Optimal. Leaf size=19 \[ \sinh (x)-\frac {1}{3} \tan ^{-1}(\sinh (x))-\frac {1}{3} \tan ^{-1}(2 \sinh (x)) \]
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Rubi [A] time = 0.04, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1279, 1163, 203} \[ \sinh (x)-\frac {1}{3} \tan ^{-1}(\sinh (x))-\frac {1}{3} \tan ^{-1}(2 \sinh (x)) \]
Antiderivative was successfully verified.
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Rule 203
Rule 1163
Rule 1279
Rubi steps
\begin {align*} \int \sinh (x) \tanh (3 x) \, dx &=-\operatorname {Subst}\left (\int \frac {x^2 \left (-3-4 x^2\right )}{1+5 x^2+4 x^4} \, dx,x,\sinh (x)\right )\\ &=\sinh (x)+\frac {1}{4} \operatorname {Subst}\left (\int \frac {-4-8 x^2}{1+5 x^2+4 x^4} \, dx,x,\sinh (x)\right )\\ &=\sinh (x)-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{1+4 x^2} \, dx,x,\sinh (x)\right )-\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{4+4 x^2} \, dx,x,\sinh (x)\right )\\ &=-\frac {1}{3} \tan ^{-1}(\sinh (x))-\frac {1}{3} \tan ^{-1}(2 \sinh (x))+\sinh (x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 19, normalized size = 1.00 \[ \sinh (x)-\frac {1}{3} \tan ^{-1}(\sinh (x))-\frac {1}{3} \tan ^{-1}(2 \sinh (x)) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 76, normalized size = 4.00 \[ \frac {2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \arctan \left (-\frac {\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}{\cosh \relax (x) - \sinh \relax (x)}\right ) - 6 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) + 3 \, \cosh \relax (x)^{2} + 6 \, \cosh \relax (x) \sinh \relax (x) + 3 \, \sinh \relax (x)^{2} - 3}{6 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 43, normalized size = 2.26 \[ -\frac {1}{3} \, \pi - \frac {1}{3} \, \arctan \left ({\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right ) - \frac {1}{3} \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right ) - \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.29, size = 60, normalized size = 3.16 \[ \frac {{\mathrm e}^{x}}{2}-\frac {{\mathrm e}^{-x}}{2}+\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{3}-\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{3}+\frac {i \ln \left ({\mathrm e}^{2 x}-i {\mathrm e}^{x}-1\right )}{6}-\frac {i \ln \left ({\mathrm e}^{2 x}+i {\mathrm e}^{x}-1\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 46, normalized size = 2.42 \[ \frac {1}{3} \, \arctan \left (\sqrt {3} + 2 \, e^{\left (-x\right )}\right ) + \frac {1}{3} \, \arctan \left (-\sqrt {3} + 2 \, e^{\left (-x\right )}\right ) + \frac {2}{3} \, \arctan \left (e^{\left (-x\right )}\right ) - \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.44, size = 23, normalized size = 1.21 \[ \frac {{\mathrm {e}}^x}{2}-\mathrm {atan}\left ({\mathrm {e}}^x\right )-\frac {\mathrm {atan}\left ({\mathrm {e}}^{3\,x}\right )}{3}-\frac {{\mathrm {e}}^{-x}}{2} \]
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 \[ \int \sinh {\relax (x )} \tanh {\left (3 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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