Optimal. Leaf size=17 \[ -\frac {F\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 419
Rule 1095
Rule 1106
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*}
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Mathematica [C] time = 0.27, 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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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x}}{x^{4} - 4 \, x^{3} + 8 \, x^{2} - 8 \, x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-{\left (x^{2} - 2 \, x + 4\right )} {\left (x - 2\right )} x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 200, normalized size = 11.76 \[ \frac {2 \left (-1-i \sqrt {3}\right ) \sqrt {\frac {\left (i \sqrt {3}-1\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \left (x -2\right )^{2} \sqrt {\frac {x -1+i \sqrt {3}}{\left (1-i \sqrt {3}\right ) \left (x -2\right )}}\, \sqrt {\frac {x -1-i \sqrt {3}}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \EllipticF \left (\sqrt {\frac {\left (i \sqrt {3}-1\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}, \sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-1-i \sqrt {3}\right )}{\left (i \sqrt {3}-1\right ) \left (1-i \sqrt {3}\right )}}\right )}{\left (i \sqrt {3}-1\right ) \sqrt {-\left (x -2\right ) \left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right ) x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-{\left (x^{2} - 2 \, x + 4\right )} {\left (x - 2\right )} x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {1}{\sqrt {-x\,\left (x-2\right )\,\left (x^2-2\,x+4\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x \left (2 - x\right ) \left (x^{2} - 2 x + 4\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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