Optimal. Leaf size=21 \[ \frac {1}{6} \log \left (3-4 \cosh ^2(x)\right )-\frac {1}{3} \log (\cosh (x)) \]
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Rubi [A] time = 0.03, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {4357, 266, 36, 29, 31} \[ \frac {1}{6} \log \left (3-4 \cosh ^2(x)\right )-\frac {1}{3} \log (\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 4357
Rubi steps
\begin {align*} \int \text {sech}(3 x) \sinh (x) \, dx &=\operatorname {Subst}\left (\int \frac {1}{x \left (-3+4 x^2\right )} \, dx,x,\cosh (x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x (-3+4 x)} \, dx,x,\cosh ^2(x)\right )\\ &=-\left (\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\cosh ^2(x)\right )\right )+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{-3+4 x} \, dx,x,\cosh ^2(x)\right )\\ &=-\frac {1}{3} \log (\cosh (x))+\frac {1}{6} \log \left (3-4 \cosh ^2(x)\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 17, normalized size = 0.81 \[ -\frac {1}{3} \tanh ^{-1}\left (\frac {1}{3} \left (8 \sinh ^2(x)+5\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 52, normalized size = 2.48 \[ \frac {1}{6} \, \log \left (\frac {2 \, \cosh \relax (x)^{2} + 2 \, \sinh \relax (x)^{2} - 1}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}\right ) - \frac {1}{3} \, \log \left (\frac {2 \, \cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.11, size = 41, normalized size = 1.95 \[ \frac {1}{6} \, \log \left (\sqrt {3} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{6} \, \log \left (-\sqrt {3} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{3} \, \log \left (e^{\left (2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 26, normalized size = 1.24 \[ -\frac {\ln \left (1+{\mathrm e}^{2 x}\right )}{3}+\frac {\ln \left (1-{\mathrm e}^{2 x}+{\mathrm e}^{4 x}\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 45, normalized size = 2.14 \[ \frac {1}{6} \, \log \left (\sqrt {3} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) + \frac {1}{6} \, \log \left (-\sqrt {3} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) - \frac {1}{3} \, \log \left (e^{\left (-2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.46, size = 27, normalized size = 1.29 \[ \frac {\ln \left ({\mathrm {e}}^{2\,x}-{\mathrm {e}}^{4\,x}-1\right )}{6}-\frac {\ln \left (3\,{\mathrm {e}}^{2\,x}+3\right )}{3} \]
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 \[ \int \sinh {\relax (x )} \operatorname {sech}{\left (3 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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