Optimal. Leaf size=102 \[ \frac{\log \left (2 \sqrt [3]{2} \tanh ^2(x)+2^{2/3} \sqrt [3]{3} \tanh (x)+3^{2/3}\right )}{6\ 6^{2/3}}-\frac{\log \left (\sqrt [3]{3}-2^{2/3} \tanh (x)\right )}{3\ 6^{2/3}}+\frac{\tan ^{-1}\left (\frac{2\ 2^{2/3} \tanh (x)+\sqrt [3]{3}}{3^{5/6}}\right )}{3\ 2^{2/3} \sqrt [6]{3}} \]
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Rubi [A] time = 0.112152, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used = {3675, 200, 31, 634, 617, 204, 628} \[ \frac{\log \left (2 \sqrt [3]{2} \tanh ^2(x)+2^{2/3} \sqrt [3]{3} \tanh (x)+3^{2/3}\right )}{6\ 6^{2/3}}-\frac{\log \left (\sqrt [3]{3}-2^{2/3} \tanh (x)\right )}{3\ 6^{2/3}}+\frac{\tan ^{-1}\left (\frac{2\ 2^{2/3} \tanh (x)+\sqrt [3]{3}}{3^{5/6}}\right )}{3\ 2^{2/3} \sqrt [6]{3}} \]
Antiderivative was successfully verified.
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Rule 3675
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{\text{sech}^2(x)}{3-4 \tanh ^3(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{3-4 x^3} \, dx,x,\tanh (x)\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{3}-2^{2/3} x} \, dx,x,\tanh (x)\right )}{3\ 3^{2/3}}+\frac{\operatorname{Subst}\left (\int \frac{2 \sqrt [3]{3}+2^{2/3} x}{3^{2/3}+2^{2/3} \sqrt [3]{3} x+2 \sqrt [3]{2} x^2} \, dx,x,\tanh (x)\right )}{3\ 3^{2/3}}\\ &=-\frac{\log \left (\sqrt [3]{3}-2^{2/3} \tanh (x)\right )}{3\ 6^{2/3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{3^{2/3}+2^{2/3} \sqrt [3]{3} x+2 \sqrt [3]{2} x^2} \, dx,x,\tanh (x)\right )}{2 \sqrt [3]{3}}+\frac{\operatorname{Subst}\left (\int \frac{2^{2/3} \sqrt [3]{3}+4 \sqrt [3]{2} x}{3^{2/3}+2^{2/3} \sqrt [3]{3} x+2 \sqrt [3]{2} x^2} \, dx,x,\tanh (x)\right )}{6\ 6^{2/3}}\\ &=-\frac{\log \left (\sqrt [3]{3}-2^{2/3} \tanh (x)\right )}{3\ 6^{2/3}}+\frac{\log \left (3^{2/3}+2^{2/3} \sqrt [3]{3} \tanh (x)+2 \sqrt [3]{2} \tanh ^2(x)\right )}{6\ 6^{2/3}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2\ 2^{2/3} \tanh (x)}{\sqrt [3]{3}}\right )}{6^{2/3}}\\ &=\frac{\tan ^{-1}\left (\frac{3+2\ 6^{2/3} \tanh (x)}{3 \sqrt{3}}\right )}{3\ 2^{2/3} \sqrt [6]{3}}-\frac{\log \left (\sqrt [3]{3}-2^{2/3} \tanh (x)\right )}{3\ 6^{2/3}}+\frac{\log \left (3^{2/3}+2^{2/3} \sqrt [3]{3} \tanh (x)+2 \sqrt [3]{2} \tanh ^2(x)\right )}{6\ 6^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.105728, size = 74, normalized size = 0.73 \[ \frac{\log \left (2 \sqrt [3]{6} \tanh ^2(x)+6^{2/3} \tanh (x)+3\right )-2 \log \left (3-6^{2/3} \tanh (x)\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2\ 6^{2/3} \tanh (x)+3}{3 \sqrt{3}}\right )}{6\ 6^{2/3}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.052, size = 34, normalized size = 0.3 \begin{align*}{\frac{1}{3}\sum _{{\it \_R}={\it RootOf} \left ( 36\,{{\it \_Z}}^{3}+1 \right ) }{\it \_R}\,\ln \left ( -24\,\tanh \left ( x/2 \right ){{\it \_R}}^{2}+ \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) } \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 \frac{\operatorname{sech}\left (x\right )^{2}}{4 \, \tanh \left (x\right )^{3} - 3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.1398, size = 1177, normalized size = 11.54 \begin{align*} \text{result too large to display} \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 \frac{\operatorname{sech}^{2}{\left (x \right )}}{4 \tanh ^{3}{\left (x \right )} - 3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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