Optimal. Leaf size=17 \[ -\frac{F\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0118359, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {1106, 1095, 419} \[ -\frac{F\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1106
Rule 1095
Rule 419
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{(2-x) x \left (4-2 x+x^2\right )}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{3-2 x^2-x^4}} \, dx,x,-1+x\right )\\ &=2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2-2 x^2} \sqrt{6+2 x^2}} \, dx,x,-1+x\right )\\ &=-\frac{F\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{\sqrt{3}}\\ \end{align*}
Mathematica [C] time = 0.262311, size = 100, normalized size = 5.88 \[ -\frac{\sqrt [3]{-1} (x-2)^2 \sqrt{\frac{x \left (x+i \sqrt{3}-1\right )}{(x-2)^2}} \sqrt{\frac{-\sqrt [3]{-1} x+x-2}{x-2}} F\left (\sin ^{-1}\left (\sqrt{-\frac{(-1)^{2/3} x}{x-2}}\right )|(-1)^{2/3}\right )}{\sqrt{-x \left (x^3-4 x^2+8 x-8\right )}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.025, size = 200, normalized size = 11.8 \begin{align*} 2\,{\frac{ \left ( -i\sqrt{3}-1 \right ) \left ( -2+x \right ) ^{2}}{ \left ( i\sqrt{3}-1 \right ) \sqrt{-x \left ( -2+x \right ) \left ( x-1+i\sqrt{3} \right ) \left ( x-1-i\sqrt{3} \right ) }}\sqrt{{\frac{ \left ( i\sqrt{3}-1 \right ) x}{ \left ( 1+i\sqrt{3} \right ) \left ( -2+x \right ) }}}\sqrt{{\frac{x-1+i\sqrt{3}}{ \left ( 1-i\sqrt{3} \right ) \left ( -2+x \right ) }}}\sqrt{{\frac{x-1-i\sqrt{3}}{ \left ( 1+i\sqrt{3} \right ) \left ( -2+x \right ) }}}{\it EllipticF} \left ( \sqrt{{\frac{ \left ( i\sqrt{3}-1 \right ) x}{ \left ( 1+i\sqrt{3} \right ) \left ( -2+x \right ) }}},\sqrt{{\frac{ \left ( 1+i\sqrt{3} \right ) \left ( -i\sqrt{3}-1 \right ) }{ \left ( i\sqrt{3}-1 \right ) \left ( 1-i\sqrt{3} \right ) }}} \right ) } \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{1}{\sqrt{-{\left (x^{2} - 2 \, x + 4\right )}{\left (x - 2\right )} x}}\,{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{-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x}}{x^{4} - 4 \, x^{3} + 8 \, x^{2} - 8 \, x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x \left (2 - x\right ) \left (x^{2} - 2 x + 4\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-{\left (x^{2} - 2 \, x + 4\right )}{\left (x - 2\right )} x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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