Optimal. Leaf size=45 \[ \cosh (x)+\frac {1}{6} \log (1-2 \cosh (x))+\frac {1}{6} \log (1-\cosh (x))-\frac {1}{6} \log (\cosh (x)+1)-\frac {1}{6} \log (2 \cosh (x)+1) \]
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Rubi [A] time = 0.06, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {1279, 1161, 616, 31} \[ \cosh (x)+\frac {1}{6} \log (1-2 \cosh (x))+\frac {1}{6} \log (1-\cosh (x))-\frac {1}{6} \log (\cosh (x)+1)-\frac {1}{6} \log (2 \cosh (x)+1) \]
Antiderivative was successfully verified.
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Rule 31
Rule 616
Rule 1161
Rule 1279
Rubi steps
\begin {align*} \int \cosh (x) \coth (3 x) \, dx &=-\operatorname {Subst}\left (\int \frac {x^2 \left (3-4 x^2\right )}{1-5 x^2+4 x^4} \, dx,x,\cosh (x)\right )\\ &=\cosh (x)+\frac {1}{4} \operatorname {Subst}\left (\int \frac {-4+8 x^2}{1-5 x^2+4 x^4} \, dx,x,\cosh (x)\right )\\ &=\cosh (x)+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2}-\frac {x}{2}+x^2} \, dx,x,\cosh (x)\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2}+\frac {x}{2}+x^2} \, dx,x,\cosh (x)\right )\\ &=\cosh (x)+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{-1+x} \, dx,x,\cosh (x)\right )+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2}+x} \, dx,x,\cosh (x)\right )-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{\frac {1}{2}+x} \, dx,x,\cosh (x)\right )-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\cosh (x)\right )\\ &=\cosh (x)+\frac {1}{6} \log (1-2 \cosh (x))+\frac {1}{6} \log (1-\cosh (x))-\frac {1}{6} \log (1+\cosh (x))-\frac {1}{6} \log (1+2 \cosh (x))\\ \end {align*}
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Mathematica [A] time = 0.02, size = 47, normalized size = 1.04 \[ \cosh (x)+\frac {1}{3} \log \left (\sinh \left (\frac {x}{2}\right )\right )-\frac {1}{3} \log \left (\cosh \left (\frac {x}{2}\right )\right )+\frac {1}{6} \log (1-2 \cosh (x))-\frac {1}{6} \log (2 \cosh (x)+1) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 104, normalized size = 2.31 \[ \frac {3 \, \cosh \relax (x)^{2} - {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\frac {2 \, \cosh \relax (x) + 1}{\cosh \relax (x) - \sinh \relax (x)}\right ) + {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\frac {2 \, \cosh \relax (x) - 1}{\cosh \relax (x) - \sinh \relax (x)}\right ) - 2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + 2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) + 6 \, \cosh \relax (x) \sinh \relax (x) + 3 \, \sinh \relax (x)^{2} + 3}{6 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 55, normalized size = 1.22 \[ \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} - \frac {1}{6} \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) - \frac {1}{6} \, \log \left (e^{\left (-x\right )} + e^{x} + 1\right ) + \frac {1}{6} \, \log \left (e^{\left (-x\right )} + e^{x} - 1\right ) + \frac {1}{6} \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 50, normalized size = 1.11 \[ \frac {{\mathrm e}^{x}}{2}+\frac {{\mathrm e}^{-x}}{2}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{3}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{3}+\frac {\ln \left (1-{\mathrm e}^{x}+{\mathrm e}^{2 x}\right )}{6}-\frac {\ln \left (1+{\mathrm e}^{x}+{\mathrm e}^{2 x}\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 57, normalized size = 1.27 \[ \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} - \frac {1}{6} \, \log \left (e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) - \frac {1}{3} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {1}{3} \, \log \left (e^{\left (-x\right )} - 1\right ) + \frac {1}{6} \, \log \left (-e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 57, normalized size = 1.27 \[ \frac {\ln \left (6-6\,{\mathrm {e}}^x\right )}{3}-\frac {\ln \left (-6\,{\mathrm {e}}^x-6\right )}{3}+\frac {{\mathrm {e}}^{-x}}{2}+\frac {\ln \left ({\mathrm {e}}^x-{\mathrm {e}}^{2\,x}-1\right )}{6}-\frac {\ln \left (-{\mathrm {e}}^{2\,x}-{\mathrm {e}}^x-1\right )}{6}+\frac {{\mathrm {e}}^x}{2} \]
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 \[ \int \cosh {\relax (x )} \coth {\left (3 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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