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 - 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[-I/4, 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

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Rubi [F]  time = 0.0202924, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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 = 0.465129, 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[-I/4, p, -p, 1 - I/4, E^((2*I
)*a)*x^(4*I), -(E^((2*I)*a)*x^(4*I))])/((4 - I)*AppellF1[-I/4, 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))]))

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Maple [F]  time = 0.357, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( a+2\,\ln \left ( x \right ) \right ) \right ) ^{p}\, dx \end{align*}

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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cot \left (a + 2 \, \log \left (x\right )\right )^{p}\,{d x} \end{align*}

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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\cot \left (a + 2 \, \log \left (x\right )\right )^{p}, x\right ) \end{align*}

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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cot ^{p}{\left (a + 2 \log{\left (x \right )} \right )}\, dx \end{align*}

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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cot \left (a + 2 \, \log \left (x\right )\right )^{p}\,{d x} \end{align*}

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)