Optimal. Leaf size=24 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{\sqrt {a}} \]
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Rubi [A]
time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {272, 65, 214}
\begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x} \, dx &=-\left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\frac {\text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^2}}\right )}{b}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{\sqrt {a}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(50\) vs. \(2(24)=48\).
time = 0.01, size = 50, normalized size = 2.08 \begin {gather*} \frac {\sqrt {b+a x^2} \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b+a x^2}}\right )}{\sqrt {a} \sqrt {a+\frac {b}{x^2}} x} \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. \(45\) vs.
\(2(18)=36\).
time = 0.02, size = 46, normalized size = 1.92
method | result | size |
default | \(\frac {\sqrt {a \,x^{2}+b}\, \ln \left (x \sqrt {a}+\sqrt {a \,x^{2}+b}\right )}{\sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x \sqrt {a}}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs.
\(2 (18) = 36\).
time = 0.53, size = 37, normalized size = 1.54 \begin {gather*} -\frac {\log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{2}}} + \sqrt {a}}\right )}{2 \, \sqrt {a}} \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. 42 vs.
\(2 (18) = 36\).
time = 0.41, size = 80, normalized size = 3.33 \begin {gather*} \left [\frac {\log \left (-2 \, a x^{2} - 2 \, \sqrt {a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} - b\right )}{2 \, \sqrt {a}}, -\frac {\sqrt {-a} \arctan \left (\frac {\sqrt {-a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right )}{a}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.50, size = 17, normalized size = 0.71 \begin {gather*} \frac {\operatorname {asinh}{\left (\frac {\sqrt {a} x}{\sqrt {b}} \right )}}{\sqrt {a}} \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. 38 vs.
\(2 (18) = 36\).
time = 1.44, size = 38, normalized size = 1.58 \begin {gather*} \frac {\log \left ({\left | b \right |}\right ) \mathrm {sgn}\left (x\right )}{2 \, \sqrt {a}} - \frac {\log \left ({\left | -\sqrt {a} x + \sqrt {a x^{2} + b} \right |}\right )}{\sqrt {a} \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.31, size = 18, normalized size = 0.75 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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