Optimal. Leaf size=110 \[ \cosh (x)+\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (-4 \cosh (x)-\sqrt{5}+1\right )+\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (-4 \cosh (x)+\sqrt{5}+1\right )-\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (4 \cosh (x)-\sqrt{5}+1\right )-\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (4 \cosh (x)+\sqrt{5}+1\right )-\frac{1}{5} \tanh ^{-1}(\cosh (x)) \]
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Rubi [A] time = 0.176159, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {2075, 207, 632, 31} \[ \cosh (x)+\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (-4 \cosh (x)-\sqrt{5}+1\right )+\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (-4 \cosh (x)+\sqrt{5}+1\right )-\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (4 \cosh (x)-\sqrt{5}+1\right )-\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (4 \cosh (x)+\sqrt{5}+1\right )-\frac{1}{5} \tanh ^{-1}(\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 2075
Rule 207
Rule 632
Rule 31
Rubi steps
\begin{align*} \int \cosh (x) \coth (5 x) \, dx &=-\operatorname{Subst}\left (\int \frac{x^2 \left (5-20 x^2+16 x^4\right )}{1-13 x^2+28 x^4-16 x^6} \, dx,x,\cosh (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (-1-\frac{1}{5 \left (-1+x^2\right )}-\frac{2 (1+x)}{5 \left (-1-2 x+4 x^2\right )}+\frac{2 (-1+x)}{5 \left (-1+2 x+4 x^2\right )}\right ) \, dx,x,\cosh (x)\right )\\ &=\cosh (x)+\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\cosh (x)\right )+\frac{2}{5} \operatorname{Subst}\left (\int \frac{1+x}{-1-2 x+4 x^2} \, dx,x,\cosh (x)\right )-\frac{2}{5} \operatorname{Subst}\left (\int \frac{-1+x}{-1+2 x+4 x^2} \, dx,x,\cosh (x)\right )\\ &=-\frac{1}{5} \tanh ^{-1}(\cosh (x))+\cosh (x)-\frac{1}{5} \left (1-\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{5}+4 x} \, dx,x,\cosh (x)\right )+\frac{1}{5} \left (1-\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{-1+\sqrt{5}+4 x} \, dx,x,\cosh (x)\right )+\frac{1}{5} \left (1+\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-\sqrt{5}+4 x} \, dx,x,\cosh (x)\right )-\frac{1}{5} \left (1+\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{5}+4 x} \, dx,x,\cosh (x)\right )\\ &=-\frac{1}{5} \tanh ^{-1}(\cosh (x))+\cosh (x)+\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (1-\sqrt{5}-4 \cosh (x)\right )+\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (1+\sqrt{5}-4 \cosh (x)\right )-\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (1-\sqrt{5}+4 \cosh (x)\right )-\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (1+\sqrt{5}+4 \cosh (x)\right )\\ \end{align*}
Mathematica [A] time = 0.114537, size = 133, normalized size = 1.21 \[ \frac{1}{100} \left (100 \cosh (x)+20 \log \left (\sinh \left (\frac{x}{2}\right )\right )-20 \log \left (\cosh \left (\frac{x}{2}\right )\right )+\sqrt{5} \left (\sqrt{5}-5\right ) \log \left (-4 \cosh (x)-\sqrt{5}+1\right )+\sqrt{5} \left (5+\sqrt{5}\right ) \log \left (-4 \cosh (x)+\sqrt{5}+1\right )-\sqrt{5} \left (\sqrt{5}-5\right ) \log \left (4 \cosh (x)-\sqrt{5}+1\right )-\sqrt{5} \left (5+\sqrt{5}\right ) \log \left (4 \cosh (x)+\sqrt{5}+1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.083, size = 190, normalized size = 1.7 \begin{align*}{\frac{{{\rm e}^{x}}}{2}}+{\frac{{{\rm e}^{-x}}}{2}}-{\frac{\ln \left ({{\rm e}^{x}}+1 \right ) }{5}}+{\frac{\ln \left ({{\rm e}^{x}}-1 \right ) }{5}}+{\frac{\ln \left ({{\rm e}^{2\,x}}+ \left ( -{\frac{1}{2}}-{\frac{\sqrt{5}}{2}} \right ){{\rm e}^{x}}+1 \right ) }{20}}+{\frac{\ln \left ({{\rm e}^{2\,x}}+ \left ( -{\frac{1}{2}}-{\frac{\sqrt{5}}{2}} \right ){{\rm e}^{x}}+1 \right ) \sqrt{5}}{20}}+{\frac{\ln \left ({{\rm e}^{2\,x}}+ \left ({\frac{\sqrt{5}}{2}}-{\frac{1}{2}} \right ){{\rm e}^{x}}+1 \right ) }{20}}-{\frac{\ln \left ({{\rm e}^{2\,x}}+ \left ({\frac{\sqrt{5}}{2}}-{\frac{1}{2}} \right ){{\rm e}^{x}}+1 \right ) \sqrt{5}}{20}}-{\frac{\ln \left ({{\rm e}^{2\,x}}+ \left ({\frac{1}{2}}-{\frac{\sqrt{5}}{2}} \right ){{\rm e}^{x}}+1 \right ) }{20}}+{\frac{\ln \left ({{\rm e}^{2\,x}}+ \left ({\frac{1}{2}}-{\frac{\sqrt{5}}{2}} \right ){{\rm e}^{x}}+1 \right ) \sqrt{5}}{20}}-{\frac{\ln \left ({{\rm e}^{2\,x}}+ \left ({\frac{1}{2}}+{\frac{\sqrt{5}}{2}} \right ){{\rm e}^{x}}+1 \right ) }{20}}-{\frac{\ln \left ({{\rm e}^{2\,x}}+ \left ({\frac{1}{2}}+{\frac{\sqrt{5}}{2}} \right ){{\rm e}^{x}}+1 \right ) \sqrt{5}}{20}} \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*} \frac{1}{2} \,{\left (e^{\left (2 \, x\right )} + 1\right )} e^{\left (-x\right )} - \frac{1}{5} \, \int \frac{{\left (e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} + e^{x} + 1\right )} e^{x}}{e^{\left (4 \, x\right )} + e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} + e^{x} + 1}\,{d x} + \frac{1}{5} \, \int \frac{{\left (e^{\left (3 \, x\right )} - e^{\left (2 \, x\right )} + e^{x} - 1\right )} e^{x}}{e^{\left (4 \, x\right )} - e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} - e^{x} + 1}\,{d x} + \frac{3}{10} \, \int \frac{e^{\left (3 \, x\right )}}{e^{\left (4 \, x\right )} + e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} + e^{x} + 1}\,{d x} + \frac{3}{10} \, \int \frac{e^{\left (3 \, x\right )}}{e^{\left (4 \, x\right )} - e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} - e^{x} + 1}\,{d x} + \frac{1}{10} \, \int \frac{e^{\left (2 \, x\right )}}{e^{\left (4 \, x\right )} + e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} + e^{x} + 1}\,{d x} - \frac{1}{10} \, \int \frac{e^{\left (2 \, x\right )}}{e^{\left (4 \, x\right )} - e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} - e^{x} + 1}\,{d x} - \frac{1}{10} \, \int \frac{e^{x}}{e^{\left (4 \, x\right )} + e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} + e^{x} + 1}\,{d x} - \frac{1}{10} \, \int \frac{e^{x}}{e^{\left (4 \, x\right )} - e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} - e^{x} + 1}\,{d x} - \frac{1}{5} \, \log \left (e^{x} + 1\right ) + \frac{1}{5} \, \log \left (e^{x} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.23793, size = 980, normalized size = 8.91 \begin{align*} \frac{10 \, \cosh \left (x\right )^{2} +{\left (\sqrt{5} \cosh \left (x\right ) + \sqrt{5} \sinh \left (x\right )\right )} \log \left (-\frac{4 \,{\left (\sqrt{5} - 1\right )} \cosh \left (x\right ) - 4 \, \cosh \left (x\right )^{2} - 4 \, \sinh \left (x\right )^{2} + \sqrt{5} - 7}{2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1}\right ) +{\left (\sqrt{5} \cosh \left (x\right ) + \sqrt{5} \sinh \left (x\right )\right )} \log \left (-\frac{4 \,{\left (\sqrt{5} + 1\right )} \cosh \left (x\right ) - 4 \, \cosh \left (x\right )^{2} - 4 \, \sinh \left (x\right )^{2} - \sqrt{5} - 7}{2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) + 1}\right ) -{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\frac{2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) +{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\frac{2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) + 1}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) - 4 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + 4 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 20 \, \cosh \left (x\right ) \sinh \left (x\right ) + 10 \, \sinh \left (x\right )^{2} + 10}{20 \,{\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 (5 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18942, size = 212, normalized size = 1.93 \begin{align*} \frac{1}{20} \, \sqrt{5} \log \left (-\frac{\sqrt{5} - 2 \, e^{\left (-x\right )} - 2 \, e^{x} + 1}{\sqrt{5} + 2 \, e^{\left (-x\right )} + 2 \, e^{x} - 1}\right ) + \frac{1}{20} \, \sqrt{5} \log \left (-\frac{\sqrt{5} - 2 \, e^{\left (-x\right )} - 2 \, e^{x} - 1}{\sqrt{5} + 2 \, e^{\left (-x\right )} + 2 \, e^{x} + 1}\right ) + \frac{1}{2} \, e^{\left (-x\right )} + \frac{1}{2} \, e^{x} - \frac{1}{20} \, \log \left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + e^{\left (-x\right )} + e^{x} - 1\right ) + \frac{1}{20} \, \log \left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - e^{\left (-x\right )} - e^{x} - 1\right ) - \frac{1}{10} \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) + \frac{1}{10} \, \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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