Optimal. Leaf size=20 \[ -\frac {2 \sinh ^{-1}\left (\frac {\sqrt {-b x}}{\sqrt {2}}\right )}{b} \]
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Rubi [A] time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {63, 215} \[ -\frac {2 \sinh ^{-1}\left (\frac {\sqrt {-b x}}{\sqrt {2}}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 63
Rule 215
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-b x} \sqrt {2-b x}} \, dx &=-\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+x^2}} \, dx,x,\sqrt {-b x}\right )}{b}\\ &=-\frac {2 \sinh ^{-1}\left (\frac {\sqrt {-b x}}{\sqrt {2}}\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 37, normalized size = 1.85 \[ \frac {2 \sqrt {x} \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b} \sqrt {-b x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 27, normalized size = 1.35 \[ -\frac {\log \left (-b x + \sqrt {-b x + 2} \sqrt {-b x} + 1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.03, size = 23, normalized size = 1.15 \[ \frac {2 \, \log \left (\sqrt {-b x + 2} - \sqrt {-b x}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 64, normalized size = 3.20 \[ \frac {\sqrt {-\left (-b x +2\right ) b x}\, \ln \left (\frac {b^{2} x -b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}-2 b x}\right )}{\sqrt {-b x}\, \sqrt {-b x +2}\, \sqrt {b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 32, normalized size = 1.60 \[ \frac {\log \left (2 \, b^{2} x + 2 \, \sqrt {b^{2} x^{2} - 2 \, b x} b - 2 \, b\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 39, normalized size = 1.95 \[ -\frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {2}-\sqrt {2-b\,x}\right )}{\sqrt {-b\,x}\,\sqrt {-b^2}}\right )}{\sqrt {-b^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.34, size = 53, normalized size = 2.65 \[ \begin {cases} - \frac {2 \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b} & \text {for}\: \frac {\left |{b x}\right |}{2} > 1 \\- \frac {2 i \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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