Optimal. Leaf size=20 \[ \cosh (x)-\frac{\tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{3}}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0299257, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {388, 206} \[ \cosh (x)-\frac{\tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 388
Rule 206
Rubi steps
\begin{align*} \int \cosh (x) \tanh (3 x) \, dx &=\operatorname{Subst}\left (\int \frac{1-4 x^2}{3-4 x^2} \, dx,x,\cosh (x)\right )\\ &=\cosh (x)-2 \operatorname{Subst}\left (\int \frac{1}{3-4 x^2} \, dx,x,\cosh (x)\right )\\ &=-\frac{\tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{3}}\right )}{\sqrt{3}}+\cosh (x)\\ \end{align*}
Mathematica [C] time = 0.0578828, size = 55, normalized size = 2.75 \[ \cosh (x)-\frac{\tanh ^{-1}\left (\frac{2-i \tanh \left (\frac{x}{2}\right )}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{2+i \tanh \left (\frac{x}{2}\right )}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 17, normalized size = 0.9 \begin{align*} \cosh \left ( x \right ) -{\frac{\sqrt{3}}{3}{\it Artanh} \left ({\frac{2\,\cosh \left ( x \right ) \sqrt{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.69763, size = 207, normalized size = 10.35 \begin{align*} -\frac{1}{12} \, \sqrt{3} \log \left (\sqrt{3} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) + \frac{1}{12} \, \sqrt{3} \log \left (-\sqrt{3} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) - \frac{1}{12} \, \sqrt{3} \log \left (\sqrt{3} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac{1}{12} \, \sqrt{3} \log \left (-\sqrt{3} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac{1}{6} \, \arctan \left (\sqrt{3} + 2 \, e^{\left (-x\right )}\right ) + \frac{1}{6} \, \arctan \left (\sqrt{3} + 2 \, e^{x}\right ) + \frac{1}{6} \, \arctan \left (-\sqrt{3} + 2 \, e^{\left (-x\right )}\right ) + \frac{1}{6} \, \arctan \left (-\sqrt{3} + 2 \, e^{x}\right ) + \frac{1}{3} \, \arctan \left (e^{\left (-x\right )}\right ) + \frac{1}{3} \, \arctan \left (e^{x}\right ) + \frac{1}{2} \, e^{\left (-x\right )} + \frac{1}{2} \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.05551, size = 275, normalized size = 13.75 \begin{align*} \frac{3 \, \cosh \left (x\right )^{2} +{\left (\sqrt{3} \cosh \left (x\right ) + \sqrt{3} \sinh \left (x\right )\right )} \log \left (\frac{2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} - 4 \, \sqrt{3} \cosh \left (x\right ) + 5}{2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} - 1}\right ) + 6 \, \cosh \left (x\right ) \sinh \left (x\right ) + 3 \, \sinh \left (x\right )^{2} + 3}{6 \,{\left (\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh{\left (x \right )} \tanh{\left (3 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15087, size = 61, normalized size = 3.05 \begin{align*} \frac{1}{6} \, \sqrt{3} \log \left (-\frac{\sqrt{3} - e^{\left (-x\right )} - e^{x}}{\sqrt{3} + e^{\left (-x\right )} + e^{x}}\right ) + \frac{1}{2} \, e^{\left (-x\right )} + \frac{1}{2} \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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