Optimal. Leaf size=71 \[ \frac{\tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{2-\sqrt{2}}}\right )}{2 \sqrt{2 \left (2-\sqrt{2}\right )}}-\frac{\tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{2+\sqrt{2}}}\right )}{2 \sqrt{2 \left (2+\sqrt{2}\right )}} \]
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Rubi [A] time = 0.0389463, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4356, 1093, 203} \[ \frac{\tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{2-\sqrt{2}}}\right )}{2 \sqrt{2 \left (2-\sqrt{2}\right )}}-\frac{\tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{2+\sqrt{2}}}\right )}{2 \sqrt{2 \left (2+\sqrt{2}\right )}} \]
Antiderivative was successfully verified.
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Rule 4356
Rule 1093
Rule 203
Rubi steps
\begin{align*} \int \cosh (x) \text{sech}(4 x) \, dx &=\operatorname{Subst}\left (\int \frac{1}{1+8 x^2+8 x^4} \, dx,x,\sinh (x)\right )\\ &=\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{4-2 \sqrt{2}+8 x^2} \, dx,x,\sinh (x)\right )-\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{4+2 \sqrt{2}+8 x^2} \, dx,x,\sinh (x)\right )\\ &=\frac{\tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{2-\sqrt{2}}}\right )}{2 \sqrt{2 \left (2-\sqrt{2}\right )}}-\frac{\tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{2+\sqrt{2}}}\right )}{2 \sqrt{2 \left (2+\sqrt{2}\right )}}\\ \end{align*}
Mathematica [A] time = 0.0967995, size = 67, normalized size = 0.94 \[ \frac{1}{4} \sqrt{2+\sqrt{2}} \tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{2-\sqrt{2}}}\right )-\frac{\tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{2+\sqrt{2}}}\right )}{2 \sqrt{2 \left (2+\sqrt{2}\right )}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.056, size = 40, normalized size = 0.6 \begin{align*} 2\,\sum _{{\it \_R}={\it RootOf} \left ( 32768\,{{\it \_Z}}^{4}+512\,{{\it \_Z}}^{2}+1 \right ) }{\it \_R}\,\ln \left ({{\rm e}^{2\,x}}+ \left ( -4096\,{{\it \_R}}^{3}-48\,{\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*} \int \cosh \left (x\right ) \operatorname{sech}\left (4 \, x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.23634, size = 427, normalized size = 6.01 \begin{align*} -\frac{1}{2} \, \sqrt{\sqrt{2} + 2} \arctan \left (-\frac{1}{2} \,{\left ({\left (\sqrt{2} e^{\left (2 \, x\right )} - \sqrt{2}\right )} \sqrt{\sqrt{2} + 2} - \sqrt{2} \sqrt{-\sqrt{2} e^{\left (2 \, x\right )} + e^{\left (4 \, x\right )} + 1} \sqrt{\sqrt{2} + 2}\right )} e^{\left (-x\right )}\right ) + \frac{1}{2} \, \sqrt{-\sqrt{2} + 2} \arctan \left (-\frac{1}{2} \,{\left ({\left (\sqrt{2} e^{\left (2 \, x\right )} - \sqrt{2}\right )} \sqrt{-\sqrt{2} + 2} - \sqrt{2} \sqrt{\sqrt{2} e^{\left (2 \, x\right )} + e^{\left (4 \, x\right )} + 1} \sqrt{-\sqrt{2} + 2}\right )} e^{\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 \cosh{\left (x \right )} \operatorname{sech}{\left (4 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21649, size = 182, normalized size = 2.56 \begin{align*} \frac{1}{4} \, \sqrt{\sqrt{2} + 2} \arctan \left (\frac{\sqrt{\sqrt{2} + 2} + 2 \, e^{x}}{\sqrt{-\sqrt{2} + 2}}\right ) + \frac{1}{4} \, \sqrt{\sqrt{2} + 2} \arctan \left (-\frac{\sqrt{\sqrt{2} + 2} - 2 \, e^{x}}{\sqrt{-\sqrt{2} + 2}}\right ) - \frac{1}{4} \, \sqrt{-\sqrt{2} + 2} \arctan \left (\frac{\sqrt{-\sqrt{2} + 2} + 2 \, e^{x}}{\sqrt{\sqrt{2} + 2}}\right ) - \frac{1}{4} \, \sqrt{-\sqrt{2} + 2} \arctan \left (-\frac{\sqrt{-\sqrt{2} + 2} - 2 \, e^{x}}{\sqrt{\sqrt{2} + 2}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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