Optimal. Leaf size=37 \[ \frac {1}{2} \sqrt {1-x} \sqrt {x}+\frac {1}{4} \text {ArcSin}(1-2 x)+x \text {ArcSin}\left (\sqrt {x}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {4924, 12, 52, 55,
633, 222} \begin {gather*} \frac {1}{4} \text {ArcSin}(1-2 x)+x \text {ArcSin}\left (\sqrt {x}\right )+\frac {1}{2} \sqrt {1-x} \sqrt {x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 52
Rule 55
Rule 222
Rule 633
Rule 4924
Rubi steps
\begin {align*} \int \sin ^{-1}\left (\sqrt {x}\right ) \, dx &=x \sin ^{-1}\left (\sqrt {x}\right )-\int \frac {\sqrt {x}}{2 \sqrt {1-x}} \, dx\\ &=x \sin ^{-1}\left (\sqrt {x}\right )-\frac {1}{2} \int \frac {\sqrt {x}}{\sqrt {1-x}} \, dx\\ &=\frac {1}{2} \sqrt {1-x} \sqrt {x}+x \sin ^{-1}\left (\sqrt {x}\right )-\frac {1}{4} \int \frac {1}{\sqrt {1-x} \sqrt {x}} \, dx\\ &=\frac {1}{2} \sqrt {1-x} \sqrt {x}+x \sin ^{-1}\left (\sqrt {x}\right )-\frac {1}{4} \int \frac {1}{\sqrt {x-x^2}} \, dx\\ &=\frac {1}{2} \sqrt {1-x} \sqrt {x}+x \sin ^{-1}\left (\sqrt {x}\right )+\frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,1-2 x\right )\\ &=\frac {1}{2} \sqrt {1-x} \sqrt {x}+\frac {1}{4} \sin ^{-1}(1-2 x)+x \sin ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 46, normalized size = 1.24 \begin {gather*} x \text {ArcSin}\left (\sqrt {x}\right )+\frac {1}{2} \left (\sqrt {-((-1+x) x)}-2 \text {ArcTan}\left (\frac {\sqrt {x}}{-1+\sqrt {1-x}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 26, normalized size = 0.70
method | result | size |
derivativedivides | \(x \arcsin \left (\sqrt {x}\right )+\frac {\sqrt {1-x}\, \sqrt {x}}{2}-\frac {\arcsin \left (\sqrt {x}\right )}{2}\) | \(26\) |
default | \(x \arcsin \left (\sqrt {x}\right )+\frac {\sqrt {1-x}\, \sqrt {x}}{2}-\frac {\arcsin \left (\sqrt {x}\right )}{2}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 25, normalized size = 0.68 \begin {gather*} x \arcsin \left (\sqrt {x}\right ) + \frac {1}{2} \, \sqrt {x} \sqrt {-x + 1} - \frac {1}{2} \, \arcsin \left (\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.90, size = 24, normalized size = 0.65 \begin {gather*} \frac {1}{2} \, {\left (2 \, x - 1\right )} \arcsin \left (\sqrt {x}\right ) + \frac {1}{2} \, \sqrt {x} \sqrt {-x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 29, normalized size = 0.78 \begin {gather*} \frac {\sqrt {x} \sqrt {1 - x}}{2} + x \operatorname {asin}{\left (\sqrt {x} \right )} - \frac {\operatorname {asin}{\left (\sqrt {x} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 27, normalized size = 0.73 \begin {gather*} {\left (x - 1\right )} \arcsin \left (\sqrt {x}\right ) + \frac {1}{2} \, \sqrt {x} \sqrt {-x + 1} + \frac {1}{2} \, \arcsin \left (\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.74, size = 37, normalized size = 1.00 \begin {gather*} x\,\mathrm {asin}\left (\sqrt {x}\right )-\mathrm {atan}\left (\frac {\sqrt {x}}{\sqrt {1-x}-1}\right )+\frac {\sqrt {x}\,\sqrt {1-x}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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