Optimal. Leaf size=216 \[ -\frac{\log \left (\frac{\sqrt{e} x}{\sqrt{1-x^2}}-\frac{\sqrt{2} \sqrt{e x}}{\sqrt [4]{1-x^2}}+\sqrt{e}\right )}{2 \sqrt{2} \sqrt{e}}+\frac{\log \left (\frac{\sqrt{e} x}{\sqrt{1-x^2}}+\frac{\sqrt{2} \sqrt{e x}}{\sqrt [4]{1-x^2}}+\sqrt{e}\right )}{2 \sqrt{2} \sqrt{e}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{1-x^2}}\right )}{\sqrt{2} \sqrt{e}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{1-x^2}}+1\right )}{\sqrt{2} \sqrt{e}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.145457, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {125, 329, 240, 211, 1165, 628, 1162, 617, 204} \[ -\frac{\log \left (\frac{\sqrt{e} x}{\sqrt{1-x^2}}-\frac{\sqrt{2} \sqrt{e x}}{\sqrt [4]{1-x^2}}+\sqrt{e}\right )}{2 \sqrt{2} \sqrt{e}}+\frac{\log \left (\frac{\sqrt{e} x}{\sqrt{1-x^2}}+\frac{\sqrt{2} \sqrt{e x}}{\sqrt [4]{1-x^2}}+\sqrt{e}\right )}{2 \sqrt{2} \sqrt{e}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{1-x^2}}\right )}{\sqrt{2} \sqrt{e}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{1-x^2}}+1\right )}{\sqrt{2} \sqrt{e}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 125
Rule 329
Rule 240
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [4]{1-x} \sqrt{e x} \sqrt [4]{1+x}} \, dx &=\int \frac{1}{\sqrt{e x} \sqrt [4]{1-x^2}} \, dx\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{1-\frac{x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{e}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{1+\frac{x^4}{e^2}} \, dx,x,\frac{\sqrt{e x}}{\sqrt [4]{1-x^2}}\right )}{e}\\ &=\frac{\operatorname{Subst}\left (\int \frac{e-x^2}{1+\frac{x^4}{e^2}} \, dx,x,\frac{\sqrt{e x}}{\sqrt [4]{1-x^2}}\right )}{e^2}+\frac{\operatorname{Subst}\left (\int \frac{e+x^2}{1+\frac{x^4}{e^2}} \, dx,x,\frac{\sqrt{e x}}{\sqrt [4]{1-x^2}}\right )}{e^2}\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{e-\sqrt{2} \sqrt{e} x+x^2} \, dx,x,\frac{\sqrt{e x}}{\sqrt [4]{1-x^2}}\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{e+\sqrt{2} \sqrt{e} x+x^2} \, dx,x,\frac{\sqrt{e x}}{\sqrt [4]{1-x^2}}\right )-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{e}+2 x}{-e-\sqrt{2} \sqrt{e} x-x^2} \, dx,x,\frac{\sqrt{e x}}{\sqrt [4]{1-x^2}}\right )}{2 \sqrt{2} \sqrt{e}}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{e}-2 x}{-e+\sqrt{2} \sqrt{e} x-x^2} \, dx,x,\frac{\sqrt{e x}}{\sqrt [4]{1-x^2}}\right )}{2 \sqrt{2} \sqrt{e}}\\ &=-\frac{\log \left (\sqrt{e}+\frac{\sqrt{e} x}{\sqrt{1-x^2}}-\frac{\sqrt{2} \sqrt{e x}}{\sqrt [4]{1-x^2}}\right )}{2 \sqrt{2} \sqrt{e}}+\frac{\log \left (\sqrt{e}+\frac{\sqrt{e} x}{\sqrt{1-x^2}}+\frac{\sqrt{2} \sqrt{e x}}{\sqrt [4]{1-x^2}}\right )}{2 \sqrt{2} \sqrt{e}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{1-x^2}}\right )}{\sqrt{2} \sqrt{e}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{1-x^2}}\right )}{\sqrt{2} \sqrt{e}}\\ &=-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{1-x^2}}\right )}{\sqrt{2} \sqrt{e}}+\frac{\tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{1-x^2}}\right )}{\sqrt{2} \sqrt{e}}-\frac{\log \left (\sqrt{e}+\frac{\sqrt{e} x}{\sqrt{1-x^2}}-\frac{\sqrt{2} \sqrt{e x}}{\sqrt [4]{1-x^2}}\right )}{2 \sqrt{2} \sqrt{e}}+\frac{\log \left (\sqrt{e}+\frac{\sqrt{e} x}{\sqrt{1-x^2}}+\frac{\sqrt{2} \sqrt{e x}}{\sqrt [4]{1-x^2}}\right )}{2 \sqrt{2} \sqrt{e}}\\ \end{align*}
