Optimal. Leaf size=57 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}} \]
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Rubi [A] time = 0.0347115, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {266, 63, 298, 203, 206} \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 63
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt [4]{a-b x^4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x \sqrt [4]{a-b x}} \, dx,x,x^4\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{x^2}{\frac{a}{b}-\frac{x^4}{b}} \, dx,x,\sqrt [4]{a-b x^4}\right )}{b}\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{a}-x^2} \, dx,x,\sqrt [4]{a-b x^4}\right )\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{a}+x^2} \, dx,x,\sqrt [4]{a-b x^4}\right )\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}}\\ \end{align*}
Mathematica [A] time = 0.009278, size = 50, normalized size = 0.88 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )-\tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.024, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}{\frac{1}{\sqrt [4]{-b{x}^{4}+a}}}}\, dx \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 [B] time = 1.58053, size = 248, normalized size = 4.35 \begin{align*} -\frac{\arctan \left (\frac{\sqrt{\sqrt{-b x^{4} + a} + \sqrt{a}}}{a^{\frac{1}{4}}} - \frac{{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{a^{\frac{1}{4}}}\right )}{a^{\frac{1}{4}}} - \frac{\log \left ({\left (-b x^{4} + a\right )}^{\frac{1}{4}} + a^{\frac{1}{4}}\right )}{4 \, a^{\frac{1}{4}}} + \frac{\log \left ({\left (-b x^{4} + a\right )}^{\frac{1}{4}} - a^{\frac{1}{4}}\right )}{4 \, a^{\frac{1}{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.41023, size = 39, normalized size = 0.68 \begin{align*} - \frac{e^{- \frac{i \pi }{4}} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{a}{b x^{4}}} \right )}}{4 \sqrt [4]{b} x \Gamma \left (\frac{5}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15139, size = 259, normalized size = 4.54 \begin{align*} -\frac{\sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{4 \, a} - \frac{\sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{4 \, a} + \frac{\sqrt{2} \left (-a\right )^{\frac{3}{4}} \log \left (\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{-b x^{4} + a} + \sqrt{-a}\right )}{8 \, a} - \frac{\sqrt{2} \left (-a\right )^{\frac{3}{4}} \log \left (-\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{-b x^{4} + a} + \sqrt{-a}\right )}{8 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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