, trig]])
 

Maple [F]

\[\int \left (a \left (b \cot \left (d x +c \right )\right )^{p}\right )^{n}d x\]

Input:

int((a*(b*cot(d*x+c))^p)^n,x)
 

Output:

int((a*(b*cot(d*x+c))^p)^n,x)
 

Fricas [F]

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

integrate((a*(b*cot(d*x+c))^p)^n,x, algorithm="fricas")
 

Output:

integral(((b*cot(d*x + c))^p*a)^n, x)
 

Sympy [F]

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

integrate((a*(b*cot(d*x+c))**p)**n,x)
 

Output:

Integral((a*(b*cot(c + d*x))**p)**n, x)
 

Maxima [F]

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

integrate((a*(b*cot(d*x+c))^p)^n,x, algorithm="maxima")
 

Output:

integrate(((b*cot(d*x + c))^p*a)^n, x)
 

Giac [F]

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

integrate((a*(b*cot(d*x+c))^p)^n,x, algorithm="giac")
 

Output:

integrate(((b*cot(d*x + c))^p*a)^n, x)
 

Mupad [F(-1)]

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

int((a*(b*cot(c + d*x))^p)^n,x)
 

Output:

int((a*(b*cot(c + d*x))^p)^n, x)
 

Reduce [F]

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

int((a*(b*cot(d*x+c))^p)^n,x)
 

Output:

b**(n*p)*a**n*int(cot(c + d*x)**(n*p),x)