Optimal. Leaf size=61 \[ x \left (-1-e^{2 a} x^2\right )^p \left (1+e^{2 a} x^2\right )^{-p} F_1\left (\frac {1}{2};p,-p;\frac {3}{2};e^{2 a} x^2,-e^{2 a} x^2\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5653, 441, 440}
\begin {gather*} 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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 440
Rule 441
Rule 5653
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] Leaf count is larger than twice the leaf count of optimal. \(171\) vs. \(2(61)=122\).
time = 1.11, size = 171, normalized size = 2.80 \begin {gather*} \frac {3 x \left (\frac {1+e^{2 a} x^2}{-1+e^{2 a} x^2}\right )^p F_1\left (\frac {1}{2};p,-p;\frac {3}{2};e^{2 a} x^2,-e^{2 a} x^2\right )}{3 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};1+p,-p;\frac {5}{2};e^{2 a} x^2,-e^{2 a} x^2\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.84, size = 0, normalized size = 0.00 \[\int \coth ^{p}\left (a +\ln \left (x \right )\right )\, dx\]
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 + \log {\left (x \right )} \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 = -0.02 \begin {gather*} \int {\mathrm {coth}\left (a+\ln \left (x\right )\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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