Optimal. Leaf size=36 \[ \frac{1}{6} \tan ^{-1}(\sinh (x))+\frac{1}{6} \tan ^{-1}(2 \sinh (x))-\frac{\tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{3}}\right )}{2 \sqrt{3}} \]
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Rubi [A] time = 0.0454255, antiderivative size = 36, 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, 2057, 203} \[ \frac{1}{6} \tan ^{-1}(\sinh (x))+\frac{1}{6} \tan ^{-1}(2 \sinh (x))-\frac{\tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{3}}\right )}{2 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2057
Rule 203
Rubi steps
\begin{align*} \int \text{csch}(6 x) \sinh (x) \, dx &=\operatorname{Subst}\left (\int \frac{1}{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}{3+19 x^2+32 x^4+16 x^6} \, dx,x,\sinh (x)\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\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 )\\ &=\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}}\\ \end{align*}
Mathematica [A] time = 0.0325924, size = 30, normalized size = 0.83 \[ \frac{1}{6} \left (\tan ^{-1}(\sinh (x))+\tan ^{-1}(2 \sinh (x))-\sqrt{3} \tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{3}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.054, size = 92, normalized size = 2.6 \begin{align*}{\frac{i}{6}}\ln \left ({{\rm e}^{x}}+i \right ) -{\frac{i}{6}}\ln \left ({{\rm e}^{x}}-i \right ) +{\frac{i}{12}}\ln \left ({{\rm e}^{2\,x}}+i{{\rm e}^{x}}-1 \right ) -{\frac{i}{12}}\ln \left ({{\rm e}^{2\,x}}-i{{\rm e}^{x}}-1 \right ) +{\frac{i}{12}}\sqrt{3}\ln \left ({{\rm e}^{2\,x}}-i\sqrt{3}{{\rm e}^{x}}-1 \right ) -{\frac{i}{12}}\sqrt{3}\ln \left ({{\rm e}^{2\,x}}+i\sqrt{3}{{\rm e}^{x}}-1 \right ) \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}{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 ) + \int \frac{e^{\left (3 \, x\right )} + e^{x}}{6 \,{\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 = 2.15969, size = 408, normalized size = 11.33 \begin{align*} -\frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} \cosh \left (x\right ) + \frac{1}{3} \, \sqrt{3} \sinh \left (x\right )\right ) + \frac{1}{6} \, \sqrt{3} \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 ) - \frac{1}{6} \, \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 ) + \frac{1}{2} \, \arctan \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 )} \operatorname{csch}{\left (6 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19096, size = 78, normalized size = 2.17 \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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