Optimal. Leaf size=11 \[ -i \text{EllipticF}(i x,-1) \]
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Rubi [A] time = 0.0108332, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3182} \[ -i F(i x|-1) \]
Antiderivative was successfully verified.
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Rule 3182
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1-\sinh ^2(x)}} \, dx &=-i F(i x|-1)\\ \end{align*}
Mathematica [A] time = 0.0398722, size = 11, normalized size = 1. \[ -i \text{EllipticF}(i x,-1) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.147, size = 41, normalized size = 3.7 \begin{align*}{\frac{{\it EllipticF} \left ( \sinh \left ( x \right ) ,i \right ) }{\cosh \left ( x \right ) }\sqrt{- \left ( -1+ \left ( \sinh \left ( x \right ) \right ) ^{2} \right ) \left ( \cosh \left ( x \right ) \right ) ^{2}}\sqrt{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{\frac{1}{\sqrt{1- \left ( \sinh \left ( x \right ) \right ) ^{4}}}}} \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{-\sinh \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{\sqrt{-\sinh \left (x\right )^{2} + 1}}{\sinh \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{1 - \sinh ^{2}{\left (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{1}{\sqrt{-\sinh \left (x\right )^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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