Optimal. Leaf size=51 \[ \frac{e^{-2 a} 2^{-p} \left (e^{2 a} x-1\right )^{p+1} \, _2F_1\left (p,p+1;p+2;\frac{1}{2} \left (1-e^{2 a} x\right )\right )}{p+1} \]
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Rubi [F] time = 0.0468423, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \tanh ^p\left (a+\frac{\log (x)}{2}\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \tanh ^p\left (a+\frac{\log (x)}{2}\right ) \, dx &=\int \tanh ^p\left (\frac{1}{2} (2 a+\log (x))\right ) \, dx\\ \end{align*}
Mathematica [A] time = 2.90375, size = 76, normalized size = 1.49 \[ \frac{e^{-2 a} 2^{-p} \left (\frac{e^{2 a} x-1}{e^{2 a} x+1}\right )^{p+1} \left (e^{2 a} x+1\right )^{p+1} \, _2F_1\left (p,p+1;p+2;\frac{1}{2} \left (1-e^{2 a} x\right )\right )}{p+1} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.033, size = 0, normalized size = 0. \begin{align*} \int \left ( \tanh \left ( a+{\frac{\ln \left ( x \right ) }{2}} \right ) \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \tanh \left (a + \frac{1}{2} \, \log \left (x\right )\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\tanh \left (a + \frac{1}{2} \, \log \left (x\right )\right )^{p}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \tanh ^{p}{\left (a + \frac{\log{\left (x \right )}}{2} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \tanh \left (a + \frac{1}{2} \, \log \left (x\right )\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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