Optimal. Leaf size=61 \[ x \left (-e^{2 a} x^2-1\right )^p \left (e^{2 a} x^2+1\right )^{-p} F_1\left (\frac {1}{2};p,-p;\frac {3}{2};e^{2 a} x^2,-e^{2 a} x^2\right ) \]
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Rubi [F] time = 0.02, 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 = {} \[ \int \coth ^p(a+\log (x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \coth ^p(a+\log (x)) \, dx &=\int \coth ^p(a+\log (x)) \, dx\\ \end {align*}
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Mathematica [B] time = 1.76, size = 171, normalized size = 2.80 \[ \frac {3 x \left (\frac {e^{2 a} x^2+1}{e^{2 a} x^2-1}\right )^p F_1\left (\frac {1}{2};p,-p;\frac {3}{2};e^{2 a} x^2,-e^{2 a} x^2\right )}{2 e^{2 a} p x^2 \left (F_1\left (\frac {3}{2};p,1-p;\frac {5}{2};e^{2 a} x^2,-e^{2 a} x^2\right )+F_1\left (\frac {3}{2};p+1,-p;\frac {5}{2};e^{2 a} x^2,-e^{2 a} x^2\right )\right )+3 F_1\left (\frac {1}{2};p,-p;\frac {3}{2};e^{2 a} x^2,-e^{2 a} x^2\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\coth \left (a + \log \relax (x)\right )^{p}, x\right ) \]
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 \[ \int \coth \left (a + \log \relax (x)\right )^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.18, size = 0, normalized size = 0.00 \[ \int \coth ^{p}\left (a +\ln \relax (x )\right )\, dx \]
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 \[ \int \coth \left (a + \log \relax (x)\right )^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {coth}\left (a+\ln \relax (x)\right )}^p \,d x \]
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 \[ \int \coth ^{p}{\left (a + \log {\relax (x )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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