Optimal. Leaf size=17 \[ -i \text{EllipticF}\left (\frac{\pi }{2}+i x,-1\right ) \]
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Rubi [A] time = 0.0090285, 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 = {3182} \[ -i F\left (\left .i x+\frac{\pi }{2}\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 3182
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1+\cosh ^2(x)}} \, dx &=-i F\left (\left .\frac{\pi }{2}+i x\right |-1\right )\\ \end{align*}
Mathematica [A] time = 0.0343898, size = 18, normalized size = 1.06 \[ -\frac{i \text{EllipticF}\left (i x,\frac{1}{2}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.204, size = 45, normalized size = 2.7 \begin{align*}{\frac{-i{\it EllipticF} \left ( i\cosh \left ( x \right ) ,i \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 \frac{1}{\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 (\frac{1}{\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 \frac{1}{\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 \frac{1}{\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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