Integrand size = 9, antiderivative size = 120 \[ \int \cot ^p(a+2 \log (x)) \, dx=\left (1-e^{2 i a} x^{4 i}\right )^p \left (1+e^{2 i a} x^{4 i}\right )^{-p} \left (-\frac {i \left (1+e^{2 i a} x^{4 i}\right )}{1-e^{2 i a} x^{4 i}}\right )^p x \operatorname {AppellF1}\left (-\frac {i}{4},p,-p,1-\frac {i}{4},e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right ) \]
(1-exp(2*I*a)*x^(4*I))^p*(-I*(1+exp(2*I*a)*x^(4*I))/(1-exp(2*I*a)*x^(4*I)) )^p*x*AppellF1(-1/4*I,p,-p,1-1/4*I,exp(2*I*a)*x^(4*I),-exp(2*I*a)*x^(4*I)) /((1+exp(2*I*a)*x^(4*I))^p)
Time = 0.43 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.98 \[ \int \cot ^p(a+2 \log (x)) \, dx=\frac {(4-i) \left (\frac {i \left (1+e^{2 i a} x^{4 i}\right )}{-1+e^{2 i a} x^{4 i}}\right )^p x \operatorname {AppellF1}\left (-\frac {i}{4},p,-p,1-\frac {i}{4},e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right )}{(4-i) \operatorname {AppellF1}\left (-\frac {i}{4},p,-p,1-\frac {i}{4},e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right )+4 e^{2 i a} p x^{4 i} \left (\operatorname {AppellF1}\left (1-\frac {i}{4},p,1-p,2-\frac {i}{4},e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right )+\operatorname {AppellF1}\left (1-\frac {i}{4},1+p,-p,2-\frac {i}{4},e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right )\right )} \]
((4 - I)*((I*(1 + E^((2*I)*a)*x^(4*I)))/(-1 + E^((2*I)*a)*x^(4*I)))^p*x*Ap pellF1[-1/4*I, p, -p, 1 - I/4, E^((2*I)*a)*x^(4*I), -(E^((2*I)*a)*x^(4*I)) ])/((4 - I)*AppellF1[-1/4*I, p, -p, 1 - I/4, E^((2*I)*a)*x^(4*I), -(E^((2* I)*a)*x^(4*I))] + 4*E^((2*I)*a)*p*x^(4*I)*(AppellF1[1 - I/4, p, 1 - p, 2 - I/4, E^((2*I)*a)*x^(4*I), -(E^((2*I)*a)*x^(4*I))] + AppellF1[1 - I/4, 1 + p, -p, 2 - I/4, E^((2*I)*a)*x^(4*I), -(E^((2*I)*a)*x^(4*I))]))
Time = 0.27 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, 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.
\(\displaystyle \int \cot ^p(a+2 \log (x)) \, dx\) |
\(\Big \downarrow \) 5003 |
\(\displaystyle \int \left (\frac {-i e^{2 i a} x^{4 i}-i}{1-e^{2 i a} x^{4 i}}\right )^pdx\) |
\(\Big \downarrow \) 2058 |
\(\displaystyle \left (1-e^{2 i a} x^{4 i}\right )^p \left (-i e^{2 i a} x^{4 i}-i\right )^{-p} \left (-\frac {i \left (1+e^{2 i a} x^{4 i}\right )}{1-e^{2 i a} x^{4 i}}\right )^p \int \left (1-e^{2 i a} x^{4 i}\right )^{-p} \left (-i e^{2 i a} x^{4 i}-i\right )^pdx\) |
\(\Big \downarrow \) 937 |
\(\displaystyle \left (1-e^{2 i a} x^{4 i}\right )^p \left (1+e^{2 i a} x^{4 i}\right )^{-p} \left (-\frac {i \left (1+e^{2 i a} x^{4 i}\right )}{1-e^{2 i a} x^{4 i}}\right )^p \int \left (1-e^{2 i a} x^{4 i}\right )^{-p} \left (e^{2 i a} x^{4 i}+1\right )^pdx\) |
\(\Big \downarrow \) 936 |
\(\displaystyle x \left (1-e^{2 i a} x^{4 i}\right )^p \left (1+e^{2 i a} x^{4 i}\right )^{-p} \left (-\frac {i \left (1+e^{2 i a} x^{4 i}\right )}{1-e^{2 i a} x^{4 i}}\right )^p \operatorname {AppellF1}\left (-\frac {i}{4},p,-p,1-\frac {i}{4},e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right )\) |
((1 - E^((2*I)*a)*x^(4*I))^p*(((-I)*(1 + E^((2*I)*a)*x^(4*I)))/(1 - E^((2* I)*a)*x^(4*I)))^p*x*AppellF1[-1/4*I, p, -p, 1 - I/4, E^((2*I)*a)*x^(4*I), -(E^((2*I)*a)*x^(4*I))])/(1 + E^((2*I)*a)*x^(4*I))^p
3.3.7.3.1 Defintions of rubi rules used
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])
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])
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]
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]
\[\int \cot \left (a +2 \ln \left (x \right )\right )^{p}d x\]
\[ \int \cot ^p(a+2 \log (x)) \, dx=\int { \cot \left (a + 2 \, \log \left (x\right )\right )^{p} \,d x } \]
\[ \int \cot ^p(a+2 \log (x)) \, dx=\int \cot ^{p}{\left (a + 2 \log {\left (x \right )} \right )}\, dx \]
\[ \int \cot ^p(a+2 \log (x)) \, dx=\int { \cot \left (a + 2 \, \log \left (x\right )\right )^{p} \,d x } \]
\[ \int \cot ^p(a+2 \log (x)) \, dx=\int { \cot \left (a + 2 \, \log \left (x\right )\right )^{p} \,d x } \]
Timed out. \[ \int \cot ^p(a+2 \log (x)) \, dx=\int {\mathrm {cot}\left (a+2\,\ln \left (x\right )\right )}^p \,d x \]