Optimal. Leaf size=14 \[ \frac{\cosh (x) \tan ^{-1}(\sinh (x))}{\sqrt{\cosh ^2(x)}} \]
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Rubi [A] time = 0.0240236, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3176, 3207, 3770} \[ \frac{\cosh (x) \tan ^{-1}(\sinh (x))}{\sqrt{\cosh ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3207
Rule 3770
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1+\sinh ^2(x)}} \, dx &=\int \frac{1}{\sqrt{\cosh ^2(x)}} \, dx\\ &=\frac{\cosh (x) \int \text{sech}(x) \, dx}{\sqrt{\cosh ^2(x)}}\\ &=\frac{\tan ^{-1}(\sinh (x)) \cosh (x)}{\sqrt{\cosh ^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.0108901, size = 19, normalized size = 1.36 \[ \frac{2 \cosh (x) \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )}{\sqrt{\cosh ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 15, normalized size = 1.1 \begin{align*}{\frac{\arctan \left ( \sinh \left ( x \right ) \right ) }{\cosh \left ( x \right ) }\sqrt{ \left ( \cosh \left ( x \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.65678, size = 7, normalized size = 0.5 \begin{align*} 2 \, \arctan \left (e^{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03552, size = 39, normalized size = 2.79 \begin{align*} 2 \, \arctan \left (\cosh \left (x\right ) + \sinh \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 \frac{1}{\sqrt{\sinh ^{2}{\left (x \right )} + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14593, size = 7, normalized size = 0.5 \begin{align*} 2 \, \arctan \left (e^{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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