Optimal. Leaf size=388 \[ \frac {\left (1-\sqrt {4+a}\right ) \left (1+\frac {(-1+x)^2}{1-\sqrt {4+a}}\right ) (-1+x)}{\sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}+\tan ^{-1}\left (\frac {1+(-1+x)^2}{\sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}\right )-\frac {\left (1-\sqrt {4+a}\right ) \sqrt {1+\sqrt {4+a}} \left (1+\frac {(-1+x)^2}{1-\sqrt {4+a}}\right ) E\left (\tan ^{-1}\left (\frac {-1+x}{\sqrt {1+\sqrt {4+a}}}\right )|-\frac {2 \sqrt {4+a}}{1-\sqrt {4+a}}\right )}{\sqrt {\frac {1+\frac {(-1+x)^2}{1-\sqrt {4+a}}}{1+\frac {(-1+x)^2}{1+\sqrt {4+a}}}} \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}+\frac {\sqrt {1+\sqrt {4+a}} \left (1+\frac {(-1+x)^2}{1-\sqrt {4+a}}\right ) F\left (\tan ^{-1}\left (\frac {-1+x}{\sqrt {1+\sqrt {4+a}}}\right )|-\frac {2 \sqrt {4+a}}{1-\sqrt {4+a}}\right )}{\sqrt {\frac {1+\frac {(-1+x)^2}{1-\sqrt {4+a}}}{1+\frac {(-1+x)^2}{1+\sqrt {4+a}}}} \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}} \]
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Rubi [A]
time = 0.28, antiderivative size = 388, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 11, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {1694, 1687,
1216, 545, 429, 506, 422, 12, 1121, 635, 210} \begin {gather*} \text {ArcTan}\left (\frac {(x-1)^2+1}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {\sqrt {\sqrt {a+4}+1} \left (\frac {(x-1)^2}{1-\sqrt {a+4}}+1\right ) F\left (\text {ArcTan}\left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )|-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {a-(x-1)^4-2 (x-1)^2+3}}-\frac {\left (1-\sqrt {a+4}\right ) \sqrt {\sqrt {a+4}+1} \left (\frac {(x-1)^2}{1-\sqrt {a+4}}+1\right ) E\left (\text {ArcTan}\left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )|-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {\left (1-\sqrt {a+4}\right ) (x-1) \left (\frac {(x-1)^2}{1-\sqrt {a+4}}+1\right )}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 210
Rule 422
Rule 429
Rule 506
Rule 545
Rule 635
Rule 1121
Rule 1216
Rule 1687
Rule 1694
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {a+8 x-8 x^2+4 x^3-x^4}} \, dx &=\text {Subst}\left (\int \frac {(1+x)^2}{\sqrt {3+a-2 x^2-x^4}} \, dx,x,-1+x\right )\\ &=\text {Subst}\left (\int \frac {2 x}{\sqrt {3+a-2 x^2-x^4}} \, dx,x,-1+x\right )+\text {Subst}\left (\int \frac {1+x^2}{\sqrt {3+a-2 x^2-x^4}} \, dx,x,-1+x\right )\\ &=2 \text {Subst}\left (\int \frac {x}{\sqrt {3+a-2 x^2-x^4}} \, dx,x,-1+x\right )+\frac {\left (\sqrt {1-\frac {2 (-1+x)^2}{-2-2 \sqrt {4+a}}} \sqrt {1-\frac {2 (-1+x)^2}{-2+2 \sqrt {4+a}}}\right ) \text {Subst}\left (\int \frac {1+x^2}{\sqrt {1-\frac {2 x^2}{-2-2 \sqrt {4+a}}} \sqrt {1-\frac {2 x^2}{-2+2 \sqrt {4+a}}}} \, dx,x,-1+x\right )}{\sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}\\ &=\frac {\left (\sqrt {1-\frac {2 (-1+x)^2}{-2-2 \sqrt {4+a}}} \sqrt {1-\frac {2 (-1+x)^2}{-2+2 \sqrt {4+a}}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {2 x^2}{-2-2 \sqrt {4+a}}} \sqrt {1-\frac {2 x^2}{-2+2 \sqrt {4+a}}}} \, dx,x,-1+x\right )}{\sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}+\frac {\left (\sqrt {1-\frac {2 (-1+x)^2}{-2-2 \sqrt {4+a}}} \sqrt {1-\frac {2 (-1+x)^2}{-2+2 \sqrt {4+a}}}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {2 x^2}{-2-2 \sqrt {4+a}}} \sqrt {1-\frac {2 x^2}{-2+2 \sqrt {4+a}}}} \, dx,x,-1+x\right )}{\sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}+\text {Subst}\left (\int \frac {1}{\sqrt {3+a-2 x-x^2}} \, dx,x,(-1+x)^2\right )\\ &=-\frac {\left (1-\sqrt {4+a}\right ) \left (1+\frac {(1-x)^2}{1-\sqrt {4+a}}\right ) (1-x)}{\sqrt {3+a-2 (1-x)^2-(1-x)^4}}-\frac {\sqrt {1+\sqrt {4+a}} \left (1+\frac {(1-x)^2}{1-\sqrt {4+a}}\right ) F\left (\tan ^{-1}\left (\frac {1-x}{\sqrt {1+\sqrt {4+a}}}\right )|-\frac {2 \sqrt {4+a}}{1-\sqrt {4+a}}\right )}{\sqrt {\frac {1+\frac {(1-x)^2}{1-\sqrt {4+a}}}{1+\frac {(1-x)^2}{1+\sqrt {4+a}}}} \sqrt {3+a-2 (1-x)^2-(1-x)^4}}+2 \text {Subst}\left (\int \frac {1}{-4-x^2} \, dx,x,-\frac {2 \left (1+(-1+x)^2\right )}{\sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}\right )-\frac {\left (\left (1-\sqrt {4+a}\right ) \sqrt {1-\frac {2 (-1+x)^2}{-2-2 \sqrt {4+a}}} \sqrt {1-\frac {2 (-1+x)^2}{-2+2 \sqrt {4+a}}}\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {2 x^2}{-2+2 \sqrt {4+a}}}}{\left (1-\frac {2 x^2}{-2-2 \sqrt {4+a}}\right )^{3/2}} \, dx,x,-1+x\right )}{\sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}\\ &=-\frac {\left (1-\sqrt {4+a}\right ) \left (1+\frac {(1-x)^2}{1-\sqrt {4+a}}\right ) (1-x)}{\sqrt {3+a-2 (1-x)^2-(1-x)^4}}+\tan ^{-1}\left (\frac {1+(-1+x)^2}{\sqrt {3+a-2 (1-x)^2-(1-x)^4}}\right )+\frac {\left (1-\sqrt {4+a}\right ) \sqrt {1+\sqrt {4+a}} \left (1+\frac {(1-x)^2}{1-\sqrt {4+a}}\right ) E\left (\tan ^{-1}\left (\frac {1-x}{\sqrt {1+\sqrt {4+a}}}\right )|-\frac {2 \sqrt {4+a}}{1-\sqrt {4+a}}\right )}{\sqrt {\frac {1+\frac {(1-x)^2}{1-\sqrt {4+a}}}{1+\frac {(1-x)^2}{1+\sqrt {4+a}}}} \sqrt {3+a-2 (1-x)^2-(1-x)^4}}-\frac {\sqrt {1+\sqrt {4+a}} \left (1+\frac {(1-x)^2}{1-\sqrt {4+a}}\right ) F\left (\tan ^{-1}\left (\frac {1-x}{\sqrt {1+\sqrt {4+a}}}\right )|-\frac {2 \sqrt {4+a}}{1-\sqrt {4+a}}\right )}{\sqrt {\frac {1+\frac {(1-x)^2}{1-\sqrt {4+a}}}{1+\frac {(1-x)^2}{1+\sqrt {4+a}}}} \sqrt {3+a-2 (1-x)^2-(1-x)^4}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1145\) vs. \(2(388)=776\).
