Optimal. Leaf size=98 \[ \frac{\log \left (3^{2/3} \cosh ^2(x)+2^{2/3} \sqrt [3]{3} \cosh (x)+2 \sqrt [3]{2}\right )}{12 \sqrt [3]{6}}-\frac{\log \left (2^{2/3}-\sqrt [3]{3} \cosh (x)\right )}{6 \sqrt [3]{6}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{6} \cosh (x)+1}{\sqrt{3}}\right )}{2 \sqrt [3]{2} 3^{5/6}} \]
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Rubi [A] time = 0.116689, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {3223, 200, 31, 634, 617, 204, 628} \[ \frac{\log \left (3^{2/3} \cosh ^2(x)+2^{2/3} \sqrt [3]{3} \cosh (x)+2 \sqrt [3]{2}\right )}{12 \sqrt [3]{6}}-\frac{\log \left (2^{2/3}-\sqrt [3]{3} \cosh (x)\right )}{6 \sqrt [3]{6}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{6} \cosh (x)+1}{\sqrt{3}}\right )}{2 \sqrt [3]{2} 3^{5/6}} \]
Antiderivative was successfully verified.
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Rule 3223
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{\sinh (x)}{4-3 \cosh ^3(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{4-3 x^3} \, dx,x,\cosh (x)\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{2^{2/3}-\sqrt [3]{3} x} \, dx,x,\cosh (x)\right )}{6 \sqrt [3]{2}}+\frac{\operatorname{Subst}\left (\int \frac{2\ 2^{2/3}+\sqrt [3]{3} x}{2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\cosh (x)\right )}{6 \sqrt [3]{2}}\\ &=-\frac{\log \left (2^{2/3}-\sqrt [3]{3} \cosh (x)\right )}{6 \sqrt [3]{6}}+\frac{\operatorname{Subst}\left (\int \frac{1}{2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\cosh (x)\right )}{2\ 2^{2/3}}+\frac{\operatorname{Subst}\left (\int \frac{2^{2/3} \sqrt [3]{3}+2\ 3^{2/3} x}{2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\cosh (x)\right )}{12 \sqrt [3]{6}}\\ &=-\frac{\log \left (2^{2/3}-\sqrt [3]{3} \cosh (x)\right )}{6 \sqrt [3]{6}}+\frac{\log \left (2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{3} \cosh (x)+3^{2/3} \cosh ^2(x)\right )}{12 \sqrt [3]{6}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\sqrt [3]{6} \cosh (x)\right )}{2 \sqrt [3]{6}}\\ &=\frac{\tan ^{-1}\left (\frac{1+\sqrt [3]{6} \cosh (x)}{\sqrt{3}}\right )}{2 \sqrt [3]{2} 3^{5/6}}-\frac{\log \left (2^{2/3}-\sqrt [3]{3} \cosh (x)\right )}{6 \sqrt [3]{6}}+\frac{\log \left (2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{3} \cosh (x)+3^{2/3} \cosh ^2(x)\right )}{12 \sqrt [3]{6}}\\ \end{align*}
Mathematica [A] time = 0.0863516, size = 77, normalized size = 0.79 \[ \frac{1}{72} \left (6^{2/3} \left (\log \left (6^{2/3} \cosh ^2(x)+2 \sqrt [3]{6} \cosh (x)+4\right )-2 \log \left (2-\sqrt [3]{6} \cosh (x)\right )\right )+6\ 2^{2/3} \sqrt [6]{3} \tan ^{-1}\left (\frac{\sqrt [3]{6} \cosh (x)+1}{\sqrt{3}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 80, normalized size = 0.8 \begin{align*} -{\frac{\sqrt [3]{4}{3}^{{\frac{2}{3}}}}{36}\ln \left ( \cosh \left ( x \right ) -{\frac{\sqrt [3]{4}{3}^{{\frac{2}{3}}}}{3}} \right ) }+{\frac{\sqrt [3]{4}{3}^{{\frac{2}{3}}}}{72}\ln \left ( \left ( \cosh \left ( x \right ) \right ) ^{2}+{\frac{\sqrt [3]{4}{3}^{{\frac{2}{3}}}\cosh \left ( x \right ) }{3}}+{\frac{{4}^{{\frac{2}{3}}}\sqrt [3]{3}}{3}} \right ) }+{\frac{\sqrt [3]{4}\sqrt [6]{3}}{12}\arctan \left ({\frac{\sqrt{3}}{3} \left ({\frac{{4}^{{\frac{2}{3}}}\sqrt [3]{3}\cosh \left ( x \right ) }{2}}+1 \right ) } \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{\sinh \left (x\right )}{3 \, \cosh \left (x\right )^{3} - 4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.86329, size = 1143, normalized size = 11.66 \begin{align*} \frac{1}{12} \cdot 6^{\frac{1}{6}} \sqrt{2} \left (-1\right )^{\frac{1}{3}} \arctan \left (\frac{1}{12} \cdot 6^{\frac{1}{6}}{\left (6^{\frac{2}{3}} \sqrt{2} \left (-1\right )^{\frac{2}{3}} \cosh \left (x\right )^{3} + 6^{\frac{2}{3}} \sqrt{2} \left (-1\right )^{\frac{2}{3}} \sinh \left (x\right )^{3} +{\left (3 \cdot 6^{\frac{2}{3}} \sqrt{2} \left (-1\right )^{\frac{2}{3}} \cosh \left (x\right ) + 4 \cdot 6^{\frac{1}{3}} \sqrt{2}\right )} \sinh \left (x\right )^{2} + 4 \cdot 6^{\frac{1}{3}} \sqrt{2} \cosh \left (x\right )^{2} +{\left (6^{\frac{2}{3}} \sqrt{2} \left (-1\right )^{\frac{2}{3}} - 16 \, \sqrt{2} \left (-1\right )^{\frac{1}{3}}\right )} \cosh \left (x\right ) +{\left (3 \cdot 6^{\frac{2}{3}} \sqrt{2} \left (-1\right )^{\frac{2}{3}} \cosh \left (x\right )^{2} + 6^{\frac{2}{3}} \sqrt{2} \left (-1\right )^{\frac{2}{3}} + 8 \cdot 6^{\frac{1}{3}} \sqrt{2} \cosh \left (x\right ) - 16 \, \sqrt{2} \left (-1\right )^{\frac{1}{3}}\right )} \sinh \left (x\right ) + 2 \cdot 6^{\frac{1}{3}} \sqrt{2}\right )}\right ) - \frac{1}{12} \cdot 6^{\frac{1}{6}} \sqrt{2} \left (-1\right )^{\frac{1}{3}} \arctan \left (\frac{1}{12} \cdot 6^{\frac{1}{6}}{\left (6^{\frac{2}{3}} \sqrt{2} \left (-1\right )^{\frac{2}{3}} \cosh \left (x\right ) + 6^{\frac{2}{3}} \sqrt{2} \left (-1\right )^{\frac{2}{3}} \sinh \left (x\right ) + 2 \cdot 6^{\frac{1}{3}} \sqrt{2}\right )}\right ) - \frac{1}{72} \cdot 6^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} \log \left (-\frac{2 \,{\left (2 \cdot 6^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} \cosh \left (x\right ) - 3 \, \cosh \left (x\right )^{2} - 3 \, \sinh \left (x\right )^{2} - 4 \cdot 6^{\frac{1}{3}} \left (-1\right )^{\frac{2}{3}} - 3\right )}}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) + \frac{1}{36} \cdot 6^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} \log \left (\frac{2 \,{\left (6^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} + 3 \, \cosh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.8929, size = 85, normalized size = 0.87 \begin{align*} - \frac{6^{\frac{2}{3}} \log{\left (\cosh{\left (x \right )} - \frac{6^{\frac{2}{3}}}{3} \right )}}{36} + \frac{6^{\frac{2}{3}} \log{\left (36 \cosh ^{2}{\left (x \right )} + 12 \cdot 6^{\frac{2}{3}} \cosh{\left (x \right )} + 24 \sqrt [3]{6} \right )}}{72} + \frac{2^{\frac{2}{3}} \sqrt [6]{3} \operatorname{atan}{\left (\frac{\sqrt [3]{2} \cdot 3^{\frac{5}{6}} \cosh{\left (x \right )}}{3} + \frac{\sqrt{3}}{3} \right )}}{12} \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: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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