Optimal. Leaf size=162 \[ e^{-6 a} p \left (-1-e^{2 a} \sqrt [3]{x}\right )^{1+p} \left (1-e^{2 a} \sqrt [3]{x}\right )^{1-p}+e^{-4 a} \left (-1-e^{2 a} \sqrt [3]{x}\right )^{1+p} \left (1-e^{2 a} \sqrt [3]{x}\right )^{1-p} \sqrt [3]{x}-\frac {2^{-p} e^{-6 a} \left (1+2 p^2\right ) \left (-1-e^{2 a} \sqrt [3]{x}\right )^{1+p} \, _2F_1\left (p,1+p;2+p;\frac {1}{2} \left (1+e^{2 a} \sqrt [3]{x}\right )\right )}{1+p} \]
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Rubi [A]
time = 0.09, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {5653, 383, 92,
81, 71} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 81
Rule 92
Rule 383
Rule 5653
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 = 0.50, size = 142, normalized size = 0.88 \begin {gather*} \frac {e^{-6 a} \left (1+e^{2 a} \sqrt [3]{x}\right )^{1-p} \left (\frac {1+e^{2 a} \sqrt [3]{x}}{-1+e^{2 a} \sqrt [3]{x}}\right )^{-1+p} \left ((-1+p) \left (1+e^{2 a} \sqrt [3]{x}\right )^{1+p} \left (p+e^{2 a} \sqrt [3]{x}\right )-2^p \left (1+2 p^2\right ) \, _2F_1\left (1-p,-p;2-p;\frac {1}{2}-\frac {1}{2} e^{2 a} \sqrt [3]{x}\right )\right )}{-1+p} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.85, size = 0, normalized size = 0.00 \[\int \coth ^{p}\left (a +\frac {\ln \left (x \right )}{6}\right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \coth ^{p}{\left (a + \frac {\log {\left (x \right )}}{6} \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\mathrm {coth}\left (a+\frac {\ln \left (x\right )}{6}\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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