Optimal. Leaf size=16 \[ \frac{\cosh (x) \tan ^{-1}(\sinh (x))}{\sqrt{a \cosh ^2(x)}} \]
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Rubi [A] time = 0.0149485, antiderivative size = 16, 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} \[ \frac{\cosh (x) \tan ^{-1}(\sinh (x))}{\sqrt{a \cosh ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 3770
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a \cosh ^2(x)}} \, dx &=\frac{\cosh (x) \int \text{sech}(x) \, dx}{\sqrt{a \cosh ^2(x)}}\\ &=\frac{\tan ^{-1}(\sinh (x)) \cosh (x)}{\sqrt{a \cosh ^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.0065387, size = 21, normalized size = 1.31 \[ \frac{2 \cosh (x) \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )}{\sqrt{a \cosh ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.044, size = 55, normalized size = 3.4 \begin{align*} -{\frac{\cosh \left ( x \right ) }{\sinh \left ( x \right ) }\sqrt{a \left ( \sinh \left ( x \right ) \right ) ^{2}}\ln \left ( 2\,{\frac{\sqrt{-a}\sqrt{a \left ( \sinh \left ( x \right ) \right ) ^{2}}-a}{\cosh \left ( x \right ) }} \right ){\frac{1}{\sqrt{-a}}}{\frac{1}{\sqrt{a \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 = 1.69339, size = 11, normalized size = 0.69 \begin{align*} \frac{2 \, \arctan \left (e^{x}\right )}{\sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.18351, size = 552, normalized size = 34.5 \begin{align*} \left [-\frac{\sqrt{-a} \log \left (\frac{a \cosh \left (x\right )^{2} - 2 \, \sqrt{a e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} + a}{\left (\cosh \left (x\right ) e^{x} + e^{x} \sinh \left (x\right )\right )} \sqrt{-a} e^{\left (-x\right )} +{\left (a e^{\left (2 \, x\right )} + a\right )} \sinh \left (x\right )^{2} +{\left (a \cosh \left (x\right )^{2} - a\right )} e^{\left (2 \, x\right )} + 2 \,{\left (a \cosh \left (x\right ) e^{\left (2 \, x\right )} + a \cosh \left (x\right )\right )} \sinh \left (x\right ) - a}{{\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} +{\left (\cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )} + 2 \,{\left (\cosh \left (x\right ) e^{\left (2 \, x\right )} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1}\right )}{a}, \frac{2 \, \sqrt{a e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} + a} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}{a e^{\left (2 \, x\right )} + a}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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