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 = 0.017586, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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 266
Rule 63
Rule 208
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*}
Mathematica [B] time = 0.0136172, 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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Maple [B] time = 0.006, size = 49, normalized size = 1.8 \begin{align*}{\frac{1}{2\,{x}^{2}}\sqrt{a{x}^{4}+b}\ln \left ({x}^{2}\sqrt{a}+\sqrt{a{x}^{4}+b} \right ){\frac{1}{\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}}}}{\frac{1}{\sqrt{a}}}} \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 = 1.51519, size = 196, normalized size = 7.26 \begin{align*} \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 ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.39634, size = 20, normalized size = 0.74 \begin{align*} \frac{\operatorname{asinh}{\left (\frac{\sqrt{a} x^{2}}{\sqrt{b}} \right )}}{2 \sqrt{a}} \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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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