Optimal. Leaf size=17 \[ -\frac {\sinh (x) \tanh ^{-1}(\cosh (x))}{\sqrt {a \sinh ^2(x)}} \]
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Rubi [A] time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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*}
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Mathematica [A] time = 0.01, 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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fricas [B] time = 0.80, size = 110, normalized size = 6.47 \[ \left [\frac {\sqrt {a e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + a} \log \left (\frac {\cosh \relax (x) + \sinh \relax (x) - 1}{\cosh \relax (x) + \sinh \relax (x) + 1}\right )}{a e^{\left (2 \, x\right )} - a}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {a e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + a} \sqrt {-a}}{a \cosh \relax (x) e^{\left (2 \, x\right )} - a \cosh \relax (x) + {\left (a e^{\left (2 \, x\right )} - a\right )} \sinh \relax (x)}\right )}{a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 1, normalized size = 0.06 \[ 0 \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 49, normalized size = 2.88 \[ -\frac {\sinh \relax (x ) \sqrt {a \left (\cosh ^{2}\relax (x )\right )}\, \ln \left (\frac {2 \sqrt {a}\, \sqrt {a \left (\cosh ^{2}\relax (x )\right )}+2 a}{\sinh \relax (x )}\right )}{\sqrt {a}\, \cosh \relax (x ) \sqrt {a \left (\sinh ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 24, normalized size = 1.41 \[ \frac {\log \left (e^{\left (-x\right )} + 1\right )}{\sqrt {a}} - \frac {\log \left (e^{\left (-x\right )} - 1\right )}{\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {1}{\sqrt {a\,{\mathrm {sinh}\relax (x)}^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \sinh ^{2}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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