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 = 0.18, antiderivative size = 110, normalized size of antiderivative = 1.00, 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 31
Rule 207
Rule 632
Rule 2075
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*}
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Mathematica [A] time = 0.12, 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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fricas [B] time = 0.44, size = 272, normalized size = 2.47 \[ \frac {10 \, \cosh \relax (x)^{2} + {\left (\sqrt {5} \cosh \relax (x) + \sqrt {5} \sinh \relax (x)\right )} \log \left (-\frac {4 \, {\left (\sqrt {5} - 1\right )} \cosh \relax (x) - 4 \, \cosh \relax (x)^{2} - 4 \, \sinh \relax (x)^{2} + \sqrt {5} - 7}{2 \, \cosh \relax (x)^{2} + 2 \, \sinh \relax (x)^{2} + 2 \, \cosh \relax (x) + 1}\right ) + {\left (\sqrt {5} \cosh \relax (x) + \sqrt {5} \sinh \relax (x)\right )} \log \left (-\frac {4 \, {\left (\sqrt {5} + 1\right )} \cosh \relax (x) - 4 \, \cosh \relax (x)^{2} - 4 \, \sinh \relax (x)^{2} - \sqrt {5} - 7}{2 \, \cosh \relax (x)^{2} + 2 \, \sinh \relax (x)^{2} - 2 \, \cosh \relax (x) + 1}\right ) - {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\frac {2 \, \cosh \relax (x)^{2} + 2 \, \sinh \relax (x)^{2} + 2 \, \cosh \relax (x) + 1}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}\right ) + {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\frac {2 \, \cosh \relax (x)^{2} + 2 \, \sinh \relax (x)^{2} - 2 \, \cosh \relax (x) + 1}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}\right ) - 4 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + 4 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) + 20 \, \cosh \relax (x) \sinh \relax (x) + 10 \, \sinh \relax (x)^{2} + 10}{20 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 157, normalized size = 1.43 \[ \frac {1}{20} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - 2 \, e^{\left (-x\right )} - 2 \, e^{x} + 1}{\sqrt {5} + 2 \, e^{\left (-x\right )} + 2 \, e^{x} - 1}\right ) + \frac {1}{20} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - 2 \, e^{\left (-x\right )} - 2 \, e^{x} - 1}{\sqrt {5} + 2 \, e^{\left (-x\right )} + 2 \, e^{x} + 1}\right ) + \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} - \frac {1}{20} \, \log \left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + e^{\left (-x\right )} + e^{x} - 1\right ) + \frac {1}{20} \, \log \left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - e^{\left (-x\right )} - e^{x} - 1\right ) - \frac {1}{10} \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) + \frac {1}{10} \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.28, size = 190, normalized size = 1.73 \[ \frac {{\mathrm e}^{x}}{2}+\frac {{\mathrm e}^{-x}}{2}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{5}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{5}-\frac {\ln \left ({\mathrm e}^{2 x}+\left (\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) {\mathrm e}^{x}+1\right )}{20}+\frac {\ln \left ({\mathrm e}^{2 x}+\left (\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) {\mathrm e}^{x}+1\right ) \sqrt {5}}{20}-\frac {\ln \left ({\mathrm e}^{2 x}+\left (\frac {1}{2}+\frac {\sqrt {5}}{2}\right ) {\mathrm e}^{x}+1\right )}{20}-\frac {\ln \left ({\mathrm e}^{2 x}+\left (\frac {1}{2}+\frac {\sqrt {5}}{2}\right ) {\mathrm e}^{x}+1\right ) \sqrt {5}}{20}+\frac {\ln \left ({\mathrm e}^{2 x}+\left (-\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) {\mathrm e}^{x}+1\right )}{20}+\frac {\ln \left ({\mathrm e}^{2 x}+\left (-\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) {\mathrm e}^{x}+1\right ) \sqrt {5}}{20}+\frac {\ln \left ({\mathrm e}^{2 x}+\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) {\mathrm e}^{x}+1\right )}{20}-\frac {\ln \left ({\mathrm e}^{2 x}+\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) {\mathrm e}^{x}+1\right ) \sqrt {5}}{20} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, {\left (e^{\left (2 \, x\right )} + 1\right )} e^{\left (-x\right )} - \frac {1}{5} \, \int \frac {{\left (e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} + e^{x} + 1\right )} e^{x}}{e^{\left (4 \, x\right )} + e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} + e^{x} + 1}\,{d x} + \frac {1}{5} \, \int \frac {{\left (e^{\left (3 \, x\right )} - e^{\left (2 \, x\right )} + e^{x} - 1\right )} e^{x}}{e^{\left (4 \, x\right )} - e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} - e^{x} + 1}\,{d x} + \frac {3}{10} \, \int \frac {e^{\left (3 \, x\right )}}{e^{\left (4 \, x\right )} + e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} + e^{x} + 1}\,{d x} + \frac {3}{10} \, \int \frac {e^{\left (3 \, x\right )}}{e^{\left (4 \, x\right )} - e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} - e^{x} + 1}\,{d x} + \frac {1}{10} \, \int \frac {e^{\left (2 \, x\right )}}{e^{\left (4 \, x\right )} + e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} + e^{x} + 1}\,{d x} - \frac {1}{10} \, \int \frac {e^{\left (2 \, x\right )}}{e^{\left (4 \, x\right )} - e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} - e^{x} + 1}\,{d x} - \frac {1}{10} \, \int \frac {e^{x}}{e^{\left (4 \, x\right )} + e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} + e^{x} + 1}\,{d x} - \frac {1}{10} \, \int \frac {e^{x}}{e^{\left (4 \, x\right )} - e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} - e^{x} + 1}\,{d x} - \frac {1}{5} \, \log \left (e^{x} + 1\right ) + \frac {1}{5} \, \log \left (e^{x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 143, normalized size = 1.30 \[ \frac {\ln \left (10-10\,{\mathrm {e}}^x\right )}{5}-\frac {\ln \left (-10\,{\mathrm {e}}^x-10\right )}{5}+\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}-\ln \left (-{\mathrm {e}}^{2\,x}-10\,{\mathrm {e}}^x\,\left (\frac {\sqrt {5}}{20}-\frac {1}{20}\right )-1\right )\,\left (\frac {\sqrt {5}}{20}-\frac {1}{20}\right )+\ln \left (10\,{\mathrm {e}}^x\,\left (\frac {\sqrt {5}}{20}-\frac {1}{20}\right )-{\mathrm {e}}^{2\,x}-1\right )\,\left (\frac {\sqrt {5}}{20}-\frac {1}{20}\right )-\ln \left (-{\mathrm {e}}^{2\,x}-10\,{\mathrm {e}}^x\,\left (\frac {\sqrt {5}}{20}+\frac {1}{20}\right )-1\right )\,\left (\frac {\sqrt {5}}{20}+\frac {1}{20}\right )+\ln \left (10\,{\mathrm {e}}^x\,\left (\frac {\sqrt {5}}{20}+\frac {1}{20}\right )-{\mathrm {e}}^{2\,x}-1\right )\,\left (\frac {\sqrt {5}}{20}+\frac {1}{20}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh {\relax (x )} \coth {\left (5 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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