Optimal. Leaf size=15 \[ \frac{\sinh ^{-1}\left (\sqrt{2} \sinh (x)\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0185004, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4356, 215} \[ \frac{\sinh ^{-1}\left (\sqrt{2} \sinh (x)\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 4356
Rule 215
Rubi steps
\begin{align*} \int \frac{\cosh (x)}{\sqrt{\cosh (2 x)}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+2 x^2}} \, dx,x,\sinh (x)\right )\\ &=\frac{\sinh ^{-1}\left (\sqrt{2} \sinh (x)\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0107721, size = 15, normalized size = 1. \[ \frac{\sinh ^{-1}\left (\sqrt{2} \sinh (x)\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.048, size = 63, normalized size = 4.2 \begin{align*}{\frac{\sqrt{2}}{4\,\sinh \left ( x \right ) }\sqrt{ \left ( 2\, \left ( \cosh \left ( x \right ) \right ) ^{2}-1 \right ) \left ( \sinh \left ( x \right ) \right ) ^{2}}\ln \left ( \sqrt{2} \left ( \sinh \left ( x \right ) \right ) ^{2}+\sqrt{2\, \left ( \sinh \left ( x \right ) \right ) ^{4}+ \left ( \sinh \left ( x \right ) \right ) ^{2}}+{\frac{\sqrt{2}}{4}} \right ){\frac{1}{\sqrt{2\, \left ( \cosh \left ( x \right ) \right ) ^{2}-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*} \int \frac{\cosh \left (x\right )}{\sqrt{\cosh \left (2 \, x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.30835, size = 1636, normalized size = 109.07 \begin{align*} \text{result too large to display} \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 \frac{\cosh{\left (x \right )}}{\sqrt{\cosh{\left (2 x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (x\right )}{\sqrt{\cosh \left (2 \, x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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