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 = 0.0433314, antiderivative size = 19, normalized size of antiderivative = 1., 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 1279
Rule 1163
Rule 203
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*}
Mathematica [A] time = 0.028378, size = 19, normalized size = 1. \[ \sinh (x)-\frac{1}{3} \tan ^{-1}(\sinh (x))-\frac{1}{3} \tan ^{-1}(2 \sinh (x)) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.066, size = 60, normalized size = 3.2 \begin{align*}{\frac{{{\rm e}^{x}}}{2}}-{\frac{{{\rm e}^{-x}}}{2}}+{\frac{i}{3}}\ln \left ({{\rm e}^{x}}-i \right ) -{\frac{i}{3}}\ln \left ({{\rm e}^{x}}+i \right ) +{\frac{i}{6}}\ln \left ({{\rm e}^{2\,x}}-i{{\rm e}^{x}}-1 \right ) -{\frac{i}{6}}\ln \left ({{\rm e}^{2\,x}}+i{{\rm e}^{x}}-1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.61363, size = 62, normalized size = 3.26 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.01357, size = 302, normalized size = 15.89 \begin{align*} \frac{2 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \arctan \left (-\frac{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - 6 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + 3 \, \cosh \left (x\right )^{2} + 6 \, \cosh \left (x\right ) \sinh \left (x\right ) + 3 \, \sinh \left (x\right )^{2} - 3}{6 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh{\left (x \right )} \tanh{\left (3 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19821, size = 58, normalized size = 3.05 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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