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 = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 12
Rule 32
Rule 6343
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*}
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Mathematica [A] time = 0.10, 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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fricas [A] time = 0.56, size = 39, normalized size = 0.91 \[ x \log \left (\frac {x \sqrt {-\frac {x - 1}{x}} + \sqrt {x}}{x}\right ) - \sqrt {x} \sqrt {-\frac {x - 1}{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {arsech}\left (\sqrt {x}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 36, normalized size = 0.84 \[ x \,\mathrm {arcsech}\left (\sqrt {x}\right )-\sqrt {-\frac {-1+\sqrt {x}}{\sqrt {x}}}\, \sqrt {\frac {1+\sqrt {x}}{\sqrt {x}}}\, \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 19, normalized size = 0.44 \[ x \operatorname {arsech}\left (\sqrt {x}\right ) - \sqrt {x} \sqrt {\frac {1}{x} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \mathrm {acosh}\left (\frac {1}{\sqrt {x}}\right ) \,d x \]
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 \[ \int \operatorname {asech}{\left (\sqrt {x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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