Optimal. Leaf size=43 \[ x \text{sech}^{-1}\left (\sqrt{x}\right )-\frac{1-x}{\sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}} \]
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Rubi [A] time = 0.0072149, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6343, 12, 32} \[ x \text{sech}^{-1}\left (\sqrt{x}\right )-\frac{1-x}{\sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 6343
Rule 12
Rule 32
Rubi steps
\begin{align*} \int \text{sech}^{-1}\left (\sqrt{x}\right ) \, dx &=x \text{sech}^{-1}\left (\sqrt{x}\right )+\frac{\sqrt{1-x} \int \frac{1}{2 \sqrt{1-x}} \, dx}{\sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=x \text{sech}^{-1}\left (\sqrt{x}\right )+\frac{\sqrt{1-x} \int \frac{1}{\sqrt{1-x}} \, dx}{2 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=-\frac{1-x}{\sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}+x \text{sech}^{-1}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0958415, size = 67, normalized size = 1.56 \[ x \text{sech}^{-1}\left (\sqrt{x}\right )-\frac{\sqrt{\frac{1-\sqrt{x}}{\sqrt{x}+1}} \sqrt{\sqrt{x}+1} \sqrt{1-x}}{\sqrt{1-\sqrt{x}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.123, size = 36, normalized size = 0.8 \begin{align*} x{\rm arcsech} \left (\sqrt{x}\right )-\sqrt{{ \left ( 1+\sqrt{x} \right ){\frac{1}{\sqrt{x}}}}}\sqrt{-{ \left ( -1+\sqrt{x} \right ){\frac{1}{\sqrt{x}}}}}\sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.971505, size = 26, normalized size = 0.6 \begin{align*} x \operatorname{arsech}\left (\sqrt{x}\right ) - \sqrt{x} \sqrt{\frac{1}{x} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80622, size = 92, normalized size = 2.14 \begin{align*} x \log \left (\frac{x \sqrt{-\frac{x - 1}{x}} + \sqrt{x}}{x}\right ) - \sqrt{x} \sqrt{-\frac{x - 1}{x}} \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 \operatorname{asech}{\left (\sqrt{x} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{arsech}\left (\sqrt{x}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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