Optimal. Leaf size=38 \[ \sinh (x)-\frac{1}{6} \tan ^{-1}(\sinh (x))-\frac{1}{6} \tan ^{-1}(2 \sinh (x))-\frac{\tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{3}}\right )}{2 \sqrt{3}} \]
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Rubi [A] time = 0.0773604, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {12, 2073, 203} \[ \sinh (x)-\frac{1}{6} \tan ^{-1}(\sinh (x))-\frac{1}{6} \tan ^{-1}(2 \sinh (x))-\frac{\tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{3}}\right )}{2 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2073
Rule 203
Rubi steps
\begin{align*} \int \coth (6 x) \sinh (x) \, dx &=\operatorname{Subst}\left (\int \frac{1+18 x^2+48 x^4+32 x^6}{2 \left (3+19 x^2+32 x^4+16 x^6\right )} \, dx,x,\sinh (x)\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1+18 x^2+48 x^4+32 x^6}{3+19 x^2+32 x^4+16 x^6} \, dx,x,\sinh (x)\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (2-\frac{1}{3 \left (1+x^2\right )}-\frac{2}{3 \left (1+4 x^2\right )}-\frac{2}{3+4 x^2}\right ) \, dx,x,\sinh (x)\right )\\ &=\sinh (x)-\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (x)\right )-\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1+4 x^2} \, dx,x,\sinh (x)\right )-\operatorname{Subst}\left (\int \frac{1}{3+4 x^2} \, dx,x,\sinh (x)\right )\\ &=-\frac{1}{6} \tan ^{-1}(\sinh (x))-\frac{1}{6} \tan ^{-1}(2 \sinh (x))-\frac{\tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{3}}\right )}{2 \sqrt{3}}+\sinh (x)\\ \end{align*}
Mathematica [A] time = 0.0578494, size = 38, normalized size = 1. \[ \sinh (x)-\frac{1}{6} \tan ^{-1}(\sinh (x))-\frac{1}{6} \tan ^{-1}(2 \sinh (x))-\frac{\tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{3}}\right )}{2 \sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.112, size = 172, normalized size = 4.5 \begin{align*}{\frac{\sqrt{3}}{12-6\,\sqrt{3}}\arctan \left ( 2\,{\frac{\tanh \left ( x/2 \right ) }{4-2\,\sqrt{3}}} \right ) }-{\frac{2}{12-6\,\sqrt{3}}\arctan \left ( 2\,{\frac{\tanh \left ( x/2 \right ) }{4-2\,\sqrt{3}}} \right ) }-{\frac{\sqrt{3}}{12+6\,\sqrt{3}}\arctan \left ( 2\,{\frac{\tanh \left ( x/2 \right ) }{4+2\,\sqrt{3}}} \right ) }-{\frac{2}{12+6\,\sqrt{3}}\arctan \left ( 2\,{\frac{\tanh \left ( x/2 \right ) }{4+2\,\sqrt{3}}} \right ) }-{\frac{\sqrt{3}}{6}\arctan \left ({\frac{\sqrt{3}}{3}\tanh \left ({\frac{x}{2}} \right ) } \right ) }-{\frac{1}{3}\arctan \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }- \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}-{\frac{\sqrt{3}}{6}\arctan \left ( \tanh \left ({\frac{x}{2}} \right ) \sqrt{3} \right ) }- \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \,{\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )} - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, e^{x} + 1\right )}\right ) - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, e^{x} - 1\right )}\right ) - \frac{1}{3} \, \arctan \left (e^{x}\right ) - \frac{1}{2} \, \int \frac{e^{\left (3 \, x\right )} + e^{x}}{3 \,{\left (e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )} + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.23425, size = 624, normalized size = 16.42 \begin{align*} -\frac{{\left (\sqrt{3} \cosh \left (x\right ) + \sqrt{3} \sinh \left (x\right )\right )} \arctan \left (\frac{1}{3} \, \sqrt{3} \cosh \left (x\right ) + \frac{1}{3} \, \sqrt{3} \sinh \left (x\right )\right ) -{\left (\sqrt{3} \cosh \left (x\right ) + \sqrt{3} \sinh \left (x\right )\right )} \arctan \left (-\frac{\sqrt{3} \cosh \left (x\right )^{2} + 2 \, \sqrt{3} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt{3} \sinh \left (x\right )^{2} + 2 \, \sqrt{3}}{3 \,{\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}}\right ) -{\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 ) + 3 \,{\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 )} \coth{\left (6 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2043, size = 92, normalized size = 2.42 \begin{align*} -\frac{1}{6} \, \pi - \frac{1}{12} \, \sqrt{3}{\left (\pi + 2 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )} - \frac{1}{6} \, \arctan \left ({\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right ) - \frac{1}{6} \, \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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