Optimal. Leaf size=28 \[ \cosh (x)-\frac{1}{4} \tanh ^{-1}(\cosh (x))-\frac{\tanh ^{-1}\left (\sqrt{2} \cosh (x)\right )}{2 \sqrt{2}} \]
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Rubi [A] time = 0.0567339, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {1676, 1166, 207} \[ \cosh (x)-\frac{1}{4} \tanh ^{-1}(\cosh (x))-\frac{\tanh ^{-1}\left (\sqrt{2} \cosh (x)\right )}{2 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1676
Rule 1166
Rule 207
Rubi steps
\begin{align*} \int \cosh (x) \coth (4 x) \, dx &=-\operatorname{Subst}\left (\int \frac{-1+8 x^2-8 x^4}{4-12 x^2+8 x^4} \, dx,x,\cosh (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (-1+\frac{3-4 x^2}{4-12 x^2+8 x^4}\right ) \, dx,x,\cosh (x)\right )\\ &=\cosh (x)-\operatorname{Subst}\left (\int \frac{3-4 x^2}{4-12 x^2+8 x^4} \, dx,x,\cosh (x)\right )\\ &=\cosh (x)+2 \operatorname{Subst}\left (\int \frac{1}{-8+8 x^2} \, dx,x,\cosh (x)\right )+2 \operatorname{Subst}\left (\int \frac{1}{-4+8 x^2} \, dx,x,\cosh (x)\right )\\ &=-\frac{1}{4} \tanh ^{-1}(\cosh (x))-\frac{\tanh ^{-1}\left (\sqrt{2} \cosh (x)\right )}{2 \sqrt{2}}+\cosh (x)\\ \end{align*}
Mathematica [C] time = 0.224495, size = 192, normalized size = 6.86 \[ \frac{8 \sqrt{2} \cosh (x)-4 \tanh ^{-1}\left (\sqrt{2}-i \tanh \left (\frac{x}{2}\right )\right )+2 \sqrt{2} \log \left (\sinh \left (\frac{x}{2}\right )\right )-2 \sqrt{2} \log \left (\cosh \left (\frac{x}{2}\right )\right )+\log \left (\sqrt{2}-2 \cosh (x)\right )-\log \left (2 \cosh (x)+\sqrt{2}\right )-2 i \tan ^{-1}\left (\frac{\sinh \left (\frac{x}{2}\right )+\cosh \left (\frac{x}{2}\right )}{\left (1+\sqrt{2}\right ) \cosh \left (\frac{x}{2}\right )-\left (\sqrt{2}-1\right ) \sinh \left (\frac{x}{2}\right )}\right )+2 i \tan ^{-1}\left (\frac{\sinh \left (\frac{x}{2}\right )+\cosh \left (\frac{x}{2}\right )}{\left (\sqrt{2}-1\right ) \cosh \left (\frac{x}{2}\right )-\left (1+\sqrt{2}\right ) \sinh \left (\frac{x}{2}\right )}\right )}{8 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.073, size = 63, normalized size = 2.3 \begin{align*}{\frac{{{\rm e}^{x}}}{2}}+{\frac{{{\rm e}^{-x}}}{2}}-{\frac{\ln \left ({{\rm e}^{x}}+1 \right ) }{4}}+{\frac{\ln \left ({{\rm e}^{x}}-1 \right ) }{4}}+{\frac{\ln \left ( 1+{{\rm e}^{2\,x}}-{{\rm e}^{x}}\sqrt{2} \right ) \sqrt{2}}{8}}-{\frac{\ln \left ( 1+{{\rm e}^{2\,x}}+{{\rm e}^{x}}\sqrt{2} \right ) \sqrt{2}}{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.66876, size = 95, normalized size = 3.39 \begin{align*} -\frac{1}{8} \, \sqrt{2} \log \left (\sqrt{2} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) + \frac{1}{8} \, \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} - \frac{1}{4} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac{1}{4} \, \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.20536, size = 397, normalized size = 14.18 \begin{align*} \frac{4 \, \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 ) - 2 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + 2 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 8 \, \cosh \left (x\right ) \sinh \left (x\right ) + 4 \, \sinh \left (x\right )^{2} + 4}{8 \,{\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 )} \coth{\left (4 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16048, size = 90, normalized size = 3.21 \begin{align*} \frac{1}{8} \, \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} - \frac{1}{8} \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) + \frac{1}{8} \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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