3.207 \(\int \cot ^p(a+2 \log (x)) \, dx\)

Optimal. Leaf size=120 \[ x \left (1-e^{2 i a} x^{4 i}\right )^p \left (1+e^{2 i a} x^{4 i}\right )^{-p} \left (-\frac {i \left (1+e^{2 i a} x^{4 i}\right )}{1-e^{2 i a} x^{4 i}}\right )^p F_1\left (-\frac {i}{4};p,-p;1-\frac {i}{4};e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right ) \]

[Out]

(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,e
xp(2*I*a)*x^(4*I),-exp(2*I*a)*x^(4*I))/((1+exp(2*I*a)*x^(4*I))^p)

________________________________________________________________________________________

Rubi [F]  time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \cot ^p(a+2 \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Cot[a + 2*Log[x]]^p,x]

[Out]

Defer[Int][Cot[a + 2*Log[x]]^p, x]

Rubi steps

\begin {align*} \int \cot ^p(a+2 \log (x)) \, dx &=\int \cot ^p(a+2 \log (x)) \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.47, size = 238, normalized size = 1.98 \[ \frac {(4-i) x \left (\frac {i \left (1+e^{2 i a} x^{4 i}\right )}{-1+e^{2 i a} x^{4 i}}\right )^p F_1\left (-\frac {i}{4};p,-p;1-\frac {i}{4};e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right )}{(4-i) F_1\left (-\frac {i}{4};p,-p;1-\frac {i}{4};e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right )+4 e^{2 i a} p x^{4 i} \left (F_1\left (1-\frac {i}{4};p,1-p;2-\frac {i}{4};e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right )+F_1\left (1-\frac {i}{4};p+1,-p;2-\frac {i}{4};e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cot[a + 2*Log[x]]^p,x]

[Out]

((4 - I)*((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))])/((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))]))

________________________________________________________________________________________

fricas [F]  time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cot \left (a + 2 \, \log \relax (x)\right )^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+2*log(x))^p,x, algorithm="fricas")

[Out]

integral(cot(a + 2*log(x))^p, x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cot \left (a + 2 \, \log \relax (x)\right )^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+2*log(x))^p,x, algorithm="giac")

[Out]

integrate(cot(a + 2*log(x))^p, x)

________________________________________________________________________________________

maple [F]  time = 0.30, size = 0, normalized size = 0.00 \[ \int \cot ^{p}\left (a +2 \ln \relax (x )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(a+2*ln(x))^p,x)

[Out]

int(cot(a+2*ln(x))^p,x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cot \left (a + 2 \, \log \relax (x)\right )^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+2*log(x))^p,x, algorithm="maxima")

[Out]

integrate(cot(a + 2*log(x))^p, x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {cot}\left (a+2\,\ln \relax (x)\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(a + 2*log(x))^p,x)

[Out]

int(cot(a + 2*log(x))^p, x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cot ^{p}{\left (a + 2 \log {\relax (x )} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+2*ln(x))**p,x)

[Out]

Integral(cot(a + 2*log(x))**p, x)

________________________________________________________________________________________