Optimal. Leaf size=27 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{2 \sqrt {a}} \]
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Rubi [A] time = 0.02, antiderivative size = 27, 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 = {266, 63, 208} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{2 \sqrt {a}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+\frac {b}{x^4}} x} \, dx &=-\left (\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^4}\right )\right )\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^4}}\right )}{2 b}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{2 \sqrt {a}}\\ \end {align*}
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Mathematica [B] time = 0.03, size = 55, normalized size = 2.04 \[ \frac {\sqrt {a x^4+b} \tanh ^{-1}\left (\frac {\sqrt {a} x^2}{\sqrt {a x^4+b}}\right )}{2 \sqrt {a} x^2 \sqrt {a+\frac {b}{x^4}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.12, size = 80, normalized size = 2.96 \[ \left [\frac {\log \left (-2 \, a x^{4} - 2 \, \sqrt {a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}} - b\right )}{4 \, \sqrt {a}}, -\frac {\sqrt {-a} \arctan \left (\frac {\sqrt {-a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{a x^{4} + b}\right )}{2 \, a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 34, normalized size = 1.26 \[ -\frac {\log \left ({\left | -\sqrt {a} x^{2} + \sqrt {a x^{4} + b} \right |}\right )}{2 \, \sqrt {a}} + \frac {\log \left ({\left | b \right |}\right )}{4 \, \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 49, normalized size = 1.81 \[ \frac {\sqrt {a \,x^{4}+b}\, \ln \left (\sqrt {a}\, x^{2}+\sqrt {a \,x^{4}+b}\right )}{2 \sqrt {\frac {a \,x^{4}+b}{x^{4}}}\, \sqrt {a}\, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.97, size = 37, normalized size = 1.37 \[ -\frac {\log \left (\frac {\sqrt {a + \frac {b}{x^{4}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{4}}} + \sqrt {a}}\right )}{4 \, \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.30, size = 19, normalized size = 0.70 \[ \frac {\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{2\,\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.45, size = 20, normalized size = 0.74 \[ \frac {\operatorname {asinh}{\left (\frac {\sqrt {a} x^{2}}{\sqrt {b}} \right )}}{2 \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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