Optimal. Leaf size=190 \[ -\frac {e^{-8 a} 2^{2-p} p \left (p^2+2\right ) \left (e^{2 a} \sqrt [4]{x}-1\right )^{p+1} \, _2F_1\left (p,p+1;p+2;\frac {1}{2} \left (1-e^{2 a} \sqrt [4]{x}\right )\right )}{3 (p+1)}+\frac {1}{3} e^{-12 a} \left (e^{2 a} \sqrt [4]{x}-1\right )^{p+1} \left (e^{4 a} \left (2 p^2+3\right )-2 e^{6 a} p \sqrt [4]{x}\right ) \left (e^{2 a} \sqrt [4]{x}+1\right )^{1-p}+e^{-4 a} \sqrt {x} \left (e^{2 a} \sqrt [4]{x}-1\right )^{p+1} \left (e^{2 a} \sqrt [4]{x}+1\right )^{1-p} \]
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Rubi [F] time = 0.05, 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 = {} \[ \int \tanh ^p\left (a+\frac {\log (x)}{8}\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \tanh ^p\left (a+\frac {\log (x)}{8}\right ) \, dx &=\int \tanh ^p\left (\frac {1}{8} (8 a+\log (x))\right ) \, dx\\ \end {align*}
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Mathematica [C] time = 3.41, size = 177, normalized size = 0.93 \[ \frac {5 x \left (\frac {e^{2 a} \sqrt [4]{x}-1}{e^{2 a} \sqrt [4]{x}+1}\right )^p F_1\left (4;-p,p;5;e^{2 a} \sqrt [4]{x},-e^{2 a} \sqrt [4]{x}\right )}{5 F_1\left (4;-p,p;5;e^{2 a} \sqrt [4]{x},-e^{2 a} \sqrt [4]{x}\right )-e^{2 a} p \sqrt [4]{x} \left (F_1\left (5;1-p,p;6;e^{2 a} \sqrt [4]{x},-e^{2 a} \sqrt [4]{x}\right )+F_1\left (5;-p,p+1;6;e^{2 a} \sqrt [4]{x},-e^{2 a} \sqrt [4]{x}\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 2.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\tanh \left (a + \frac {1}{8} \, \log \relax (x)\right )^{p}, x\right ) \]
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 \[ \int \tanh \left (a + \frac {1}{8} \, \log \relax (x)\right )^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \tanh ^{p}\left (a +\frac {\ln \relax (x )}{8}\right )\, dx \]
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 \[ \int \tanh \left (a + \frac {1}{8} \, \log \relax (x)\right )^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {tanh}\left (a+\frac {\ln \relax (x)}{8}\right )}^p \,d x \]
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 \[ \int \tanh ^{p}{\left (a + \frac {\log {\relax (x )}}{8} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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