Optimal. Leaf size=85 \[ -\frac{\tan ^{-1}\left (\sqrt{2} \sinh (x)\right )}{3 \sqrt{2}}+\frac{\tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{2-\sqrt{3}}}\right )}{6 \sqrt{2-\sqrt{3}}}+\frac{\tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{2+\sqrt{3}}}\right )}{6 \sqrt{2+\sqrt{3}}} \]
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Rubi [A] time = 0.0576444, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {4356, 2057, 203, 1166} \[ -\frac{\tan ^{-1}\left (\sqrt{2} \sinh (x)\right )}{3 \sqrt{2}}+\frac{\tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{2-\sqrt{3}}}\right )}{6 \sqrt{2-\sqrt{3}}}+\frac{\tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{2+\sqrt{3}}}\right )}{6 \sqrt{2+\sqrt{3}}} \]
Antiderivative was successfully verified.
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Rule 4356
Rule 2057
Rule 203
Rule 1166
Rubi steps
\begin{align*} \int \cosh (x) \text{sech}(6 x) \, dx &=\operatorname{Subst}\left (\int \frac{1}{1+18 x^2+48 x^4+32 x^6} \, dx,x,\sinh (x)\right )\\ &=\operatorname{Subst}\left (\int \left (-\frac{1}{3 \left (1+2 x^2\right )}+\frac{4 \left (1+2 x^2\right )}{3 \left (1+16 x^2+16 x^4\right )}\right ) \, dx,x,\sinh (x)\right )\\ &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1+2 x^2} \, dx,x,\sinh (x)\right )\right )+\frac{4}{3} \operatorname{Subst}\left (\int \frac{1+2 x^2}{1+16 x^2+16 x^4} \, dx,x,\sinh (x)\right )\\ &=-\frac{\tan ^{-1}\left (\sqrt{2} \sinh (x)\right )}{3 \sqrt{2}}+\frac{4}{3} \operatorname{Subst}\left (\int \frac{1}{8-4 \sqrt{3}+16 x^2} \, dx,x,\sinh (x)\right )+\frac{4}{3} \operatorname{Subst}\left (\int \frac{1}{8+4 \sqrt{3}+16 x^2} \, dx,x,\sinh (x)\right )\\ &=-\frac{\tan ^{-1}\left (\sqrt{2} \sinh (x)\right )}{3 \sqrt{2}}+\frac{\tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{2-\sqrt{3}}}\right )}{6 \sqrt{2-\sqrt{3}}}+\frac{\tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{2+\sqrt{3}}}\right )}{6 \sqrt{2+\sqrt{3}}}\\ \end{align*}
Mathematica [A] time = 0.0843409, size = 81, normalized size = 0.95 \[ \frac{1}{6} \left (-\sqrt{2} \tan ^{-1}\left (\sqrt{2} \sinh (x)\right )+\sqrt{2+\sqrt{3}} \tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{2-\sqrt{3}}}\right )+\sqrt{2-\sqrt{3}} \tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{2+\sqrt{3}}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.083, size = 83, normalized size = 1. \begin{align*}{\frac{i}{12}}\sqrt{2}\ln \left ({{\rm e}^{2\,x}}-i\sqrt{2}{{\rm e}^{x}}-1 \right ) -{\frac{i}{12}}\sqrt{2}\ln \left ({{\rm e}^{2\,x}}+i\sqrt{2}{{\rm e}^{x}}-1 \right ) +2\,\sum _{{\it \_R}={\it RootOf} \left ( 331776\,{{\it \_Z}}^{4}+2304\,{{\it \_Z}}^{2}+1 \right ) }{\it \_R}\,\ln \left ({{\rm e}^{2\,x}}+ \left ( 13824\,{{\it \_R}}^{3}+96\,{\it \_R} \right ){{\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{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, e^{x}\right )}\right ) - \frac{1}{6} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, e^{x}\right )}\right ) + \int \frac{e^{\left (7 \, x\right )} + e^{\left (5 \, x\right )} + e^{\left (3 \, x\right )} + e^{x}}{3 \,{\left (e^{\left (8 \, x\right )} - e^{\left (4 \, 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 = 2.38837, size = 489, normalized size = 5.75 \begin{align*} -\frac{1}{3} \, \sqrt{\sqrt{3} + 2} \arctan \left (-{\left (\sqrt{\sqrt{3} + 2}{\left (e^{\left (2 \, x\right )} - 1\right )} - \sqrt{-\sqrt{3} e^{\left (2 \, x\right )} + e^{\left (4 \, x\right )} + 1} \sqrt{\sqrt{3} + 2}\right )} e^{\left (-x\right )}\right ) - \frac{1}{3} \, \sqrt{-\sqrt{3} + 2} \arctan \left (-{\left (\sqrt{-\sqrt{3} + 2}{\left (e^{\left (2 \, x\right )} - 1\right )} - \sqrt{\sqrt{3} e^{\left (2 \, x\right )} + e^{\left (4 \, x\right )} + 1} \sqrt{-\sqrt{3} + 2}\right )} e^{\left (-x\right )}\right ) - \frac{1}{6} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} e^{\left (3 \, x\right )} + \frac{1}{2} \, \sqrt{2} e^{x}\right ) - \frac{1}{6} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} e^{x}\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 \cosh{\left (x \right )} \operatorname{sech}{\left (6 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.20785, size = 239, normalized size = 2.81 \begin{align*} \frac{1}{12} \,{\left (\sqrt{6} - \sqrt{2}\right )} \arctan \left (\frac{\sqrt{6} - \sqrt{2} + 4 \, e^{x}}{\sqrt{6} + \sqrt{2}}\right ) + \frac{1}{12} \,{\left (\sqrt{6} - \sqrt{2}\right )} \arctan \left (-\frac{\sqrt{6} - \sqrt{2} - 4 \, e^{x}}{\sqrt{6} + \sqrt{2}}\right ) + \frac{1}{12} \,{\left (\sqrt{6} + \sqrt{2}\right )} \arctan \left (\frac{\sqrt{6} + \sqrt{2} + 4 \, e^{x}}{\sqrt{6} - \sqrt{2}}\right ) + \frac{1}{12} \,{\left (\sqrt{6} + \sqrt{2}\right )} \arctan \left (-\frac{\sqrt{6} + \sqrt{2} - 4 \, e^{x}}{\sqrt{6} - \sqrt{2}}\right ) - \frac{1}{6} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, e^{x}\right )}\right ) - \frac{1}{6} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, e^{x}\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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