Optimal. Leaf size=17 \[ -i E\left (\left .i x+\frac{\pi }{2}\right |-1\right ) \]
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Rubi [A] time = 0.0094512, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3177} \[ -i E\left (\left .i x+\frac{\pi }{2}\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 3177
Rubi steps
\begin{align*} \int \sqrt{1+\cosh ^2(x)} \, dx &=-i E\left (\left .\frac{\pi }{2}+i x\right |-1\right )\\ \end{align*}
Mathematica [A] time = 0.0223248, size = 18, normalized size = 1.06 \[ -i \sqrt{2} E\left (i x\left |\frac{1}{2}\right .\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.236, size = 58, normalized size = 3.4 \begin{align*}{\frac{-i \left ( 2\,{\it EllipticF} \left ( i\cosh \left ( x \right ) ,i \right ) -{\it EllipticE} \left ( i\cosh \left ( x \right ) ,i \right ) \right ) }{\sinh \left ( x \right ) }\sqrt{ \left ( 1+ \left ( \cosh \left ( x \right ) \right ) ^{2} \right ) \left ( \sinh \left ( x \right ) \right ) ^{2}}\sqrt{- \left ( \sinh \left ( x \right ) \right ) ^{2}}{\frac{1}{\sqrt{ \left ( \cosh \left ( x \right ) \right ) ^{4}-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 \sqrt{\cosh \left (x\right )^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{\cosh \left (x\right )^{2} + 1}, 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 \sqrt{\cosh ^{2}{\left (x \right )} + 1}\, 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 \sqrt{\cosh \left (x\right )^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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