Optimal. Leaf size=19 \[ \sinh (x)-\frac {\tan ^{-1}\left (\sqrt {2} \sinh (x)\right )}{\sqrt {2}} \]
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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, 203} \[ \sinh (x)-\frac {\tan ^{-1}\left (\sqrt {2} \sinh (x)\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 321
Rubi steps
\begin {align*} \int \sinh (x) \tanh (2 x) \, dx &=-\operatorname {Subst}\left (\int -\frac {2 x^2}{1+2 x^2} \, dx,x,\sinh (x)\right )\\ &=2 \operatorname {Subst}\left (\int \frac {x^2}{1+2 x^2} \, dx,x,\sinh (x)\right )\\ &=\sinh (x)-\operatorname {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\sinh (x)\right )\\ &=-\frac {\tan ^{-1}\left (\sqrt {2} \sinh (x)\right )}{\sqrt {2}}+\sinh (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 19, normalized size = 1.00 \[ \sinh (x)-\frac {\tan ^{-1}\left (\sqrt {2} \sinh (x)\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 115, normalized size = 6.05 \[ -\frac {{\left (\sqrt {2} \cosh \relax (x) + \sqrt {2} \sinh \relax (x)\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} \cosh \relax (x) + \frac {1}{2} \, \sqrt {2} \sinh \relax (x)\right ) - {\left (\sqrt {2} \cosh \relax (x) + \sqrt {2} \sinh \relax (x)\right )} \arctan \left (-\frac {\sqrt {2} \cosh \relax (x)^{2} + 2 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x) + \sqrt {2} \sinh \relax (x)^{2} + \sqrt {2}}{2 \, {\left (\cosh \relax (x) - \sinh \relax (x)\right )}}\right ) - \cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) - \sinh \relax (x)^{2} + 1}{2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 36, normalized size = 1.89 \[ -\frac {1}{4} \, \sqrt {2} {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )} - \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.26, size = 54, normalized size = 2.84 \[ \frac {{\mathrm e}^{x}}{2}-\frac {{\mathrm e}^{-x}}{2}+\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-i \sqrt {2}\, {\mathrm e}^{x}-1\right )}{4}-\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+i \sqrt {2}\, {\mathrm e}^{x}-1\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 53, normalized size = 2.79 \[ \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, e^{\left (-x\right )}\right )}\right ) + \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, e^{\left (-x\right )}\right )}\right ) - \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.47, size = 47, normalized size = 2.47 \[ \frac {{\mathrm {e}}^x}{2}-\frac {{\mathrm {e}}^{-x}}{2}-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\mathrm {e}}^x}{2}+\frac {\sqrt {2}\,{\mathrm {e}}^{3\,x}}{2}\right )}{2}-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\mathrm {e}}^x}{2}\right )}{2} \]
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 \[ \int \sinh {\relax (x )} \tanh {\left (2 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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