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 = 0.0244563, antiderivative size = 30, normalized size of antiderivative = 1., 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 335
Rule 275
Rule 217
Rule 206
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*}
Mathematica [A] time = 0.0150563, 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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Maple [B] time = 0.008, size = 52, normalized size = 1.7 \begin{align*} -{\frac{1}{2\,{x}^{2}}\sqrt{a{x}^{4}+b}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{4}+b}+b}{{x}^{2}}} \right ){\frac{1}{\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}}}}{\frac{1}{\sqrt{b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52684, size = 188, normalized size = 6.27 \begin{align*} \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 ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.65845, size = 22, normalized size = 0.73 \begin{align*} - \frac{\operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x^{2}} \right )}}{2 \sqrt{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a + \frac{b}{x^{4}}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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