\(\int \cot ^p(a+b \log (x)) \, dx\) [204]

Optimal result

Integrand size = 9, antiderivative size = 142 \[ \int \cot ^p(a+b \log (x)) \, dx=x \left (1-e^{2 i a} x^{2 i b}\right )^p \left (1+e^{2 i a} x^{2 i b}\right )^{-p} \left (-\frac {i \left (1+e^{2 i a} x^{2 i b}\right )}{1-e^{2 i a} x^{2 i b}}\right )^p \operatorname {AppellF1}\left (-\frac {i}{2 b},p,-p,1-\frac {i}{2 b},e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right ) \] Output:

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

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(330\) vs. \(2(142)=284\).

Time = 0.48 (sec) , antiderivative size = 330, normalized size of antiderivative = 2.32 \[ \int \cot ^p(a+b \log (x)) \, dx=\frac {(-i+2 b) x \left (\frac {i \left (1+e^{2 i a} x^{2 i b}\right )}{-1+e^{2 i a} x^{2 i b}}\right )^p \operatorname {AppellF1}\left (-\frac {i}{2 b},p,-p,1-\frac {i}{2 b},e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right )}{2 b e^{2 i a} p x^{2 i b} \operatorname {AppellF1}\left (1-\frac {i}{2 b},p,1-p,2-\frac {i}{2 b},e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right )+2 b e^{2 i a} p x^{2 i b} \operatorname {AppellF1}\left (1-\frac {i}{2 b},1+p,-p,2-\frac {i}{2 b},e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right )+(-i+2 b) \operatorname {AppellF1}\left (-\frac {i}{2 b},p,-p,1-\frac {i}{2 b},e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right )} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, 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.

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

\(\Big \downarrow \) 5003

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 937

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

\(\Big \downarrow \) 936

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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