Optimal. Leaf size=38 \[ (x+1) \left (\sqrt {\frac {1}{x+1}} \sqrt {\frac {x}{x+1}}+\cos ^{-1}\left (\sqrt {\frac {x}{x+1}}\right )\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.50, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4841, 12, 6719, 50, 63, 203} \[ \sqrt {\frac {x}{(x+1)^2}} (x+1)+x \cos ^{-1}\left (\sqrt {\frac {x}{x+1}}\right )-\frac {\sqrt {\frac {x}{(x+1)^2}} (x+1) \tan ^{-1}\left (\sqrt {x}\right )}{\sqrt {x}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 50
Rule 63
Rule 203
Rule 4841
Rule 6719
Rubi steps
\begin {align*} \int \cos ^{-1}\left (\sqrt {\frac {x}{1+x}}\right ) \, dx &=x \cos ^{-1}\left (\sqrt {\frac {x}{1+x}}\right )+\int \frac {1}{2} \sqrt {\frac {x}{(1+x)^2}} \, dx\\ &=x \cos ^{-1}\left (\sqrt {\frac {x}{1+x}}\right )+\frac {1}{2} \int \sqrt {\frac {x}{(1+x)^2}} \, dx\\ &=x \cos ^{-1}\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 \cos ^{-1}\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 \cos ^{-1}\left (\sqrt {\frac {x}{1+x}}\right )-\frac {\left (\sqrt {\frac {x}{(1+x)^2}} (1+x)\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x}}\\ &=\sqrt {\frac {x}{(1+x)^2}} (1+x)+x \cos ^{-1}\left (\sqrt {\frac {x}{1+x}}\right )-\frac {\sqrt {\frac {x}{(1+x)^2}} (1+x) \tan ^{-1}\left (\sqrt {x}\right )}{\sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 49, normalized size = 1.29 \[ x \cos ^{-1}\left (\sqrt {\frac {x}{x+1}}\right )+\frac {\sqrt {\frac {x}{(x+1)^2}} (x+1) \left (\sqrt {x}-\tan ^{-1}\left (\sqrt {x}\right )\right )}{\sqrt {x}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos ^{-1}\left (\sqrt {\frac {x}{1+x}}\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.83, size = 30, normalized size = 0.79 \[ {\left (x + 1\right )} \arccos \left (\sqrt {\frac {x}{x + 1}}\right ) + \sqrt {x + 1} \sqrt {\frac {x}{x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 45, normalized size = 1.18
method | result | size |
default | \(x \arccos \left (\sqrt {\frac {x}{1+x}}\right )-\frac {\sqrt {x}\, \sqrt {\frac {1}{1+x}}\, \left (-\sqrt {x}+\arctan \left (\sqrt {x}\right )\right )}{\sqrt {\frac {x}{1+x}}}\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.97, size = 78, normalized size = 2.05 \[ -\frac {\arccos \left (\sqrt {\frac {x}{x + 1}}\right )}{\frac {x}{x + 1} - 1} - \frac {\sqrt {-\frac {x}{x + 1} + 1}}{2 \, {\left (\sqrt {\frac {x}{x + 1}} + 1\right )}} - \frac {\sqrt {-\frac {x}{x + 1} + 1}}{2 \, {\left (\sqrt {\frac {x}{x + 1}} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \mathrm {acos}\left (\sqrt {\frac {x}{x+1}}\right ) \,d x \]
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 \[ \int \operatorname {acos}{\left (\sqrt {\frac {x}{x + 1}} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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