Optimal. Leaf size=162 \[ -\frac {e^{-6 a} 2^{-p} \left (2 p^2+1\right ) \left (-e^{2 a} \sqrt [3]{x}-1\right )^{p+1} \, _2F_1\left (p,p+1;p+2;\frac {1}{2} \left (e^{2 a} \sqrt [3]{x}+1\right )\right )}{p+1}+e^{-6 a} p \left (-e^{2 a} \sqrt [3]{x}-1\right )^{p+1} \left (1-e^{2 a} \sqrt [3]{x}\right )^{1-p}+e^{-4 a} \sqrt [3]{x} \left (-e^{2 a} \sqrt [3]{x}-1\right )^{p+1} \left (1-e^{2 a} \sqrt [3]{x}\right )^{1-p} \]
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Rubi [F] time = 0.05, 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 = {} \[ \int \coth ^p\left (a+\frac {\log (x)}{6}\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \coth ^p\left (a+\frac {\log (x)}{6}\right ) \, dx &=\int \coth ^p\left (\frac {1}{6} (6 a+\log (x))\right ) \, dx\\ \end {align*}
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Mathematica [A] time = 0.60, size = 142, normalized size = 0.88 \[ \frac {e^{-6 a} \left (e^{2 a} \sqrt [3]{x}+1\right )^{1-p} \left (\frac {e^{2 a} \sqrt [3]{x}+1}{e^{2 a} \sqrt [3]{x}-1}\right )^{p-1} \left ((p-1) \left (e^{2 a} \sqrt [3]{x}+1\right )^{p+1} \left (e^{2 a} \sqrt [3]{x}+p\right )-2^p \left (2 p^2+1\right ) \, _2F_1\left (1-p,-p;2-p;\frac {1}{2}-\frac {1}{2} e^{2 a} \sqrt [3]{x}\right )\right )}{p-1} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\coth \left (a + \frac {1}{6} \, \log \relax (x)\right )^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \coth \left (a + \frac {1}{6} \, \log \relax (x)\right )^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.15, size = 0, normalized size = 0.00 \[ \int \coth ^{p}\left (a +\frac {\ln \relax (x )}{6}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \coth \left (a + \frac {1}{6} \, \log \relax (x)\right )^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {coth}\left (a+\frac {\ln \relax (x)}{6}\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \coth ^{p}{\left (a + \frac {\log {\relax (x )}}{6} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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