Optimal. Leaf size=30 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {b}}{x^2 \sqrt {a+\frac {b}{x^4}}}\right )}{2 \sqrt {b}} \]
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Rubi [A] time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {335, 275, 217, 206} \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {b}}{x^2 \sqrt {a+\frac {b}{x^4}}}\right )}{2 \sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 275
Rule 335
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+\frac {b}{x^4}} x^3} \, dx &=-\operatorname {Subst}\left (\int \frac {x}{\sqrt {a+b x^4}} \, dx,x,\frac {1}{x}\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x^4}} x^2}\right )\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x^4}} x^2}\right )}{2 \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 52, normalized size = 1.73 \[ -\frac {\sqrt {a x^4+b} \tanh ^{-1}\left (\frac {\sqrt {a x^4+b}}{\sqrt {b}}\right )}{2 \sqrt {b} x^2 \sqrt {a+\frac {b}{x^4}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 77, normalized size = 2.57 \[ \left [\frac {\log \left (\frac {a x^{4} - 2 \, \sqrt {b} x^{2} \sqrt {\frac {a x^{4} + b}{x^{4}}} + 2 \, b}{x^{4}}\right )}{4 \, \sqrt {b}}, \frac {\sqrt {-b} \arctan \left (\frac {\sqrt {-b} x^{2} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{b}\right )}{2 \, b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 41, normalized size = 1.37 \[ \frac {\arctan \left (\frac {\sqrt {a x^{4} + b}}{\sqrt {-b}}\right )}{2 \, \sqrt {-b}} - \frac {\arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right )}{2 \, \sqrt {-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 52, normalized size = 1.73 \[ -\frac {\sqrt {a \,x^{4}+b}\, \ln \left (\frac {2 b +2 \sqrt {a \,x^{4}+b}\, \sqrt {b}}{x^{2}}\right )}{2 \sqrt {\frac {a \,x^{4}+b}{x^{4}}}\, \sqrt {b}\, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.97, size = 45, normalized size = 1.50 \[ \frac {\log \left (\frac {\sqrt {a + \frac {b}{x^{4}}} x^{2} - \sqrt {b}}{\sqrt {a + \frac {b}{x^{4}}} x^{2} + \sqrt {b}}\right )}{4 \, \sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{x^3\,\sqrt {a+\frac {b}{x^4}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.63, size = 22, normalized size = 0.73 \[ - \frac {\operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} x^{2}} \right )}}{2 \sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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