Optimal. Leaf size=8 \[ -2 \text{sech}\left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.188749, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6715, 2606, 8} \[ -2 \text{sech}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Rule 6715
Rule 2606
Rule 8
Rubi steps
\begin{align*} \int \frac{\text{sech}\left (\sqrt{x}\right ) \tanh \left (\sqrt{x}\right )}{\sqrt{x}} \, dx &=2 \operatorname{Subst}\left (\int \text{sech}(x) \tanh (x) \, dx,x,\sqrt{x}\right )\\ &=-\left (2 \operatorname{Subst}\left (\int 1 \, dx,x,\text{sech}\left (\sqrt{x}\right )\right )\right )\\ &=-2 \text{sech}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0156439, size = 8, normalized size = 1. \[ -2 \text{sech}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 7, normalized size = 0.9 \begin{align*} -2\,{\rm sech} \left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.04147, size = 20, normalized size = 2.5 \begin{align*} -\frac{4}{e^{\left (-\sqrt{x}\right )} + e^{\sqrt{x}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.76584, size = 146, normalized size = 18.25 \begin{align*} -\frac{4 \,{\left (\cosh \left (\sqrt{x}\right ) + \sinh \left (\sqrt{x}\right )\right )}}{\cosh \left (\sqrt{x}\right )^{2} + 2 \, \cosh \left (\sqrt{x}\right ) \sinh \left (\sqrt{x}\right ) + \sinh \left (\sqrt{x}\right )^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.431821, size = 8, normalized size = 1. \begin{align*} - 2 \operatorname{sech}{\left (\sqrt{x} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.12011, size = 20, normalized size = 2.5 \begin{align*} -\frac{4}{e^{\left (-\sqrt{x}\right )} + e^{\sqrt{x}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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