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 = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1120, 1109,
430} \begin {gather*} -\frac {F\left (\text {ArcSin}(1-x)\left |-\frac {1}{3}\right .\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 1109
Rule 1120
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {8 x-8 x^2+4 x^3-x^4}} \, dx &=\text {Subst}\left (\int \frac {1}{\sqrt {3-2 x^2-x^4}} \, dx,x,-1+x\right )\\ &=2 \text {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] Result contains complex when optimal does not.
time = 30.17, size = 156, normalized size = 9.18 \begin {gather*} \frac {\sqrt {-i+\sqrt {3}+\frac {4 i}{x}} \sqrt {-\frac {i (-2+x)}{\left (-i+\sqrt {3}\right ) x}} x \left (-4+x-i \sqrt {3} x\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {i+\sqrt {3}-\frac {4 i}{x}}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{-i+\sqrt {3}}\right )}{\sqrt {2} \sqrt {i+\sqrt {3}-\frac {4 i}{x}} \sqrt {-x \left (-8+8 x-4 x^2+x^3\right )}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 199 vs. \(2 (15 ) = 30\).
time = 0.52, size = 200, normalized size = 11.76
method | result | size |
default | \(\frac {2 \left (-1-i \sqrt {3}\right ) \sqrt {\frac {\left (-1+i \sqrt {3}\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \left (x -2\right )^{2} \sqrt {\frac {x -1+i \sqrt {3}}{\left (1-i \sqrt {3}\right ) \left (x -2\right )}}\, \sqrt {\frac {x -1-i \sqrt {3}}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \EllipticF \left (\sqrt {\frac {\left (-1+i \sqrt {3}\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}, \sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-1-i \sqrt {3}\right )}{\left (-1+i \sqrt {3}\right ) \left (1-i \sqrt {3}\right )}}\right )}{\left (-1+i \sqrt {3}\right ) \sqrt {-x \left (x -2\right ) \left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right )}}\) | \(200\) |
elliptic | \(\frac {2 \left (-1-i \sqrt {3}\right ) \sqrt {\frac {\left (-1+i \sqrt {3}\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \left (x -2\right )^{2} \sqrt {\frac {x -1+i \sqrt {3}}{\left (1-i \sqrt {3}\right ) \left (x -2\right )}}\, \sqrt {\frac {x -1-i \sqrt {3}}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \EllipticF \left (\sqrt {\frac {\left (-1+i \sqrt {3}\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}, \sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-1-i \sqrt {3}\right )}{\left (-1+i \sqrt {3}\right ) \left (1-i \sqrt {3}\right )}}\right )}{\left (-1+i \sqrt {3}\right ) \sqrt {-x \left (x -2\right ) \left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right )}}\) | \(200\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.09, size = 16, normalized size = 0.94 \begin {gather*} -\frac {1}{2} \, \sqrt {2} {\rm weierstrassPInverse}\left (-\frac {2}{3}, \frac {7}{54}, -\frac {x - 3}{3 \, x}\right ) \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {- x^{4} + 4 x^{3} - 8 x^{2} + 8 x}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {-x^4+4\,x^3-8\,x^2+8\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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