Optimal. Leaf size=120 \[ x \left (1-e^{2 i a} x^{2 i}\right )^p \left (1+e^{2 i a} x^{2 i}\right )^{-p} \left (-\frac{i \left (1+e^{2 i a} x^{2 i}\right )}{1-e^{2 i a} x^{2 i}}\right )^p F_1\left (-\frac{i}{2};p,-p;1-\frac{i}{2};e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right ) \]
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Rubi [F] time = 0.0200752, 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 = {} \[ \int \cot ^p(a+\log (x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \cot ^p(a+\log (x)) \, dx &=\int \cot ^p(a+\log (x)) \, dx\\ \end{align*}
Mathematica [A] time = 0.479071, size = 238, normalized size = 1.98 \[ \frac{(2-i) x \left (\frac{i \left (1+e^{2 i a} x^{2 i}\right )}{-1+e^{2 i a} x^{2 i}}\right )^p F_1\left (-\frac{i}{2};p,-p;1-\frac{i}{2};e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right )}{(2-i) F_1\left (-\frac{i}{2};p,-p;1-\frac{i}{2};e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right )+2 e^{2 i a} p x^{2 i} \left (F_1\left (1-\frac{i}{2};p,1-p;2-\frac{i}{2};e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right )+F_1\left (1-\frac{i}{2};p+1,-p;2-\frac{i}{2};e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.322, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( a+\ln \left ( x \right ) \right ) \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cot \left (a + \log \left (x\right )\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\cot \left (a + \log \left (x\right )\right )^{p}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cot ^{p}{\left (a + \log{\left (x \right )} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cot \left (a + \log \left (x\right )\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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