Optimal. Leaf size=60 \[ \frac{\sqrt [4]{1-\frac{1}{x^2}} \sqrt{e x} E\left (\left .\frac{1}{2} \csc ^{-1}(x)\right |2\right )}{\sqrt [4]{1-x^2}}-\frac{e \left (1-x^2\right )^{3/4}}{\sqrt{e x}} \]
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Rubi [A] time = 0.0224061, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {125, 315, 317, 335, 228} \[ \frac{\sqrt [4]{1-\frac{1}{x^2}} \sqrt{e x} E\left (\left .\frac{1}{2} \csc ^{-1}(x)\right |2\right )}{\sqrt [4]{1-x^2}}-\frac{e \left (1-x^2\right )^{3/4}}{\sqrt{e x}} \]
Antiderivative was successfully verified.
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Rule 125
Rule 315
Rule 317
Rule 335
Rule 228
Rubi steps
\begin{align*} \int \frac{\sqrt{e x}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx &=\int \frac{\sqrt{e x}}{\sqrt [4]{1-x^2}} \, dx\\ &=-\frac{e \left (1-x^2\right )^{3/4}}{\sqrt{e x}}-\frac{1}{2} e^2 \int \frac{1}{(e x)^{3/2} \sqrt [4]{1-x^2}} \, dx\\ &=-\frac{e \left (1-x^2\right )^{3/4}}{\sqrt{e x}}-\frac{\left (\sqrt [4]{1-\frac{1}{x^2}} \sqrt{e x}\right ) \int \frac{1}{\sqrt [4]{1-\frac{1}{x^2}} x^2} \, dx}{2 \sqrt [4]{1-x^2}}\\ &=-\frac{e \left (1-x^2\right )^{3/4}}{\sqrt{e x}}+\frac{\left (\sqrt [4]{1-\frac{1}{x^2}} \sqrt{e x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{1-x^2}} \, dx,x,\frac{1}{x}\right )}{2 \sqrt [4]{1-x^2}}\\ &=-\frac{e \left (1-x^2\right )^{3/4}}{\sqrt{e x}}+\frac{\sqrt [4]{1-\frac{1}{x^2}} \sqrt{e x} E\left (\left .\frac{1}{2} \csc ^{-1}(x)\right |2\right )}{\sqrt [4]{1-x^2}}\\ \end{align*}
Mathematica [C] time = 0.004759, size = 25, normalized size = 0.42 \[ \frac{2}{3} x \sqrt{e x} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};x^2\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.037, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{ex}{\frac{1}{\sqrt [4]{1-x}}}{\frac{1}{\sqrt [4]{1+x}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e x}}{{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{e x}{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}}}{x^{2} - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 10.0919, size = 105, normalized size = 1.75 \begin{align*} \frac{i \sqrt{e}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{8}, \frac{3}{8} & 0, \frac{1}{4}, \frac{1}{2}, 1 \\- \frac{1}{2}, - \frac{1}{8}, 0, \frac{3}{8}, \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{x^{2}}} \right )} e^{\frac{i \pi }{4}}}{4 \pi \Gamma \left (\frac{1}{4}\right )} - \frac{\sqrt{e}{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{3}{4}, - \frac{5}{8}, - \frac{1}{4}, - \frac{1}{8}, \frac{1}{4}, 1 & \\- \frac{5}{8}, - \frac{1}{8} & - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0 \end{matrix} \middle |{\frac{1}{x^{2}}} \right )}}{4 \pi \Gamma \left (\frac{1}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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