Optimal. Leaf size=69 \[ -\frac {1}{16} \log \left (1-\tanh ^{\frac {2}{3}}(8 x)\right )+\frac {1}{32} \log \left (\tanh ^{\frac {4}{3}}(8 x)+\tanh ^{\frac {2}{3}}(8 x)+1\right )-\frac {1}{16} \sqrt {3} \tan ^{-1}\left (\frac {2 \tanh ^{\frac {2}{3}}(8 x)+1}{\sqrt {3}}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.125, Rules used = {3476, 329, 275, 292, 31, 634, 618, 204, 628} \[ -\frac {1}{16} \log \left (1-\tanh ^{\frac {2}{3}}(8 x)\right )+\frac {1}{32} \log \left (\tanh ^{\frac {4}{3}}(8 x)+\tanh ^{\frac {2}{3}}(8 x)+1\right )-\frac {1}{16} \sqrt {3} \tan ^{-1}\left (\frac {2 \tanh ^{\frac {2}{3}}(8 x)+1}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 275
Rule 292
Rule 329
Rule 618
Rule 628
Rule 634
Rule 3476
Rubi steps
\begin {align*} \int \sqrt [3]{\tanh (8 x)} \, dx &=-\left (\frac {1}{8} \operatorname {Subst}\left (\int \frac {\sqrt [3]{x}}{-1+x^2} \, dx,x,\tanh (8 x)\right )\right )\\ &=-\left (\frac {3}{8} \operatorname {Subst}\left (\int \frac {x^3}{-1+x^6} \, dx,x,\sqrt [3]{\tanh (8 x)}\right )\right )\\ &=-\left (\frac {3}{16} \operatorname {Subst}\left (\int \frac {x}{-1+x^3} \, dx,x,\tanh ^{\frac {2}{3}}(8 x)\right )\right )\\ &=-\left (\frac {1}{16} \operatorname {Subst}\left (\int \frac {1}{-1+x} \, dx,x,\tanh ^{\frac {2}{3}}(8 x)\right )\right )+\frac {1}{16} \operatorname {Subst}\left (\int \frac {-1+x}{1+x+x^2} \, dx,x,\tanh ^{\frac {2}{3}}(8 x)\right )\\ &=-\frac {1}{16} \log \left (1-\tanh ^{\frac {2}{3}}(8 x)\right )+\frac {1}{32} \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\tanh ^{\frac {2}{3}}(8 x)\right )-\frac {3}{32} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\tanh ^{\frac {2}{3}}(8 x)\right )\\ &=-\frac {1}{16} \log \left (1-\tanh ^{\frac {2}{3}}(8 x)\right )+\frac {1}{32} \log \left (1+\tanh ^{\frac {2}{3}}(8 x)+\tanh ^{\frac {4}{3}}(8 x)\right )+\frac {3}{16} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \tanh ^{\frac {2}{3}}(8 x)\right )\\ &=-\frac {1}{16} \sqrt {3} \tan ^{-1}\left (\frac {1+2 \tanh ^{\frac {2}{3}}(8 x)}{\sqrt {3}}\right )-\frac {1}{16} \log \left (1-\tanh ^{\frac {2}{3}}(8 x)\right )+\frac {1}{32} \log \left (1+\tanh ^{\frac {2}{3}}(8 x)+\tanh ^{\frac {4}{3}}(8 x)\right )\\ \end {align*}
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Mathematica [C] time = 0.03, size = 26, normalized size = 0.38 \[ \frac {3}{32} \tanh ^{\frac {4}{3}}(8 x) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\tanh ^2(8 x)\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 179, normalized size = 2.59 \[ -\frac {1}{16} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {\sinh \left (8 \, x\right )}{\cosh \left (8 \, x\right )}\right )^{\frac {2}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{16} \, \log \left (\left (\frac {\sinh \left (8 \, x\right )}{\cosh \left (8 \, x\right )}\right )^{\frac {2}{3}} - 1\right ) + \frac {1}{32} \, \log \left (\frac {\cosh \left (8 \, x\right )^{2} + 2 \, \cosh \left (8 \, x\right ) \sinh \left (8 \, x\right ) + \sinh \left (8 \, x\right )^{2} + {\left (\cosh \left (8 \, x\right )^{2} + 2 \, \cosh \left (8 \, x\right ) \sinh \left (8 \, x\right ) + \sinh \left (8 \, x\right )^{2} + 1\right )} \left (\frac {\sinh \left (8 \, x\right )}{\cosh \left (8 \, x\right )}\right )^{\frac {2}{3}} + {\left (\cosh \left (8 \, x\right )^{2} + 2 \, \cosh \left (8 \, x\right ) \sinh \left (8 \, x\right ) + \sinh \left (8 \, x\right )^{2} - 1\right )} \left (\frac {\sinh \left (8 \, x\right )}{\cosh \left (8 \, x\right )}\right )^{\frac {1}{3}} + 1}{\cosh \left (8 \, x\right )^{2} + 2 \, \cosh \left (8 \, x\right ) \sinh \left (8 \, x\right ) + \sinh \left (8 \, x\right )^{2} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 110, normalized size = 1.59 \[ -\frac {1}{16} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {e^{\left (16 \, x\right )} - 1}{e^{\left (16 \, x\right )} + 1}\right )^{\frac {2}{3}} + 1\right )}\right ) + \frac {1}{32} \, \log \left (\left (\frac {e^{\left (16 \, x\right )} - 1}{e^{\left (16 \, x\right )} + 1}\right )^{\frac {2}{3}} + \frac {\left (\frac {e^{\left (16 \, x\right )} - 1}{e^{\left (16 \, x\right )} + 1}\right )^{\frac {1}{3}} {\left (e^{\left (16 \, x\right )} - 1\right )}}{e^{\left (16 \, x\right )} + 1} + 1\right ) - \frac {1}{16} \, \log \left ({\left | \left (\frac {e^{\left (16 \, x\right )} - 1}{e^{\left (16 \, x\right )} + 1}\right )^{\frac {2}{3}} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 102, normalized size = 1.48 \[ \frac {\ln \left (\tanh ^{\frac {2}{3}}\left (8 x \right )+\tanh ^{\frac {1}{3}}\left (8 x \right )+1\right )}{32}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (1+2 \left (\tanh ^{\frac {1}{3}}\left (8 x \right )\right )\right ) \sqrt {3}}{3}\right )}{16}-\frac {\ln \left (\tanh ^{\frac {1}{3}}\left (8 x \right )-1\right )}{16}+\frac {\ln \left (\tanh ^{\frac {2}{3}}\left (8 x \right )-\left (\tanh ^{\frac {1}{3}}\left (8 x \right )\right )+1\right )}{32}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 \left (\tanh ^{\frac {1}{3}}\left (8 x \right )\right )-1\right ) \sqrt {3}}{3}\right )}{16}-\frac {\ln \left (\tanh ^{\frac {1}{3}}\left (8 x \right )+1\right )}{16} \]
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 \[ \int \tanh \left (8 \, x\right )^{\frac {1}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.32, size = 71, normalized size = 1.03 \[ -\frac {\ln \left (81\,{\mathrm {tanh}\left (8\,x\right )}^{2/3}-81\right )}{16}-\ln \left (162\,{\mathrm {tanh}\left (8\,x\right )}^{2/3}\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )-81\right )\,\left (-\frac {1}{32}+\frac {\sqrt {3}\,1{}\mathrm {i}}{32}\right )+\ln \left (-162\,{\mathrm {tanh}\left (8\,x\right )}^{2/3}\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )-81\right )\,\left (\frac {1}{32}+\frac {\sqrt {3}\,1{}\mathrm {i}}{32}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.70, size = 63, normalized size = 0.91 \[ - \frac {\log {\left (\tanh ^{\frac {2}{3}}{\left (8 x \right )} - 1 \right )}}{16} + \frac {\log {\left (\tanh ^{\frac {4}{3}}{\left (8 x \right )} + \tanh ^{\frac {2}{3}}{\left (8 x \right )} + 1 \right )}}{32} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} \left (\tanh ^{\frac {2}{3}}{\left (8 x \right )} + \frac {1}{2}\right )}{3} \right )}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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