Optimal. Leaf size=179 \[ \frac {1}{2} \tan ^{-1}\left (\frac {1+(-1+x)^2}{\sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}\right )+\frac {\sqrt {1+\sqrt {4+a}} \left (1+\frac {(-1+x)^2}{1-\sqrt {4+a}}\right ) F\left (\tan ^{-1}\left (\frac {-1+x}{\sqrt {1+\sqrt {4+a}}}\right )|-\frac {2 \sqrt {4+a}}{1-\sqrt {4+a}}\right )}{\sqrt {\frac {1+\frac {(-1+x)^2}{1-\sqrt {4+a}}}{1+\frac {(-1+x)^2}{1+\sqrt {4+a}}}} \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}} \]
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Rubi [A]
time = 0.11, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1694, 1687,
1118, 429, 1121, 635, 210} \begin {gather*} \frac {1}{2} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 429
Rule 635
Rule 1118
Rule 1121
Rule 1687
Rule 1694
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {a+8 x-8 x^2+4 x^3-x^4}} \, dx &=\text {Subst}\left (\int \frac {1+x}{\sqrt {3+a-2 x^2-x^4}} \, dx,x,-1+x\right )\\ &=\text {Subst}\left (\int \frac {1}{\sqrt {3+a-2 x^2-x^4}} \, dx,x,-1+x\right )+\text {Subst}\left (\int \frac {x}{\sqrt {3+a-2 x^2-x^4}} \, dx,x,-1+x\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {3+a-2 x-x^2}} \, dx,x,(-1+x)^2\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}{\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 {\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}}+\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 {1}{2} \tan ^{-1}\left (\frac {1+(-1+x)^2}{\sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}\right )-\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. \(813\) vs. \(2(179)=358\).
time = 11.79, size = 813, normalized size = 4.54 \begin {gather*} \frac {2 \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 (-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}}\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 )-2 \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 )}{\sqrt {-1-\sqrt {4+a}} \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 {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. \(787\) vs.
\(2(213)=426\).
time = 0.04, size = 788, normalized size = 4.40
method | result | size |
default | \(-\frac {\left (\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}\right ) \sqrt {\frac {\left (-\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}\right ) \left (x -1-\sqrt {-1+\sqrt {4+a}}\right )}{\left (-\sqrt {-1-\sqrt {4+a}}-\sqrt {-1+\sqrt {4+a}}\right ) \left (x -1+\sqrt {-1+\sqrt {4+a}}\right )}}\, \left (x -1+\sqrt {-1+\sqrt {4+a}}\right )^{2} \sqrt {-\frac {2 \sqrt {-1+\sqrt {4+a}}\, \left (x -1-\sqrt {-1-\sqrt {4+a}}\right )}{\left (\sqrt {-1-\sqrt {4+a}}-\sqrt {-1+\sqrt {4+a}}\right ) \left (x -1+\sqrt {-1+\sqrt {4+a}}\right )}}\, \sqrt {-\frac {2 \sqrt {-1+\sqrt {4+a}}\, \left (x -1+\sqrt {-1-\sqrt {4+a}}\right )}{\left (-\sqrt {-1-\sqrt {4+a}}-\sqrt {-1+\sqrt {4+a}}\right ) \left (x -1+\sqrt {-1+\sqrt {4+a}}\right )}}\, \left (\left (1-\sqrt {-1+\sqrt {4+a}}\right ) \EllipticF \left (\sqrt {\frac {\left (-\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}\right ) \left (x -1-\sqrt {-1+\sqrt {4+a}}\right )}{\left (-\sqrt {-1-\sqrt {4+a}}-\sqrt {-1+\sqrt {4+a}}\right ) \left (x -1+\sqrt {-1+\sqrt {4+a}}\right )}}, \sqrt {\frac {\left (-\sqrt {-1-\sqrt {4+a}}-\sqrt {-1+\sqrt {4+a}}\right ) \left (\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}\right )}{\left (-\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}\right ) \left (\sqrt {-1-\sqrt {4+a}}-\sqrt {-1+\sqrt {4+a}}\right )}}\right )+2 \sqrt {-1+\sqrt {4+a}}\, \EllipticPi \left (\sqrt {\frac {\left (-\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}\right ) \left (x -1-\sqrt {-1+\sqrt {4+a}}\right )}{\left (-\sqrt {-1-\sqrt {4+a}}-\sqrt {-1+\sqrt {4+a}}\right ) \left (x -1+\sqrt {-1+\sqrt {4+a}}\right )}}, \frac {-\sqrt {-1-\sqrt {4+a}}-\sqrt {-1+\sqrt {4+a}}}{-\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}}, \sqrt {\frac {\left (-\sqrt {-1-\sqrt {4+a}}-\sqrt {-1+\sqrt {4+a}}\right ) \left (\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}\right )}{\left (-\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}\right ) \left (\sqrt {-1-\sqrt {4+a}}-\sqrt {-1+\sqrt {4+a}}\right )}}\right )\right )}{\left (-\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}\right ) \sqrt {-1+\sqrt {4+a}}\, \sqrt {-\left (x -1-\sqrt {-1+\sqrt {4+a}}\right ) \left (x -1+\sqrt {-1+\sqrt {4+a}}\right ) \left (x -1-\sqrt {-1-\sqrt {4+a}}\right ) \left (x -1+\sqrt {-1-\sqrt {4+a}}\right )}}\) | \(788\) |
elliptic | \(-\frac {\left (\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}\right ) \sqrt {\frac {\left (-\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}\right ) \left (x -1-\sqrt {-1+\sqrt {4+a}}\right )}{\left (-\sqrt {-1-\sqrt {4+a}}-\sqrt {-1+\sqrt {4+a}}\right ) \left (x -1+\sqrt {-1+\sqrt {4+a}}\right )}}\, \left (x -1+\sqrt {-1+\sqrt {4+a}}\right )^{2} \sqrt {-\frac {2 \sqrt {-1+\sqrt {4+a}}\, \left (x -1-\sqrt {-1-\sqrt {4+a}}\right )}{\left (\sqrt {-1-\sqrt {4+a}}-\sqrt {-1+\sqrt {4+a}}\right ) \left (x -1+\sqrt {-1+\sqrt {4+a}}\right )}}\, \sqrt {-\frac {2 \sqrt {-1+\sqrt {4+a}}\, \left (x -1+\sqrt {-1-\sqrt {4+a}}\right )}{\left (-\sqrt {-1-\sqrt {4+a}}-\sqrt {-1+\sqrt {4+a}}\right ) \left (x -1+\sqrt {-1+\sqrt {4+a}}\right )}}\, \left (\left (1-\sqrt {-1+\sqrt {4+a}}\right ) \EllipticF \left (\sqrt {\frac {\left (-\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}\right ) \left (x -1-\sqrt {-1+\sqrt {4+a}}\right )}{\left (-\sqrt {-1-\sqrt {4+a}}-\sqrt {-1+\sqrt {4+a}}\right ) \left (x -1+\sqrt {-1+\sqrt {4+a}}\right )}}, \sqrt {\frac {\left (-\sqrt {-1-\sqrt {4+a}}-\sqrt {-1+\sqrt {4+a}}\right ) \left (\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}\right )}{\left (-\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}\right ) \left (\sqrt {-1-\sqrt {4+a}}-\sqrt {-1+\sqrt {4+a}}\right )}}\right )+2 \sqrt {-1+\sqrt {4+a}}\, \EllipticPi \left (\sqrt {\frac {\left (-\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}\right ) \left (x -1-\sqrt {-1+\sqrt {4+a}}\right )}{\left (-\sqrt {-1-\sqrt {4+a}}-\sqrt {-1+\sqrt {4+a}}\right ) \left (x -1+\sqrt {-1+\sqrt {4+a}}\right )}}, \frac {-\sqrt {-1-\sqrt {4+a}}-\sqrt {-1+\sqrt {4+a}}}{-\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}}, \sqrt {\frac {\left (-\sqrt {-1-\sqrt {4+a}}-\sqrt {-1+\sqrt {4+a}}\right ) \left (\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}\right )}{\left (-\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}\right ) \left (\sqrt {-1-\sqrt {4+a}}-\sqrt {-1+\sqrt {4+a}}\right )}}\right )\right )}{\left (-\sqrt {-1-\sqrt {4+a}}+\sqrt {-1+\sqrt {4+a}}\right ) \sqrt {-1+\sqrt {4+a}}\, \sqrt {-\left (x -1-\sqrt {-1+\sqrt {4+a}}\right ) \left (x -1+\sqrt {-1+\sqrt {4+a}}\right ) \left (x -1-\sqrt {-1-\sqrt {4+a}}\right ) \left (x -1+\sqrt {-1-\sqrt {4+a}}\right )}}\) | \(788\) |
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}{\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 = -0.01 \begin {gather*} \int \frac {x}{\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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