Optimal. Leaf size=17 \[ -\frac{\sinh (x) \tanh ^{-1}(\cosh (x))}{\sqrt{a \sinh ^2(x)}} \]
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Rubi [A] time = 0.0133405, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3207, 3770} \[ -\frac{\sinh (x) \tanh ^{-1}(\cosh (x))}{\sqrt{a \sinh ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 3770
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a \sinh ^2(x)}} \, dx &=\frac{\sinh (x) \int \text{csch}(x) \, dx}{\sqrt{a \sinh ^2(x)}}\\ &=-\frac{\tanh ^{-1}(\cosh (x)) \sinh (x)}{\sqrt{a \sinh ^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.0056344, size = 20, normalized size = 1.18 \[ \frac{\sinh (x) \log \left (\tanh \left (\frac{x}{2}\right )\right )}{\sqrt{a \sinh ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.053, size = 49, normalized size = 2.9 \begin{align*} -{\frac{\sinh \left ( x \right ) }{\cosh \left ( x \right ) }\sqrt{a \left ( \cosh \left ( x \right ) \right ) ^{2}}\ln \left ( 2\,{\frac{\sqrt{a}\sqrt{a \left ( \cosh \left ( x \right ) \right ) ^{2}}+a}{\sinh \left ( x \right ) }} \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{a \left ( \sinh \left ( x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.80917, size = 32, normalized size = 1.88 \begin{align*} \frac{\log \left (e^{\left (-x\right )} + 1\right )}{\sqrt{a}} - \frac{\log \left (e^{\left (-x\right )} - 1\right )}{\sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.71426, size = 312, normalized size = 18.35 \begin{align*} \left [\frac{\sqrt{a e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + a} \log \left (\frac{\cosh \left (x\right ) + \sinh \left (x\right ) - 1}{\cosh \left (x\right ) + \sinh \left (x\right ) + 1}\right )}{a e^{\left (2 \, x\right )} - a}, \frac{2 \, \sqrt{-a} \arctan \left (\frac{\sqrt{a e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + a} \sqrt{-a}}{a \cosh \left (x\right ) e^{\left (2 \, x\right )} - a \cosh \left (x\right ) +{\left (a e^{\left (2 \, x\right )} - a\right )} \sinh \left (x\right )}\right )}{a}\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{a \sinh ^{2}{\left (x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.26213, size = 61, normalized size = 3.59 \begin{align*} -\frac{\log \left (e^{x} + 1\right )}{\sqrt{a} \mathrm{sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )} + \frac{\log \left ({\left | e^{x} - 1 \right |}\right )}{\sqrt{a} \mathrm{sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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