Optimal. Leaf size=20 \[ -\frac {\csc ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {b}}\right )}{\sqrt {b}} \]
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Rubi [A]
time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {25, 342, 222}
\begin {gather*} -\frac {\csc ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {b}}\right )}{\sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 25
Rule 222
Rule 342
Rubi steps
\begin {align*} \int \frac {\sqrt {2-\frac {b}{x^2}}}{-b+2 x^2} \, dx &=\int \frac {1}{\sqrt {2-\frac {b}{x^2}} x^2} \, dx\\ &=-\text {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\csc ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {b}}\right )}{\sqrt {b}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(52\) vs. \(2(20)=40\).
time = 0.02, size = 52, normalized size = 2.60 \begin {gather*} \frac {\sqrt {2-\frac {b}{x^2}} x \tan ^{-1}\left (\frac {\sqrt {-b+2 x^2}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {-b+2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(60\) vs.
\(2(14)=28\).
time = 0.22, size = 61, normalized size = 3.05
method | result | size |
default | \(-\frac {\sqrt {-\frac {-2 x^{2}+b}{x^{2}}}\, x \ln \left (\frac {-2 b +2 \sqrt {-b}\, \sqrt {2 x^{2}-b}}{x}\right )}{\sqrt {2 x^{2}-b}\, \sqrt {-b}}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs.
\(2 (14) = 28\).
time = 0.39, size = 84, normalized size = 4.20 \begin {gather*} \left [-\frac {\sqrt {-b} \log \left (-\frac {x^{2} - \sqrt {-b} x \sqrt {\frac {2 \, x^{2} - b}{x^{2}}} - b}{x^{2}}\right )}{2 \, b}, -\frac {\arctan \left (\frac {\sqrt {b} x \sqrt {\frac {2 \, x^{2} - b}{x^{2}}}}{2 \, x^{2} - b}\right )}{\sqrt {b}}\right ] \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {- \frac {b}{x^{2}} + 2}}{- b + 2 x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 40 vs.
\(2 (14) = 28\).
time = 4.23, size = 40, normalized size = 2.00 \begin {gather*} \frac {\arctan \left (\frac {\sqrt {2 \, x^{2} - b}}{\sqrt {b}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {b}} - \frac {\arctan \left (\frac {\sqrt {-b}}{\sqrt {b}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.47, size = 21, normalized size = 1.05 \begin {gather*} -\frac {\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {-b}}{2\,x}\right )}{\sqrt {-b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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