time = 13.98, size = 1145, normalized size = 2.95 \begin {gather*} \frac {\left (-1+\sqrt {-1-\sqrt {4+a}}+x\right ) \left (-1-\sqrt {-1+\sqrt {4+a}}+x\right ) \left (-1+\sqrt {-1+\sqrt {4+a}}+x\right )+\frac {2 \left (\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}\right ) \left (1+\sqrt {-1-\sqrt {4+a}}-x\right )^2 \sqrt {\frac {\sqrt {-1-\sqrt {4+a}} \left (1+\sqrt {-1+\sqrt {4+a}}-x\right )}{\left (\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}\right ) \left (1+\sqrt {-1-\sqrt {4+a}}-x\right )}} \sqrt {\frac {\left (\sqrt {-1-\sqrt {4+a}}-\sqrt {-1+\sqrt {4+a}}\right ) \left (-1+\sqrt {-1-\sqrt {4+a}}+x\right )}{\left (\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}\right ) \left (1+\sqrt {-1-\sqrt {4+a}}-x\right )}} \sqrt {-\frac {\sqrt {-1-\sqrt {4+a}} \left (-1+\sqrt {-1+\sqrt {4+a}}+x\right )}{\left (\sqrt {-1-\sqrt {4+a}}-\sqrt {-1+\sqrt {4+a}}\right ) \left (1+\sqrt {-1-\sqrt {4+a}}-x\right )}} \left (\left (1+\sqrt {-1-\sqrt {4+a}} \sqrt {-1+\sqrt {4+a}}\right ) E\left (\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {-1-\sqrt {4+a}}-\sqrt {-1+\sqrt {4+a}}\right ) \left (-1+\sqrt {-1-\sqrt {4+a}}+x\right )}{\left (\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}\right ) \left (1+\sqrt {-1-\sqrt {4+a}}-x\right )}}\right )|\frac {\left (\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}\right )^2}{\left (\sqrt {-1-\sqrt {4+a}}-\sqrt {-1+\sqrt {4+a}}\right )^2}\right )-\left (1+2 \sqrt {-1-\sqrt {4+a}}+\sqrt {-1-\sqrt {4+a}} \sqrt {-1+\sqrt {4+a}}\right ) F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {-1-\sqrt {4+a}}-\sqrt {-1+\sqrt {4+a}}\right ) \left (-1+\sqrt {-1-\sqrt {4+a}}+x\right )}{\left (\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}\right ) \left (1+\sqrt {-1-\sqrt {4+a}}-x\right )}}\right )|\frac {\left (\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}\right )^2}{\left (\sqrt {-1-\sqrt {4+a}}-\sqrt {-1+\sqrt {4+a}}\right )^2}\right )+4 \sqrt {-1-\sqrt {4+a}} \Pi \left (\frac {\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}}{-\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}};\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {-1-\sqrt {4+a}}-\sqrt {-1+\sqrt {4+a}}\right ) \left (-1+\sqrt {-1-\sqrt {4+a}}+x\right )}{\left (\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}\right ) \left (1+\sqrt {-1-\sqrt {4+a}}-x\right )}}\right )|\frac {\left (\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}\right )^2}{\left (\sqrt {-1-\sqrt {4+a}}-\sqrt {-1+\sqrt {4+a}}\right )^2}\right )\right )}{1+\sqrt {4+a}+\sqrt {-1-\sqrt {4+a}} \sqrt {-1+\sqrt {4+a}}}}{\sqrt {a-x \left (-8+8 x-4 x^2+x^3\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1146\) vs.
\(2(454)=908\).
time = 0.04, size = 1147, normalized size = 2.96 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 [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\sqrt {a - 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 = -0.00 \begin {gather*} \int \frac {x^2}{\sqrt {-x^4+4\,x^3-8\,x^2+8\,x+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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