Optimal. Leaf size=190 \[ \frac {1}{3} e^{-12 a} \left (-1+e^{2 a} \sqrt [4]{x}\right )^{1+p} \left (1+e^{2 a} \sqrt [4]{x}\right )^{1-p} \left (e^{4 a} \left (3+2 p^2\right )-2 e^{6 a} p \sqrt [4]{x}\right )+e^{-4 a} \left (-1+e^{2 a} \sqrt [4]{x}\right )^{1+p} \left (1+e^{2 a} \sqrt [4]{x}\right )^{1-p} \sqrt {x}-\frac {2^{2-p} e^{-8 a} p \left (2+p^2\right ) \left (-1+e^{2 a} \sqrt [4]{x}\right )^{1+p} \, _2F_1\left (p,1+p;2+p;\frac {1}{2} \left (1-e^{2 a} \sqrt [4]{x}\right )\right )}{3 (1+p)} \]
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Rubi [A]
time = 0.10, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {5652, 383, 102,
152, 71} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 102
Rule 152
Rule 383
Rule 5652
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] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 2.24, size = 177, normalized size = 0.93 \begin {gather*} \frac {5 \left (\frac {-1+e^{2 a} \sqrt [4]{x}}{1+e^{2 a} \sqrt [4]{x}}\right )^p x 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,1+p;6;e^{2 a} \sqrt [4]{x},-e^{2 a} \sqrt [4]{x}\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.63, size = 0, normalized size = 0.00 \[\int \tanh ^{p}\left (a +\frac {\ln \left (x \right )}{8}\right )\, dx\]
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 \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 + \frac {\log {\left (x \right )}}{8} \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 = -0.01 \begin {gather*} \int {\mathrm {tanh}\left (a+\frac {\ln \left (x\right )}{8}\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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