Optimal. Leaf size=13 \[ \coth (x) \sqrt{a \sinh ^2(x)} \]
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Rubi [A] time = 0.0128127, antiderivative size = 13, 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, 2638} \[ \coth (x) \sqrt{a \sinh ^2(x)} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 2638
Rubi steps
\begin{align*} \int \sqrt{a \sinh ^2(x)} \, dx &=\left (\text{csch}(x) \sqrt{a \sinh ^2(x)}\right ) \int \sinh (x) \, dx\\ &=\coth (x) \sqrt{a \sinh ^2(x)}\\ \end{align*}
Mathematica [A] time = 0.004513, size = 13, normalized size = 1. \[ \coth (x) \sqrt{a \sinh ^2(x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 15, normalized size = 1.2 \begin{align*}{a\sinh \left ( x \right ) \cosh \left ( x \right ){\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 = 1.82899, size = 23, normalized size = 1.77 \begin{align*} -\frac{1}{2} \, \sqrt{a} e^{\left (-x\right )} - \frac{1}{2} \, \sqrt{a} e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.74253, size = 216, normalized size = 16.62 \begin{align*} \frac{{\left (2 \, \cosh \left (x\right ) e^{x} \sinh \left (x\right ) + e^{x} \sinh \left (x\right )^{2} +{\left (\cosh \left (x\right )^{2} + 1\right )} e^{x}\right )} \sqrt{a e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + a} e^{\left (-x\right )}}{2 \,{\left (\cosh \left (x\right ) e^{\left (2 \, x\right )} +{\left (e^{\left (2 \, x\right )} - 1\right )} \sinh \left (x\right ) - \cosh \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.609714, size = 19, normalized size = 1.46 \begin{align*} \frac{\sqrt{a} \sqrt{\sinh ^{2}{\left (x \right )}} \cosh{\left (x \right )}}{\sinh{\left (x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19944, size = 46, normalized size = 3.54 \begin{align*} \frac{1}{2} \,{\left (e^{\left (-x\right )} \mathrm{sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) + e^{x} \mathrm{sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )\right )} \sqrt{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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