Optimal. Leaf size=85 \[ \frac{\tanh ^{-1}\left (\sqrt{2} \cosh (x)\right )}{3 \sqrt{2}}-\frac{\tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{2-\sqrt{3}}}\right )}{6 \sqrt{2-\sqrt{3}}}-\frac{\tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{2+\sqrt{3}}}\right )}{6 \sqrt{2+\sqrt{3}}} \]
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Rubi [A] time = 0.0791248, 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 = {4357, 2057, 207, 1166} \[ \frac{\tanh ^{-1}\left (\sqrt{2} \cosh (x)\right )}{3 \sqrt{2}}-\frac{\tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{2-\sqrt{3}}}\right )}{6 \sqrt{2-\sqrt{3}}}-\frac{\tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{2+\sqrt{3}}}\right )}{6 \sqrt{2+\sqrt{3}}} \]
Antiderivative was successfully verified.
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Rule 4357
Rule 2057
Rule 207
Rule 1166
Rubi steps
\begin{align*} \int \text{sech}(6 x) \sinh (x) \, dx &=\operatorname{Subst}\left (\int \frac{1}{-1+18 x^2-48 x^4+32 x^6} \, dx,x,\cosh (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,\cosh (x)\right )\\ &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{-1+2 x^2} \, dx,x,\cosh (x)\right )\right )+\frac{4}{3} \operatorname{Subst}\left (\int \frac{-1+2 x^2}{1-16 x^2+16 x^4} \, dx,x,\cosh (x)\right )\\ &=\frac{\tanh ^{-1}\left (\sqrt{2} \cosh (x)\right )}{3 \sqrt{2}}+\frac{4}{3} \operatorname{Subst}\left (\int \frac{1}{-8-4 \sqrt{3}+16 x^2} \, dx,x,\cosh (x)\right )+\frac{4}{3} \operatorname{Subst}\left (\int \frac{1}{-8+4 \sqrt{3}+16 x^2} \, dx,x,\cosh (x)\right )\\ &=\frac{\tanh ^{-1}\left (\sqrt{2} \cosh (x)\right )}{3 \sqrt{2}}-\frac{\tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{2-\sqrt{3}}}\right )}{6 \sqrt{2-\sqrt{3}}}-\frac{\tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{2+\sqrt{3}}}\right )}{6 \sqrt{2+\sqrt{3}}}\\ \end{align*}
Mathematica [C] time = 0.26666, size = 385, normalized size = 4.53 \[ \frac{\sqrt{2} \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^4+1\& ,\frac{\text{$\#$1}^6 x-\text{$\#$1}^4 x+\text{$\#$1}^2 x+2 \text{$\#$1}^6 \log \left (-\text{$\#$1} \sinh \left (\frac{x}{2}\right )+\text{$\#$1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )-2 \text{$\#$1}^4 \log \left (-\text{$\#$1} \sinh \left (\frac{x}{2}\right )+\text{$\#$1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )+2 \text{$\#$1}^2 \log \left (-\text{$\#$1} \sinh \left (\frac{x}{2}\right )+\text{$\#$1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )-2 \log \left (-\text{$\#$1} \sinh \left (\frac{x}{2}\right )+\text{$\#$1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )-x}{2 \text{$\#$1}^7-\text{$\#$1}^3}\& \right ]+8 \tanh ^{-1}\left (\sqrt{2}-i \tanh \left (\frac{x}{2}\right )\right )-2 \log \left (\sqrt{2}-2 \cosh (x)\right )+2 \log \left (2 \cosh (x)+\sqrt{2}\right )+4 i \tan ^{-1}\left (\frac{\sinh \left (\frac{x}{2}\right )+\cosh \left (\frac{x}{2}\right )}{\left (1+\sqrt{2}\right ) \cosh \left (\frac{x}{2}\right )-\left (\sqrt{2}-1\right ) \sinh \left (\frac{x}{2}\right )}\right )-4 i \tan ^{-1}\left (\frac{\sinh \left (\frac{x}{2}\right )+\cosh \left (\frac{x}{2}\right )}{\left (\sqrt{2}-1\right ) \cosh \left (\frac{x}{2}\right )-\left (1+\sqrt{2}\right ) \sinh \left (\frac{x}{2}\right )}\right )}{24 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.068, size = 78, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( 1+{{\rm e}^{2\,x}}+{{\rm e}^{x}}\sqrt{2} \right ) \sqrt{2}}{12}}-{\frac{\ln \left ( 1+{{\rm e}^{2\,x}}-{{\rm e}^{x}}\sqrt{2} \right ) \sqrt{2}}{12}}+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}{12} \, \sqrt{2} \log \left (\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac{1}{12} \, \sqrt{2} \log \left (-\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\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 = 2.33375, size = 906, normalized size = 10.66 \begin{align*} \frac{1}{12} \, \sqrt{\sqrt{3} + 2} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} +{\left ({\left (\sqrt{3} - 2\right )} \cosh \left (x\right ) +{\left (\sqrt{3} - 2\right )} \sinh \left (x\right )\right )} \sqrt{\sqrt{3} + 2} + 1\right ) - \frac{1}{12} \, \sqrt{\sqrt{3} + 2} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} -{\left ({\left (\sqrt{3} - 2\right )} \cosh \left (x\right ) +{\left (\sqrt{3} - 2\right )} \sinh \left (x\right )\right )} \sqrt{\sqrt{3} + 2} + 1\right ) - \frac{1}{12} \, \sqrt{-\sqrt{3} + 2} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} +{\left ({\left (\sqrt{3} + 2\right )} \cosh \left (x\right ) +{\left (\sqrt{3} + 2\right )} \sinh \left (x\right )\right )} \sqrt{-\sqrt{3} + 2} + 1\right ) + \frac{1}{12} \, \sqrt{-\sqrt{3} + 2} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} -{\left ({\left (\sqrt{3} + 2\right )} \cosh \left (x\right ) +{\left (\sqrt{3} + 2\right )} \sinh \left (x\right )\right )} \sqrt{-\sqrt{3} + 2} + 1\right ) + \frac{1}{12} \, \sqrt{2} \log \left (\frac{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} + 2 \, \sqrt{2} \cosh \left (x\right ) + 2}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2}}\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{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 = 1.24073, size = 208, normalized size = 2.45 \begin{align*} -\frac{1}{24} \,{\left (\sqrt{6} - \sqrt{2}\right )} \log \left (\frac{1}{2} \,{\left (\sqrt{6} + \sqrt{2}\right )} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac{1}{24} \,{\left (\sqrt{6} - \sqrt{2}\right )} \log \left (-\frac{1}{2} \,{\left (\sqrt{6} + \sqrt{2}\right )} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac{1}{24} \,{\left (\sqrt{6} + \sqrt{2}\right )} \log \left (\frac{1}{2} \,{\left (\sqrt{6} - \sqrt{2}\right )} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac{1}{24} \,{\left (\sqrt{6} + \sqrt{2}\right )} \log \left (-\frac{1}{2} \,{\left (\sqrt{6} - \sqrt{2}\right )} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac{1}{12} \, \sqrt{2} \log \left (\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac{1}{12} \, \sqrt{2} \log \left (-\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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