Optimal. Leaf size=52 \[ -\frac{e^{-2 a} 2^{-p} \left (-e^{2 a} x-1\right )^{p+1} \, _2F_1\left (p,p+1;p+2;\frac{1}{2} \left (e^{2 a} x+1\right )\right )}{p+1} \]
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Rubi [F] time = 0.046479, 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 = {} \[ \int \coth ^p\left (a+\frac{\log (x)}{2}\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \coth ^p\left (a+\frac{\log (x)}{2}\right ) \, dx &=\int \coth ^p\left (\frac{1}{2} (2 a+\log (x))\right ) \, dx\\ \end{align*}
Mathematica [A] time = 0.391243, size = 83, normalized size = 1.6 \[ -\frac{e^{-2 a} 2^p \left (e^{2 a} x+1\right )^{1-p} \left (\frac{e^{2 a} x+1}{e^{2 a} x-1}\right )^{p-1} \, _2F_1\left (1-p,-p;2-p;\frac{1}{2}-\frac{1}{2} e^{2 a} x\right )}{p-1} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm coth} \left (a+{\frac{\ln \left ( x \right ) }{2}}\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 \coth \left (a + \frac{1}{2} \, \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 (\coth \left (a + \frac{1}{2} \, \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 \coth ^{p}{\left (a + \frac{\log{\left (x \right )}}{2} \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 \coth \left (a + \frac{1}{2} \, \log \left (x\right )\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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