\(\int \cot ^p(a+\log (x)) \, dx\) [206]

Optimal result

Integrand size = 7, antiderivative size = 120 \[ \int \cot ^p(a+\log (x)) \, dx=\left (1-e^{2 i a} x^{2 i}\right )^p \left (1+e^{2 i a} x^{2 i}\right )^{-p} \left (-\frac {i \left (1+e^{2 i a} x^{2 i}\right )}{1-e^{2 i a} x^{2 i}}\right )^p x \operatorname {AppellF1}\left (-\frac {i}{2},p,-p,1-\frac {i}{2},e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right ) \] Output:

(1-exp(2*I*a)*x^(2*I))^p*(-I*(1+exp(2*I*a)*x^(2*I))/(1-exp(2*I*a)*x^(2*I)) 
)^p*x*AppellF1(-1/2*I,p,-p,1-1/2*I,exp(2*I*a)*x^(2*I),-exp(2*I*a)*x^(2*I)) 
/((1+exp(2*I*a)*x^(2*I))^p)
 

Mathematica [A] (warning: unable to verify)

Time = 0.36 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.98 \[ \int \cot ^p(a+\log (x)) \, dx=\frac {(2-i) \left (\frac {i \left (1+e^{2 i a} x^{2 i}\right )}{-1+e^{2 i a} x^{2 i}}\right )^p x \operatorname {AppellF1}\left (-\frac {i}{2},p,-p,1-\frac {i}{2},e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right )}{(2-i) \operatorname {AppellF1}\left (-\frac {i}{2},p,-p,1-\frac {i}{2},e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right )+2 e^{2 i a} p x^{2 i} \left (\operatorname {AppellF1}\left (1-\frac {i}{2},p,1-p,2-\frac {i}{2},e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right )+\operatorname {AppellF1}\left (1-\frac {i}{2},1+p,-p,2-\frac {i}{2},e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right )\right )} \] Input:

Integrate[Cot[a + Log[x]]^p,x]
 

Output:

((2 - I)*((I*(1 + E^((2*I)*a)*x^(2*I)))/(-1 + E^((2*I)*a)*x^(2*I)))^p*x*Ap 
pellF1[-1/2*I, p, -p, 1 - I/2, E^((2*I)*a)*x^(2*I), -(E^((2*I)*a)*x^(2*I)) 
])/((2 - I)*AppellF1[-1/2*I, p, -p, 1 - I/2, E^((2*I)*a)*x^(2*I), -(E^((2* 
I)*a)*x^(2*I))] + 2*E^((2*I)*a)*p*x^(2*I)*(AppellF1[1 - I/2, p, 1 - p, 2 - 
 I/2, E^((2*I)*a)*x^(2*I), -(E^((2*I)*a)*x^(2*I))] + AppellF1[1 - I/2, 1 + 
 p, -p, 2 - I/2, E^((2*I)*a)*x^(2*I), -(E^((2*I)*a)*x^(2*I))]))
 

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5003, 2058, 937, 936}

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.

\(\displaystyle \int \cot ^p(a+\log (x)) \, dx\)

\(\Big \downarrow \) 5003

\(\displaystyle \int \left (\frac {-i e^{2 i a} x^{2 i}-i}{1-e^{2 i a} x^{2 i}}\right )^pdx\)

\(\Big \downarrow \) 2058

\(\displaystyle \left (1-e^{2 i a} x^{2 i}\right )^p \left (-i e^{2 i a} x^{2 i}-i\right )^{-p} \left (-\frac {i \left (1+e^{2 i a} x^{2 i}\right )}{1-e^{2 i a} x^{2 i}}\right )^p \int \left (1-e^{2 i a} x^{2 i}\right )^{-p} \left (-i e^{2 i a} x^{2 i}-i\right )^pdx\)

\(\Big \downarrow \) 937

\(\displaystyle \left (1-e^{2 i a} x^{2 i}\right )^p \left (1+e^{2 i a} x^{2 i}\right )^{-p} \left (-\frac {i \left (1+e^{2 i a} x^{2 i}\right )}{1-e^{2 i a} x^{2 i}}\right )^p \int \left (1-e^{2 i a} x^{2 i}\right )^{-p} \left (e^{2 i a} x^{2 i}+1\right )^pdx\)

\(\Big \downarrow \) 936

\(\displaystyle x \left (1-e^{2 i a} x^{2 i}\right )^p \left (1+e^{2 i a} x^{2 i}\right )^{-p} \left (-\frac {i \left (1+e^{2 i a} x^{2 i}\right )}{1-e^{2 i a} x^{2 i}}\right )^p \operatorname {AppellF1}\left (-\frac {i}{2},p,-p,1-\frac {i}{2},e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right )\)

Input:

Int[Cot[a + Log[x]]^p,x]
 

Output:

((1 - E^((2*I)*a)*x^(2*I))^p*(((-I)*(1 + E^((2*I)*a)*x^(2*I)))/(1 - E^((2* 
I)*a)*x^(2*I)))^p*x*AppellF1[-1/2*I, p, -p, 1 - I/2, E^((2*I)*a)*x^(2*I), 
-(E^((2*I)*a)*x^(2*I))])/(1 + E^((2*I)*a)*x^(2*I))^p
 

Defintions of rubi rules used

rule 936

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[a^p*c^q*x*AppellF1[1/n, -p, -q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c) 
], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] 
 && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 937

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]) 
  Int[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, p, q 
}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])
 

rule 2058

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ 
(r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + 
 b*x^n)^(p*q)*(c + d*x^n)^(p*r))]   Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* 
r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
 

rule 5003

Int[Cot[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Int[((-I - I*E^( 
2*I*a*d)*x^(2*I*b*d))/(1 - E^(2*I*a*d)*x^(2*I*b*d)))^p, x] /; FreeQ[{a, b, 
d, p}, x]
 

Maple [F]

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

Input:

int(cot(a+ln(x))^p,x)
 

Output:

int(cot(a+ln(x))^p,x)
 

Fricas [F]

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

integrate(cot(a+log(x))^p,x, algorithm="fricas")
 

Output:

integral(cot(a + log(x))^p, x)
 

Sympy [F]

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

integrate(cot(a+ln(x))**p,x)
 

Output:

Integral(cot(a + log(x))**p, x)
 

Maxima [F]

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

integrate(cot(a+log(x))^p,x, algorithm="maxima")
 

Output:

integrate(cot(a + log(x))^p, x)
 

Giac [F]

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

integrate(cot(a+log(x))^p,x, algorithm="giac")
 

Output:

integrate(cot(a + log(x))^p, x)
 

Mupad [F(-1)]

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

int(cot(a + log(x))^p,x)
 

Output:

int(cot(a + log(x))^p, x)
 

Reduce [F]

\[ \int \cot ^p(a+\log (x)) \, dx=\cot \left (\mathrm {log}\left (x \right )+a \right )^{p} x +\left (\int \frac {\cot \left (\mathrm {log}\left (x \right )+a \right )^{p}}{\cot \left (\mathrm {log}\left (x \right )+a \right )}d x \right ) p +\left (\int \cot \left (\mathrm {log}\left (x \right )+a \right )^{p} \cot \left (\mathrm {log}\left (x \right )+a \right )d x \right ) p \] Input:

int(cot(a+log(x))^p,x)
 

Output:

cot(log(x) + a)**p*x + int(cot(log(x) + a)**p/cot(log(x) + a),x)*p + int(c 
ot(log(x) + a)**p*cot(log(x) + a),x)*p