Integrand size = 12, antiderivative size = 38 \[ \int \arccos \left (\sqrt {\frac {x}{1+x}}\right ) \, dx=(1+x) \left (\sqrt {\frac {1}{1+x}} \sqrt {\frac {x}{1+x}}+\arccos \left (\sqrt {\frac {x}{1+x}}\right )\right ) \]
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Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.50, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4925, 12, 6851, 52, 65, 209} \[ \int \arccos \left (\sqrt {\frac {x}{1+x}}\right ) \, dx=x \arccos \left (\sqrt {\frac {x}{x+1}}\right )-\frac {\sqrt {\frac {x}{(x+1)^2}} (x+1) \arctan \left (\sqrt {x}\right )}{\sqrt {x}}+\sqrt {\frac {x}{(x+1)^2}} (x+1) \]
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Rule 12
Rule 52
Rule 65
Rule 209
Rule 4925
Rule 6851
Rubi steps \begin{align*} \text {integral}& = x \arccos \left (\sqrt {\frac {x}{1+x}}\right )+\int \frac {1}{2} \sqrt {\frac {x}{(1+x)^2}} \, dx \\ & = x \arccos \left (\sqrt {\frac {x}{1+x}}\right )+\frac {1}{2} \int \sqrt {\frac {x}{(1+x)^2}} \, dx \\ & = x \arccos \left (\sqrt {\frac {x}{1+x}}\right )+\frac {\left (\sqrt {\frac {x}{(1+x)^2}} (1+x)\right ) \int \frac {\sqrt {x}}{1+x} \, dx}{2 \sqrt {x}} \\ & = \sqrt {\frac {x}{(1+x)^2}} (1+x)+x \arccos \left (\sqrt {\frac {x}{1+x}}\right )-\frac {\left (\sqrt {\frac {x}{(1+x)^2}} (1+x)\right ) \int \frac {1}{\sqrt {x} (1+x)} \, dx}{2 \sqrt {x}} \\ & = \sqrt {\frac {x}{(1+x)^2}} (1+x)+x \arccos \left (\sqrt {\frac {x}{1+x}}\right )-\frac {\left (\sqrt {\frac {x}{(1+x)^2}} (1+x)\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x}} \\ & = \sqrt {\frac {x}{(1+x)^2}} (1+x)+x \arccos \left (\sqrt {\frac {x}{1+x}}\right )-\frac {\sqrt {\frac {x}{(1+x)^2}} (1+x) \arctan \left (\sqrt {x}\right )}{\sqrt {x}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.29 \[ \int \arccos \left (\sqrt {\frac {x}{1+x}}\right ) \, dx=x \arccos \left (\sqrt {\frac {x}{1+x}}\right )+\frac {\sqrt {\frac {x}{(1+x)^2}} (1+x) \left (\sqrt {x}-\arctan \left (\sqrt {x}\right )\right )}{\sqrt {x}} \]
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Time = 0.05 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.18
method | result | size |
default | \(x \arccos \left (\sqrt {\frac {x}{1+x}}\right )-\frac {\sqrt {x}\, \sqrt {\frac {1}{1+x}}\, \left (\arctan \left (\sqrt {x}\right )-\sqrt {x}\right )}{\sqrt {\frac {x}{1+x}}}\) | \(45\) |
parts | \(x \arccos \left (\sqrt {\frac {x}{1+x}}\right )-\frac {\sqrt {x}\, \sqrt {\frac {1}{1+x}}\, \left (\arctan \left (\sqrt {x}\right )-\sqrt {x}\right )}{\sqrt {\frac {x}{1+x}}}\) | \(45\) |
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none
Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.79 \[ \int \arccos \left (\sqrt {\frac {x}{1+x}}\right ) \, dx={\left (x + 1\right )} \arccos \left (\sqrt {\frac {x}{x + 1}}\right ) + \sqrt {x + 1} \sqrt {\frac {x}{x + 1}} \]
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Time = 4.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.53 \[ \int \arccos \left (\sqrt {\frac {x}{1+x}}\right ) \, dx=x \operatorname {acos}{\left (\sqrt {\frac {x}{x + 1}} \right )} - \begin {cases} - \frac {\sqrt {\frac {x}{x + 1}}}{\sqrt {- \frac {x}{x + 1} + 1}} + \operatorname {asin}{\left (\sqrt {\frac {x}{x + 1}} \right )} & \text {for}\: \sqrt {\frac {x}{x + 1}} > -1 \wedge \sqrt {\frac {x}{x + 1}} < 1 \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (30) = 60\).
Time = 0.30 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.05 \[ \int \arccos \left (\sqrt {\frac {x}{1+x}}\right ) \, dx=-\frac {\arccos \left (\sqrt {\frac {x}{x + 1}}\right )}{\frac {x}{x + 1} - 1} - \frac {\sqrt {-\frac {x}{x + 1} + 1}}{2 \, {\left (\sqrt {\frac {x}{x + 1}} + 1\right )}} - \frac {\sqrt {-\frac {x}{x + 1} + 1}}{2 \, {\left (\sqrt {\frac {x}{x + 1}} - 1\right )}} \]
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Exception generated. \[ \int \arccos \left (\sqrt {\frac {x}{1+x}}\right ) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \arccos \left (\sqrt {\frac {x}{1+x}}\right ) \, dx=\int \mathrm {acos}\left (\sqrt {\frac {x}{x+1}}\right ) \,d x \]
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