Optimal. Leaf size=20 \[ \sinh (x)-\frac {\tan ^{-1}\left (\frac {2 \sinh (x)}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.03, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {388, 203} \[ \sinh (x)-\frac {\tan ^{-1}\left (\frac {2 \sinh (x)}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 388
Rubi steps
\begin {align*} \int \coth (3 x) \sinh (x) \, dx &=\operatorname {Subst}\left (\int \frac {1+4 x^2}{3+4 x^2} \, dx,x,\sinh (x)\right )\\ &=\sinh (x)-2 \operatorname {Subst}\left (\int \frac {1}{3+4 x^2} \, dx,x,\sinh (x)\right )\\ &=-\frac {\tan ^{-1}\left (\frac {2 \sinh (x)}{\sqrt {3}}\right )}{\sqrt {3}}+\sinh (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 20, normalized size = 1.00 \[ \sinh (x)-\frac {\tan ^{-1}\left (\frac {2 \sinh (x)}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 118, normalized size = 5.90 \[ -\frac {2 \, {\left (\sqrt {3} \cosh \relax (x) + \sqrt {3} \sinh \relax (x)\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} \cosh \relax (x) + \frac {1}{3} \, \sqrt {3} \sinh \relax (x)\right ) - 2 \, {\left (\sqrt {3} \cosh \relax (x) + \sqrt {3} \sinh \relax (x)\right )} \arctan \left (-\frac {\sqrt {3} \cosh \relax (x)^{2} + 2 \, \sqrt {3} \cosh \relax (x) \sinh \relax (x) + \sqrt {3} \sinh \relax (x)^{2} + 2 \, \sqrt {3}}{3 \, {\left (\cosh \relax (x) - \sinh \relax (x)\right )}}\right ) - 3 \, \cosh \relax (x)^{2} - 6 \, \cosh \relax (x) \sinh \relax (x) - 3 \, \sinh \relax (x)^{2} + 3}{6 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 36, normalized size = 1.80 \[ -\frac {1}{6} \, \sqrt {3} {\left (\pi + 2 \, \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )} - \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.22, size = 51, normalized size = 2.55 \[ -\frac {1}{\tanh \left (\frac {x}{2}\right )+1}-\frac {\sqrt {3}\, \arctan \left (\frac {\tanh \left (\frac {x}{2}\right ) \sqrt {3}}{3}\right )}{3}-\frac {\sqrt {3}\, \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {3}\right )}{3}-\frac {1}{\tanh \left (\frac {x}{2}\right )-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 49, normalized size = 2.45 \[ \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{\left (-x\right )} + 1\right )}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{\left (-x\right )} - 1\right )}\right ) - \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 47, normalized size = 2.35 \[ \frac {{\mathrm {e}}^x}{2}-\frac {{\mathrm {e}}^{-x}}{2}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,{\mathrm {e}}^x}{3}+\frac {\sqrt {3}\,{\mathrm {e}}^{3\,x}}{3}\right )}{3}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,{\mathrm {e}}^x}{3}\right )}{3} \]
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 \[ \int \sinh {\relax (x )} \coth {\left (3 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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