Optimal. Leaf size=19 \[ \cosh (x)-\frac {\tanh ^{-1}\left (\sqrt {2} \cosh (x)\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.03, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 207
Rule 321
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*}
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Mathematica [C] time = 0.18, 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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fricas [B] time = 0.45, size = 73, normalized size = 3.84 \[ \frac {2 \, \cosh \relax (x)^{2} + {\left (\sqrt {2} \cosh \relax (x) + \sqrt {2} \sinh \relax (x)\right )} \log \left (\frac {\cosh \relax (x)^{2} + \sinh \relax (x)^{2} - 2 \, \sqrt {2} \cosh \relax (x) + 2}{\cosh \relax (x)^{2} + \sinh \relax (x)^{2}}\right ) + 4 \, \cosh \relax (x) \sinh \relax (x) + 2 \, \sinh \relax (x)^{2} + 2}{4 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.11, size = 45, normalized size = 2.37 \[ \frac {1}{4} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} - e^{x}}{\sqrt {2} + e^{\left (-x\right )} + e^{x}}\right ) + \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 16, normalized size = 0.84 \[ \cosh \relax (x )-\frac {\arctanh \left (\cosh \relax (x ) \sqrt {2}\right ) \sqrt {2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 52, normalized size = 2.74 \[ -\frac {1}{4} \, \sqrt {2} \log \left (\sqrt {2} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) + \frac {1}{4} \, \sqrt {2} \log \left (-\sqrt {2} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) + \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.45, size = 48, normalized size = 2.53 \[ \frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}-\frac {\sqrt {2}\,\ln \left ({\mathrm {e}}^{2\,x}+\sqrt {2}\,{\mathrm {e}}^x+1\right )}{4}+\frac {\sqrt {2}\,\ln \left ({\mathrm {e}}^{2\,x}-\sqrt {2}\,{\mathrm {e}}^x+1\right )}{4} \]
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 \[ \int \cosh {\relax (x )} \tanh {\left (2 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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