Optimal. Leaf size=19 \[ \cosh (x)-\frac{\tanh ^{-1}\left (\sqrt{2} \cosh (x)\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.031566, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {12, 321, 207} \[ \cosh (x)-\frac{\tanh ^{-1}\left (\sqrt{2} \cosh (x)\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 321
Rule 207
Rubi steps
\begin{align*} \int \cosh (x) \tanh (2 x) \, dx &=\operatorname{Subst}\left (\int \frac{2 x^2}{-1+2 x^2} \, dx,x,\cosh (x)\right )\\ &=2 \operatorname{Subst}\left (\int \frac{x^2}{-1+2 x^2} \, dx,x,\cosh (x)\right )\\ &=\cosh (x)+\operatorname{Subst}\left (\int \frac{1}{-1+2 x^2} \, dx,x,\cosh (x)\right )\\ &=-\frac{\tanh ^{-1}\left (\sqrt{2} \cosh (x)\right )}{\sqrt{2}}+\cosh (x)\\ \end{align*}
Mathematica [C] time = 0.17027, size = 164, normalized size = 8.63 \[ \frac{4 \sqrt{2} \cosh (x)-4 \tanh ^{-1}\left (\sqrt{2}-i \tanh \left (\frac{x}{2}\right )\right )+\log \left (\sqrt{2}-2 \cosh (x)\right )-\log \left (2 \cosh (x)+\sqrt{2}\right )-2 i \tan ^{-1}\left (\frac{\sinh \left (\frac{x}{2}\right )+\cosh \left (\frac{x}{2}\right )}{\left (1+\sqrt{2}\right ) \cosh \left (\frac{x}{2}\right )-\left (\sqrt{2}-1\right ) \sinh \left (\frac{x}{2}\right )}\right )+2 i \tan ^{-1}\left (\frac{\sinh \left (\frac{x}{2}\right )+\cosh \left (\frac{x}{2}\right )}{\left (\sqrt{2}-1\right ) \cosh \left (\frac{x}{2}\right )-\left (1+\sqrt{2}\right ) \sinh \left (\frac{x}{2}\right )}\right )}{4 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 16, normalized size = 0.8 \begin{align*} \cosh \left ( x \right ) -{\frac{{\it Artanh} \left ( \cosh \left ( x \right ) \sqrt{2} \right ) \sqrt{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.7082, size = 70, normalized size = 3.68 \begin{align*} -\frac{1}{4} \, \sqrt{2} \log \left (\sqrt{2} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) + \frac{1}{4} \, \sqrt{2} \log \left (-\sqrt{2} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\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 = 2.11186, size = 259, normalized size = 13.63 \begin{align*} \frac{2 \, \cosh \left (x\right )^{2} +{\left (\sqrt{2} \cosh \left (x\right ) + \sqrt{2} \sinh \left (x\right )\right )} \log \left (\frac{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 2 \, \sqrt{2} \cosh \left (x\right ) + 2}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2}}\right ) + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, \sinh \left (x\right )^{2} + 2}{4 \,{\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 (2 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14523, size = 61, normalized size = 3.21 \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - e^{\left (-x\right )} - e^{x}}{\sqrt{2} + 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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