Optimal. Leaf size=28 \[ \sinh (x)-\frac{1}{4} \tan ^{-1}(\sinh (x))-\frac{\tan ^{-1}\left (\sqrt{2} \sinh (x)\right )}{2 \sqrt{2}} \]
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Rubi [A] time = 0.0531672, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {1676, 1166, 203} \[ \sinh (x)-\frac{1}{4} \tan ^{-1}(\sinh (x))-\frac{\tan ^{-1}\left (\sqrt{2} \sinh (x)\right )}{2 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1676
Rule 1166
Rule 203
Rubi steps
\begin{align*} \int \coth (4 x) \sinh (x) \, dx &=\operatorname{Subst}\left (\int \frac{1+8 x^2+8 x^4}{4+12 x^2+8 x^4} \, dx,x,\sinh (x)\right )\\ &=\operatorname{Subst}\left (\int \left (1-\frac{3+4 x^2}{4+12 x^2+8 x^4}\right ) \, dx,x,\sinh (x)\right )\\ &=\sinh (x)-\operatorname{Subst}\left (\int \frac{3+4 x^2}{4+12 x^2+8 x^4} \, dx,x,\sinh (x)\right )\\ &=\sinh (x)-2 \operatorname{Subst}\left (\int \frac{1}{4+8 x^2} \, dx,x,\sinh (x)\right )-2 \operatorname{Subst}\left (\int \frac{1}{8+8 x^2} \, dx,x,\sinh (x)\right )\\ &=-\frac{1}{4} \tan ^{-1}(\sinh (x))-\frac{\tan ^{-1}\left (\sqrt{2} \sinh (x)\right )}{2 \sqrt{2}}+\sinh (x)\\ \end{align*}
Mathematica [A] time = 0.029861, size = 28, normalized size = 1. \[ \sinh (x)-\frac{1}{4} \tan ^{-1}(\sinh (x))-\frac{\tan ^{-1}\left (\sqrt{2} \sinh (x)\right )}{2 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.092, size = 143, normalized size = 5.1 \begin{align*} -{\frac{\sqrt{2}}{4+4\,\sqrt{2}}\arctan \left ( 2\,{\frac{\tanh \left ( x/2 \right ) }{2+2\,\sqrt{2}}} \right ) }-{\frac{1}{2+2\,\sqrt{2}}\arctan \left ( 2\,{\frac{\tanh \left ( x/2 \right ) }{2+2\,\sqrt{2}}} \right ) }+{\frac{\sqrt{2}}{-4+4\,\sqrt{2}}\arctan \left ( 2\,{\frac{\tanh \left ( x/2 \right ) }{-2+2\,\sqrt{2}}} \right ) }-{\frac{1}{-2+2\,\sqrt{2}}\arctan \left ( 2\,{\frac{\tanh \left ( x/2 \right ) }{-2+2\,\sqrt{2}}} \right ) }-{\frac{1}{2}\arctan \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }- \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}- \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.56177, size = 81, normalized size = 2.89 \begin{align*} \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, e^{\left (-x\right )}\right )}\right ) + \frac{1}{4} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, e^{\left (-x\right )}\right )}\right ) + \frac{1}{2} \, \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 = 2.31449, size = 493, normalized size = 17.61 \begin{align*} -\frac{{\left (\sqrt{2} \cosh \left (x\right ) + \sqrt{2} \sinh \left (x\right )\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} \cosh \left (x\right ) + \frac{1}{2} \, \sqrt{2} \sinh \left (x\right )\right ) -{\left (\sqrt{2} \cosh \left (x\right ) + \sqrt{2} \sinh \left (x\right )\right )} \arctan \left (-\frac{\sqrt{2} \cosh \left (x\right )^{2} + 2 \, \sqrt{2} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt{2} \sinh \left (x\right )^{2} + \sqrt{2}}{2 \,{\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}}\right ) + 2 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - 2 \, \cosh \left (x\right )^{2} - 4 \, \cosh \left (x\right ) \sinh \left (x\right ) - 2 \, \sinh \left (x\right )^{2} + 2}{4 \,{\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 )} \coth{\left (4 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21276, size = 73, normalized size = 2.61 \begin{align*} -\frac{1}{8} \, \pi - \frac{1}{8} \, \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}{4} \, \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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