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 = {5652, 441, 440}
\begin {gather*} x \left (1-e^{2 a} x^2\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 5652
Rubi steps
\begin {align*} \int \tanh ^p(a+\log (x)) \, dx &=\int \tanh ^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 = 1.17, 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};1-p,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 )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.64, size = 0, normalized size = 0.00 \[\int \tanh ^{p}\left (a +\ln \left (x \right )\right )\, dx\]
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 \tanh ^{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 {tanh}\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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