Integrand size = 12, antiderivative size = 25 \[ \int \frac {\arccos \left (\sqrt {x}\right )}{\sqrt {x}} \, dx=-2 \sqrt {1-x}+2 \sqrt {x} \arccos \left (\sqrt {x}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6847, 4716, 267} \[ \int \frac {\arccos \left (\sqrt {x}\right )}{\sqrt {x}} \, dx=2 \sqrt {x} \arccos \left (\sqrt {x}\right )-2 \sqrt {1-x} \]
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Rule 267
Rule 4716
Rule 6847
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \arccos (x) \, dx,x,\sqrt {x}\right ) \\ & = 2 \sqrt {x} \arccos \left (\sqrt {x}\right )+2 \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2}} \, dx,x,\sqrt {x}\right ) \\ & = -2 \sqrt {1-x}+2 \sqrt {x} \arccos \left (\sqrt {x}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {\arccos \left (\sqrt {x}\right )}{\sqrt {x}} \, dx=-2 \sqrt {1-x}+2 \sqrt {x} \arccos \left (\sqrt {x}\right ) \]
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Time = 0.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(-2 \sqrt {1-x}+2 \arccos \left (\sqrt {x}\right ) \sqrt {x}\) | \(20\) |
default | \(-2 \sqrt {1-x}+2 \arccos \left (\sqrt {x}\right ) \sqrt {x}\) | \(20\) |
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Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {\arccos \left (\sqrt {x}\right )}{\sqrt {x}} \, dx=2 \, \sqrt {x} \arccos \left (\sqrt {x}\right ) - 2 \, \sqrt {-x + 1} \]
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Time = 0.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {\arccos \left (\sqrt {x}\right )}{\sqrt {x}} \, dx=2 \sqrt {x} \operatorname {acos}{\left (\sqrt {x} \right )} - 2 \sqrt {1 - x} \]
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none
Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {\arccos \left (\sqrt {x}\right )}{\sqrt {x}} \, dx=2 \, \sqrt {x} \arccos \left (\sqrt {x}\right ) - 2 \, \sqrt {-x + 1} \]
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Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {\arccos \left (\sqrt {x}\right )}{\sqrt {x}} \, dx=2 \, \sqrt {x} \arccos \left (\sqrt {x}\right ) - 2 \, \sqrt {-x + 1} \]
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Time = 0.44 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {\arccos \left (\sqrt {x}\right )}{\sqrt {x}} \, dx=2\,\sqrt {x}\,\mathrm {acos}\left (\sqrt {x}\right )-2\,\sqrt {1-x} \]
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