Optimal. Leaf size=31 \[ \frac {\sqrt {-x-1} \sqrt {x}}{\sqrt {-x}}+x \text {csch}^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 31, 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 = {6344, 12, 32} \[ \frac {\sqrt {-x-1} \sqrt {x}}{\sqrt {-x}}+x \text {csch}^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 32
Rule 6344
Rubi steps
\begin {align*} \int \text {csch}^{-1}\left (\sqrt {x}\right ) \, dx &=x \text {csch}^{-1}\left (\sqrt {x}\right )-\frac {\sqrt {x} \int \frac {1}{2 \sqrt {-1-x}} \, dx}{\sqrt {-x}}\\ &=x \text {csch}^{-1}\left (\sqrt {x}\right )-\frac {\sqrt {x} \int \frac {1}{\sqrt {-1-x}} \, dx}{2 \sqrt {-x}}\\ &=\frac {\sqrt {-1-x} \sqrt {x}}{\sqrt {-x}}+x \text {csch}^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 0.77 \[ \sqrt {\frac {1}{x}+1} \sqrt {x}+x \text {csch}^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 36, normalized size = 1.16 \[ x \log \left (\frac {x \sqrt {\frac {x + 1}{x}} + \sqrt {x}}{x}\right ) + \sqrt {x} \sqrt {\frac {x + 1}{x}} \]
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 \[ \int \operatorname {arcsch}\left (\sqrt {x}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 24, normalized size = 0.77 \[ x \,\mathrm {arccsch}\left (\sqrt {x}\right )+\frac {1+x}{\sqrt {\frac {1+x}{x}}\, \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 18, normalized size = 0.58 \[ x \operatorname {arcsch}\left (\sqrt {x}\right ) + \sqrt {x} \sqrt {\frac {1}{x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.57, size = 18, normalized size = 0.58 \[ x\,\mathrm {asinh}\left (\frac {1}{\sqrt {x}}\right )+\sqrt {x}\,\sqrt {\frac {1}{x}+1} \]
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 \[ \int \operatorname {acsch}{\left (\sqrt {x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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