Optimal. Leaf size=62 \[ \frac {1}{3} \sqrt {3-2 (-1+x)^2-(-1+x)^4} (-1+x)+\frac {2 E\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )}{\sqrt {3}}-\frac {4 F\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )}{\sqrt {3}} \]
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Rubi [A]
time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {1120, 1105,
1194, 538, 435, 430} \begin {gather*} -\frac {4 F\left (\text {ArcSin}(1-x)\left |-\frac {1}{3}\right .\right )}{\sqrt {3}}+\frac {2 E\left (\text {ArcSin}(1-x)\left |-\frac {1}{3}\right .\right )}{\sqrt {3}}+\frac {1}{3} \sqrt {-(x-1)^4-2 (x-1)^2+3} (x-1) \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 435
Rule 538
Rule 1105
Rule 1120
Rule 1194
Rubi steps
\begin {align*} \int \sqrt {8 x-8 x^2+4 x^3-x^4} \, dx &=\text {Subst}\left (\int \sqrt {3-2 x^2-x^4} \, dx,x,-1+x\right )\\ &=\frac {1}{3} \sqrt {3-2 (-1+x)^2-(-1+x)^4} (-1+x)+\frac {1}{3} \text {Subst}\left (\int \frac {6-2 x^2}{\sqrt {3-2 x^2-x^4}} \, dx,x,-1+x\right )\\ &=\frac {1}{3} \sqrt {3-2 (-1+x)^2-(-1+x)^4} (-1+x)+\frac {2}{3} \text {Subst}\left (\int \frac {6-2 x^2}{\sqrt {2-2 x^2} \sqrt {6+2 x^2}} \, dx,x,-1+x\right )\\ &=\frac {1}{3} \sqrt {3-2 (-1+x)^2-(-1+x)^4} (-1+x)-\frac {2}{3} \text {Subst}\left (\int \frac {\sqrt {6+2 x^2}}{\sqrt {2-2 x^2}} \, dx,x,-1+x\right )+8 \text {Subst}\left (\int \frac {1}{\sqrt {2-2 x^2} \sqrt {6+2 x^2}} \, dx,x,-1+x\right )\\ &=\frac {1}{3} \sqrt {3-2 (-1+x)^2-(-1+x)^4} (-1+x)+\frac {2 E\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )}{\sqrt {3}}-\frac {4 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 = 20.41, size = 256, normalized size = 4.13 \begin {gather*} -\frac {-16+24 x-24 x^2+14 x^3-5 x^4+x^5-\frac {2 i \sqrt {2} (-2+x) x \sqrt {\frac {4-2 x+x^2}{x^2}} E\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 {-\frac {i (-2+x)}{\left (-i+\sqrt {3}\right ) x}}}+8 i \sqrt {2} \sqrt {-\frac {i (-2+x)}{\left (-i+\sqrt {3}\right ) x}} x^2 \sqrt {\frac {4-2 x+x^2}{x^2}} 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 )}{3 \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. 945 vs. \(2 (54 ) = 108\).
time = 0.47, size = 946, normalized size = 15.26 Too large to display
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 [A]
time = 0.09, size = 36, normalized size = 0.58 \begin {gather*} \frac {\sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x} {\left (x^{2} - 2 \, x + 3\right )}}{3 \, {\left (x - 1\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 \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 = -0.02 \begin {gather*} \int \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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