Optimal. Leaf size=10 \[ \sinh (x)-\frac {1}{2} \tan ^{-1}(\sinh (x)) \]
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Rubi [A] time = 0.02, antiderivative size = 10, 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} \[ \sinh (x)-\frac {1}{2} \tan ^{-1}(\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 203
Rule 388
Rubi steps
\begin {align*} \int \coth (2 x) \sinh (x) \, dx &=\operatorname {Subst}\left (\int \frac {1+2 x^2}{2+2 x^2} \, dx,x,\sinh (x)\right )\\ &=\sinh (x)-\operatorname {Subst}\left (\int \frac {1}{2+2 x^2} \, dx,x,\sinh (x)\right )\\ &=-\frac {1}{2} \tan ^{-1}(\sinh (x))+\sinh (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 10, normalized size = 1.00 \[ \sinh (x)-\frac {1}{2} \tan ^{-1}(\sinh (x)) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 42, normalized size = 4.20 \[ -\frac {2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) - \cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) - \sinh \relax (x)^{2} + 1}{2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 16, normalized size = 1.60 \[ -\arctan \left (e^{x}\right ) - \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 9, normalized size = 0.90 \[ \sinh \relax (x )-\arctan \left ({\mathrm e}^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 16, normalized size = 1.60 \[ \arctan \left (e^{\left (-x\right )}\right ) - \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 16, normalized size = 1.60 \[ \frac {{\mathrm {e}}^x}{2}-\mathrm {atan}\left ({\mathrm {e}}^x\right )-\frac {{\mathrm {e}}^{-x}}{2} \]
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 \[ \int \sinh {\relax (x )} \coth {\left (2 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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