Integrand size = 15, antiderivative size = 18 \[ \int \frac {3+x}{\sqrt {1-x^2}} \, dx=-\sqrt {1-x^2}+3 \arcsin (x) \]
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Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {655, 222} \[ \int \frac {3+x}{\sqrt {1-x^2}} \, dx=3 \arcsin (x)-\sqrt {1-x^2} \]
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Rule 222
Rule 655
Rubi steps \begin{align*} \text {integral}& = -\sqrt {1-x^2}+3 \int \frac {1}{\sqrt {1-x^2}} \, dx \\ & = -\sqrt {1-x^2}+3 \sin ^{-1}(x) \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.89 \[ \int \frac {3+x}{\sqrt {1-x^2}} \, dx=-\sqrt {1-x^2}-6 \arctan \left (\frac {\sqrt {1-x^2}}{1+x}\right ) \]
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Time = 2.18 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94
method | result | size |
default | \(3 \arcsin \left (x \right )-\sqrt {-x^{2}+1}\) | \(17\) |
risch | \(\frac {x^{2}-1}{\sqrt {-x^{2}+1}}+3 \arcsin \left (x \right )\) | \(21\) |
meijerg | \(3 \arcsin \left (x \right )-\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-x^{2}+1}}{2 \sqrt {\pi }}\) | \(31\) |
trager | \(-\sqrt {-x^{2}+1}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+1}+x \right )\) | \(40\) |
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Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.67 \[ \int \frac {3+x}{\sqrt {1-x^2}} \, dx=-\sqrt {-x^{2} + 1} - 6 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67 \[ \int \frac {3+x}{\sqrt {1-x^2}} \, dx=- \sqrt {1 - x^{2}} + 3 \operatorname {asin}{\left (x \right )} \]
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Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {3+x}{\sqrt {1-x^2}} \, dx=-\sqrt {-x^{2} + 1} + 3 \, \arcsin \left (x\right ) \]
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Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {3+x}{\sqrt {1-x^2}} \, dx=-\sqrt {-x^{2} + 1} + 3 \, \arcsin \left (x\right ) \]
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Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {3+x}{\sqrt {1-x^2}} \, dx=3\,\mathrm {asin}\left (x\right )-\sqrt {1-x^2} \]
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