Optimal. Leaf size=43 \[ \frac{1}{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )-\frac{1}{2} \sqrt{a+\frac{b}{x^4}} \]
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Rubi [A] time = 0.0237749, 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} \[ \frac{1}{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )-\frac{1}{2} \sqrt{a+\frac{b}{x^4}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+\frac{b}{x^4}}}{x} \, dx &=-\left (\frac{1}{4} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\frac{1}{x^4}\right )\right )\\ &=-\frac{1}{2} \sqrt{a+\frac{b}{x^4}}-\frac{1}{4} a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x^4}\right )\\ &=-\frac{1}{2} \sqrt{a+\frac{b}{x^4}}-\frac{a \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{1}{2} \sqrt{a+\frac{b}{x^4}}+\frac{1}{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.0899466, size = 60, normalized size = 1.4 \[ \frac{1}{2} \sqrt{a+\frac{b}{x^4}} \left (\frac{\sqrt{a} x^2 \sinh ^{-1}\left (\frac{\sqrt{a} x^2}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{\frac{a x^4}{b}+1}}-1\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 80, normalized size = 1.9 \begin{align*}{\frac{1}{2\,b}\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}} \left ( a{x}^{4}\sqrt{a{x}^{4}+b}+\sqrt{a}\ln \left ({x}^{2}\sqrt{a}+\sqrt{a{x}^{4}+b} \right ){x}^{2}b- \left ( a{x}^{4}+b \right ) ^{{\frac{3}{2}}} \right ){\frac{1}{\sqrt{a{x}^{4}+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 = 1.53487, size = 269, normalized size = 6.26 \begin{align*} \left [\frac{1}{4} \, \sqrt{a} \log \left (-2 \, a x^{4} - 2 \, \sqrt{a} x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} - b\right ) - \frac{1}{2} \, \sqrt{\frac{a x^{4} + b}{x^{4}}}, -\frac{1}{2} \, \sqrt{-a} \arctan \left (\frac{\sqrt{-a} x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{a x^{4} + b}\right ) - \frac{1}{2} \, \sqrt{\frac{a x^{4} + b}{x^{4}}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.34012, size = 66, normalized size = 1.53 \begin{align*} \frac{\sqrt{a} \operatorname{asinh}{\left (\frac{\sqrt{a} x^{2}}{\sqrt{b}} \right )}}{2} - \frac{a x^{2}}{2 \sqrt{b} \sqrt{\frac{a x^{4}}{b} + 1}} - \frac{\sqrt{b}}{2 x^{2} \sqrt{\frac{a x^{4}}{b} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09501, size = 49, normalized size = 1.14 \begin{align*} -\frac{a \arctan \left (\frac{\sqrt{a + \frac{b}{x^{4}}}}{\sqrt{-a}}\right )}{2 \, \sqrt{-a}} - \frac{1}{2} \, \sqrt{a + \frac{b}{x^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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