Optimal. Leaf size=43 \[ \frac{1}{2} \sqrt{a+c x^4}-\frac{1}{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right ) \]
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Rubi [A] time = 0.0264348, antiderivative size = 43, 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 = {266, 50, 63, 208} \[ \frac{1}{2} \sqrt{a+c x^4}-\frac{1}{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+c x^4}}{x} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{\sqrt{a+c x}}{x} \, dx,x,x^4\right )\\ &=\frac{1}{2} \sqrt{a+c x^4}+\frac{1}{4} a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^4\right )\\ &=\frac{1}{2} \sqrt{a+c x^4}+\frac{a \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^4}\right )}{2 c}\\ &=\frac{1}{2} \sqrt{a+c x^4}-\frac{1}{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.0094839, size = 43, normalized size = 1. \[ \frac{1}{2} \sqrt{a+c x^4}-\frac{1}{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 41, normalized size = 1. \begin{align*}{\frac{1}{2}\sqrt{c{x}^{4}+a}}-{\frac{1}{2}\sqrt{a}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{4}+a} \right ) } \right ) } \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 = 1.52976, size = 217, normalized size = 5.05 \begin{align*} \left [\frac{1}{4} \, \sqrt{a} \log \left (\frac{c x^{4} - 2 \, \sqrt{c x^{4} + a} \sqrt{a} + 2 \, a}{x^{4}}\right ) + \frac{1}{2} \, \sqrt{c x^{4} + a}, \frac{1}{2} \, \sqrt{-a} \arctan \left (\frac{\sqrt{c x^{4} + a} \sqrt{-a}}{a}\right ) + \frac{1}{2} \, \sqrt{c x^{4} + a}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.87906, size = 66, normalized size = 1.53 \begin{align*} - \frac{\sqrt{a} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x^{2}} \right )}}{2} + \frac{a}{2 \sqrt{c} x^{2} \sqrt{\frac{a}{c x^{4}} + 1}} + \frac{\sqrt{c} x^{2}}{2 \sqrt{\frac{a}{c x^{4}} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10163, size = 49, normalized size = 1.14 \begin{align*} \frac{a \arctan \left (\frac{\sqrt{c x^{4} + a}}{\sqrt{-a}}\right )}{2 \, \sqrt{-a}} + \frac{1}{2} \, \sqrt{c x^{4} + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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