Optimal. Leaf size=18 \[ -\sqrt {-1+x}+x \sec ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 18, 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 = {5376, 12, 32}
\begin {gather*} x \sec ^{-1}\left (\sqrt {x}\right )-\sqrt {x-1} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 32
Rule 5376
Rubi steps
\begin {align*} \int \sec ^{-1}\left (\sqrt {x}\right ) \, dx &=x \sec ^{-1}\left (\sqrt {x}\right )-\int \frac {1}{2 \sqrt {-1+x}} \, dx\\ &=x \sec ^{-1}\left (\sqrt {x}\right )-\frac {1}{2} \int \frac {1}{\sqrt {-1+x}} \, dx\\ &=-\sqrt {-1+x}+x \sec ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} -\sqrt {-1+x}+x \sec ^{-1}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 25, normalized size = 1.39
method | result | size |
derivativedivides | \(x \,\mathrm {arcsec}\left (\sqrt {x}\right )-\frac {x -1}{\sqrt {\frac {x -1}{x}}\, \sqrt {x}}\) | \(25\) |
default | \(x \,\mathrm {arcsec}\left (\sqrt {x}\right )-\frac {x -1}{\sqrt {\frac {x -1}{x}}\, \sqrt {x}}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 21, normalized size = 1.17 \begin {gather*} x \operatorname {arcsec}\left (\sqrt {x}\right ) - \sqrt {x} \sqrt {-\frac {1}{x} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.99, size = 14, normalized size = 0.78 \begin {gather*} x \operatorname {arcsec}\left (\sqrt {x}\right ) - \sqrt {x - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 2.85, size = 29, normalized size = 1.61 \begin {gather*} x \operatorname {asec}{\left (\sqrt {x} \right )} - \frac {\begin {cases} 2 \sqrt {x - 1} & \text {for}\: \left |{x}\right | > 1 \\2 i \sqrt {1 - x} & \text {otherwise} \end {cases}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 41 vs.
\(2 (14) = 28\).
time = 0.45, size = 41, normalized size = 2.28 \begin {gather*} x \arccos \left (\frac {1}{\sqrt {x}}\right ) - \frac {1}{2} \, \sqrt {x} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )} + \frac {1}{2 \, \sqrt {x} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.15, size = 21, normalized size = 1.17 \begin {gather*} x\,\mathrm {acos}\left (\frac {1}{\sqrt {x}}\right )-\sqrt {x}\,\sqrt {1-\frac {1}{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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