Optimal. Leaf size=28 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a-b x^4}}{\sqrt{a}}\right )}{2 \sqrt{a}} \]
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Rubi [A] time = 0.0180761, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {266, 63, 208} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a-b x^4}}{\sqrt{a}}\right )}{2 \sqrt{a}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{a-b x^4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a-b x}} \, dx,x,x^4\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a-b x^4}\right )}{2 b}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{a-b x^4}}{\sqrt{a}}\right )}{2 \sqrt{a}}\\ \end{align*}
Mathematica [A] time = 0.0058615, size = 28, normalized size = 1. \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a-b x^4}}{\sqrt{a}}\right )}{2 \sqrt{a}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 30, normalized size = 1.1 \begin{align*} -{\frac{1}{2}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{-b{x}^{4}+a} \right ) } \right ){\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 = 1.50488, size = 163, normalized size = 5.82 \begin{align*} \left [\frac{\log \left (\frac{b x^{4} + 2 \, \sqrt{-b x^{4} + a} \sqrt{a} - 2 \, a}{x^{4}}\right )}{4 \, \sqrt{a}}, \frac{\sqrt{-a} \arctan \left (\frac{\sqrt{-b x^{4} + a} \sqrt{-a}}{a}\right )}{2 \, a}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.58102, size = 56, normalized size = 2. \begin{align*} \begin{cases} - \frac{\operatorname{acosh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2 \sqrt{a}} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x^{4}}\right |} > 1 \\\frac{i \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2 \sqrt{a}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09135, size = 32, normalized size = 1.14 \begin{align*} \frac{\arctan \left (\frac{\sqrt{-b x^{4} + a}}{\sqrt{-a}}\right )}{2 \, \sqrt{-a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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