Mathematica [A] time = 0.0354087, size = 156, normalized size = 0.72 \[ \frac{\sqrt{x} \left (-\log \left (\frac{x}{\sqrt{1-x^2}}-\frac{\sqrt{2} \sqrt{x}}{\sqrt [4]{1-x^2}}+1\right )+\log \left (\frac{x}{\sqrt{1-x^2}}+\frac{\sqrt{2} \sqrt{x}}{\sqrt [4]{1-x^2}}+1\right )-2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{x}}{\sqrt [4]{1-x^2}}\right )+2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{x}}{\sqrt [4]{1-x^2}}+1\right )\right )}{2 \sqrt{2} \sqrt{e x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt [4]{1-x}}}{\frac{1}{\sqrt{ex}}}{\frac{1}{\sqrt [4]{1+x}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.78797, size = 1278, normalized size = 5.92 \begin{align*} \sqrt{2} \frac{1}{e^{2}}^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2} \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} \frac{1}{e^{2}}^{\frac{3}{4}} - \sqrt{2}{\left (e x^{2} - e\right )} \sqrt{-\frac{\sqrt{2} \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} \frac{1}{e^{2}}^{\frac{1}{4}} + e \sqrt{x + 1} x \sqrt{-x + 1} -{\left (e^{2} x^{2} - e^{2}\right )} \sqrt{\frac{1}{e^{2}}}}{x^{2} - 1}} \frac{1}{e^{2}}^{\frac{3}{4}} - x^{2} + 1}{x^{2} - 1}\right ) + \sqrt{2} \frac{1}{e^{2}}^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2} \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} \frac{1}{e^{2}}^{\frac{3}{4}} - \sqrt{2}{\left (e x^{2} - e\right )} \sqrt{\frac{\sqrt{2} \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} \frac{1}{e^{2}}^{\frac{1}{4}} - e \sqrt{x + 1} x \sqrt{-x + 1} +{\left (e^{2} x^{2} - e^{2}\right )} \sqrt{\frac{1}{e^{2}}}}{x^{2} - 1}} \frac{1}{e^{2}}^{\frac{3}{4}} + x^{2} - 1}{x^{2} - 1}\right ) + \frac{1}{4} \, \sqrt{2} \frac{1}{e^{2}}^{\frac{1}{4}} \log \left (-\frac{\sqrt{2} \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} \frac{1}{e^{2}}^{\frac{1}{4}} + e \sqrt{x + 1} x \sqrt{-x + 1} -{\left (e^{2} x^{2} - e^{2}\right )} \sqrt{\frac{1}{e^{2}}}}{x^{2} - 1}\right ) - \frac{1}{4} \, \sqrt{2} \frac{1}{e^{2}}^{\frac{1}{4}} \log \left (\frac{\sqrt{2} \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} \frac{1}{e^{2}}^{\frac{1}{4}} - e \sqrt{x + 1} x \sqrt{-x + 1} +{\left (e^{2} x^{2} - e^{2}\right )} \sqrt{\frac{1}{e^{2}}}}{x^{2} - 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 22.1076, size = 90, normalized size = 0.42 \begin{align*} - \frac{i{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{3}{8}, \frac{7}{8} & \frac{1}{2}, \frac{3}{4}, 1, 1 \\0, \frac{3}{8}, \frac{1}{2}, \frac{7}{8}, 1, 0 & \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{x^{2}}} \right )} e^{\frac{i \pi }{4}}}{4 \pi \sqrt{e} \Gamma \left (\frac{1}{4}\right )} - \frac{{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{4}, - \frac{1}{8}, \frac{1}{4}, \frac{3}{8}, \frac{3}{4}, 1 & \\- \frac{1}{8}, \frac{3}{8} & - \frac{1}{4}, 0, \frac{1}{4}, 0 \end{matrix} \middle |{\frac{1}{x^{2}}} \right )}}{4 \pi \sqrt{e} \Gamma \left (\frac{1}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]