Optimal. Leaf size=19 \[ \frac {\sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {\pi }}\right )}{\sqrt {b}} \]
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Rubi [A] time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {216} \[ \frac {\sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {\pi }}\right )}{\sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 216
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\pi -b x^2}} \, dx &=\frac {\sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {\pi }}\right )}{\sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 19, normalized size = 1.00 \[ \frac {\sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {\pi }}\right )}{\sqrt {b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 62, normalized size = 3.26 \[ \left [-\frac {\sqrt {-b} \log \left (-\pi + 2 \, b x^{2} - 2 \, \sqrt {\pi - b x^{2}} \sqrt {-b} x\right )}{2 \, b}, -\frac {\arctan \left (-\frac {\sqrt {b} x}{\sqrt {\pi - b x^{2}}}\right )}{\sqrt {b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.05, size = 28, normalized size = 1.47 \[ -\frac {\log \left ({\left | -\sqrt {-b} x + \sqrt {\pi - b x^{2}} \right |}\right )}{\sqrt {-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 21, normalized size = 1.11 \[ \frac {\arctan \left (\frac {\sqrt {b}\, x}{\sqrt {-b \,x^{2}+\pi }}\right )}{\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.86, size = 13, normalized size = 0.68 \[ \frac {\arcsin \left (\frac {b x}{\sqrt {\pi b}}\right )}{\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 25, normalized size = 1.32 \[ \frac {\ln \left (\sqrt {\Pi -b\,x^2}+\sqrt {-b}\,x\right )}{\sqrt {-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.04, size = 46, normalized size = 2.42 \[ \begin {cases} - \frac {i \operatorname {acosh}{\left (\frac {\sqrt {b} x}{\sqrt {\pi }} \right )}}{\sqrt {b}} & \text {for}\: \frac {\left |{b x^{2}}\right |}{\pi } > 1 \\\frac {\operatorname {asin}{\left (\frac {\sqrt {b} x}{\sqrt {\pi }} \right )}}{\sqrt {b}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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