Optimal. Leaf size=179 \[ \frac {1}{2} \tan ^{-1}\left (\frac {(x-1)^2+1}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {\sqrt {\sqrt {a+4}+1} \left (\frac {(x-1)^2}{1-\sqrt {a+4}}+1\right ) F\left (\tan ^{-1}\left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )|-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {a-(x-1)^4-2 (x-1)^2+3}} \]
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Rubi [A] time = 0.15, 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 = {1680, 1673, 1104, 418, 1107, 621, 204} \[ \frac {1}{2} \tan ^{-1}\left (\frac {(x-1)^2+1}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {\sqrt {\sqrt {a+4}+1} \left (\frac {(x-1)^2}{1-\sqrt {a+4}}+1\right ) F\left (\tan ^{-1}\left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )|-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {a-(x-1)^4-2 (x-1)^2+3}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 418
Rule 621
Rule 1104
Rule 1107
Rule 1673
Rule 1680
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {a+8 x-8 x^2+4 x^3-x^4}} \, dx &=\operatorname {Subst}\left (\int \frac {1+x}{\sqrt {3+a-2 x^2-x^4}} \, dx,x,-1+x\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{\sqrt {3+a-2 x^2-x^4}} \, dx,x,-1+x\right )+\operatorname {Subst}\left (\int \frac {x}{\sqrt {3+a-2 x^2-x^4}} \, dx,x,-1+x\right )\\ &=\frac {1}{2} \operatorname {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 ) \operatorname {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}}+\operatorname {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] time = 2.93, size = 813, normalized size = 4.54 \[ \frac {2 \left (-x+\sqrt {-\sqrt {a+4}-1}+1\right ) \sqrt {\frac {\sqrt {-\sqrt {a+4}-1} \left (-x+\sqrt {\sqrt {a+4}-1}+1\right )}{\left (\sqrt {-\sqrt {a+4}-1}+\sqrt {\sqrt {a+4}-1}\right ) \left (-x+\sqrt {-\sqrt {a+4}-1}+1\right )}} \left (x+\sqrt {-\sqrt {a+4}-1}-1\right ) \sqrt {\frac {\sqrt {-\sqrt {a+4}-1} \left (x+\sqrt {\sqrt {a+4}-1}-1\right )}{\left (\sqrt {\sqrt {a+4}-1}-\sqrt {-\sqrt {a+4}-1}\right ) \left (-x+\sqrt {-\sqrt {a+4}-1}+1\right )}} \left (\left (\sqrt {-\sqrt {a+4}-1}+1\right ) F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {-\sqrt {a+4}-1}-\sqrt {\sqrt {a+4}-1}\right ) \left (x+\sqrt {-\sqrt {a+4}-1}-1\right )}{\left (\sqrt {-\sqrt {a+4}-1}+\sqrt {\sqrt {a+4}-1}\right ) \left (-x+\sqrt {-\sqrt {a+4}-1}+1\right )}}\right )|\frac {\left (\sqrt {-\sqrt {a+4}-1}+\sqrt {\sqrt {a+4}-1}\right )^2}{\left (\sqrt {-\sqrt {a+4}-1}-\sqrt {\sqrt {a+4}-1}\right )^2}\right )-2 \sqrt {-\sqrt {a+4}-1} \Pi \left (\frac {\sqrt {-\sqrt {a+4}-1}+\sqrt {\sqrt {a+4}-1}}{\sqrt {\sqrt {a+4}-1}-\sqrt {-\sqrt {a+4}-1}};\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {-\sqrt {a+4}-1}-\sqrt {\sqrt {a+4}-1}\right ) \left (x+\sqrt {-\sqrt {a+4}-1}-1\right )}{\left (\sqrt {-\sqrt {a+4}-1}+\sqrt {\sqrt {a+4}-1}\right ) \left (-x+\sqrt {-\sqrt {a+4}-1}+1\right )}}\right )|\frac {\left (\sqrt {-\sqrt {a+4}-1}+\sqrt {\sqrt {a+4}-1}\right )^2}{\left (\sqrt {-\sqrt {a+4}-1}-\sqrt {\sqrt {a+4}-1}\right )^2}\right )\right )}{\sqrt {-\sqrt {a+4}-1} \sqrt {\frac {\left (\sqrt {-\sqrt {a+4}-1}-\sqrt {\sqrt {a+4}-1}\right ) \left (x+\sqrt {-\sqrt {a+4}-1}-1\right )}{\left (\sqrt {-\sqrt {a+4}-1}+\sqrt {\sqrt {a+4}-1}\right ) \left (-x+\sqrt {-\sqrt {a+4}-1}+1\right )}} \sqrt {a-x \left (x^3-4 x^2+8 x-8\right )}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x} x}{x^{4} - 4 \, x^{3} + 8 \, x^{2} - a - 8 \, x}, x\right ) \]
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 \[ \int \frac {x}{\sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 788, normalized size = 4.40 \[ -\frac {\left (\sqrt {-1-\sqrt {a +4}}+\sqrt {-1+\sqrt {a +4}}\right ) \sqrt {\frac {\left (-\sqrt {-1-\sqrt {a +4}}+\sqrt {-1+\sqrt {a +4}}\right ) \left (x -1-\sqrt {-1+\sqrt {a +4}}\right )}{\left (-\sqrt {-1-\sqrt {a +4}}-\sqrt {-1+\sqrt {a +4}}\right ) \left (x -1+\sqrt {-1+\sqrt {a +4}}\right )}}\, \left (x -1+\sqrt {-1+\sqrt {a +4}}\right )^{2} \sqrt {-\frac {2 \sqrt {-1+\sqrt {a +4}}\, \left (x -1-\sqrt {-1-\sqrt {a +4}}\right )}{\left (\sqrt {-1-\sqrt {a +4}}-\sqrt {-1+\sqrt {a +4}}\right ) \left (x -1+\sqrt {-1+\sqrt {a +4}}\right )}}\, \sqrt {-\frac {2 \sqrt {-1+\sqrt {a +4}}\, \left (x -1+\sqrt {-1-\sqrt {a +4}}\right )}{\left (-\sqrt {-1-\sqrt {a +4}}-\sqrt {-1+\sqrt {a +4}}\right ) \left (x -1+\sqrt {-1+\sqrt {a +4}}\right )}}\, \left (\left (1-\sqrt {-1+\sqrt {a +4}}\right ) \EllipticF \left (\sqrt {\frac {\left (-\sqrt {-1-\sqrt {a +4}}+\sqrt {-1+\sqrt {a +4}}\right ) \left (x -1-\sqrt {-1+\sqrt {a +4}}\right )}{\left (-\sqrt {-1-\sqrt {a +4}}-\sqrt {-1+\sqrt {a +4}}\right ) \left (x -1+\sqrt {-1+\sqrt {a +4}}\right )}}, \sqrt {\frac {\left (-\sqrt {-1-\sqrt {a +4}}-\sqrt {-1+\sqrt {a +4}}\right ) \left (\sqrt {-1-\sqrt {a +4}}+\sqrt {-1+\sqrt {a +4}}\right )}{\left (-\sqrt {-1-\sqrt {a +4}}+\sqrt {-1+\sqrt {a +4}}\right ) \left (\sqrt {-1-\sqrt {a +4}}-\sqrt {-1+\sqrt {a +4}}\right )}}\right )+2 \sqrt {-1+\sqrt {a +4}}\, \EllipticPi \left (\sqrt {\frac {\left (-\sqrt {-1-\sqrt {a +4}}+\sqrt {-1+\sqrt {a +4}}\right ) \left (x -1-\sqrt {-1+\sqrt {a +4}}\right )}{\left (-\sqrt {-1-\sqrt {a +4}}-\sqrt {-1+\sqrt {a +4}}\right ) \left (x -1+\sqrt {-1+\sqrt {a +4}}\right )}}, \frac {-\sqrt {-1-\sqrt {a +4}}-\sqrt {-1+\sqrt {a +4}}}{-\sqrt {-1-\sqrt {a +4}}+\sqrt {-1+\sqrt {a +4}}}, \sqrt {\frac {\left (-\sqrt {-1-\sqrt {a +4}}-\sqrt {-1+\sqrt {a +4}}\right ) \left (\sqrt {-1-\sqrt {a +4}}+\sqrt {-1+\sqrt {a +4}}\right )}{\left (-\sqrt {-1-\sqrt {a +4}}+\sqrt {-1+\sqrt {a +4}}\right ) \left (\sqrt {-1-\sqrt {a +4}}-\sqrt {-1+\sqrt {a +4}}\right )}}\right )\right )}{\left (-\sqrt {-1-\sqrt {a +4}}+\sqrt {-1+\sqrt {a +4}}\right ) \sqrt {-1+\sqrt {a +4}}\, \sqrt {-\left (x -1-\sqrt {-1+\sqrt {a +4}}\right ) \left (x -1+\sqrt {-1+\sqrt {a +4}}\right ) \left (x -1-\sqrt {-1-\sqrt {a +4}}\right ) \left (x -1+\sqrt {-1-\sqrt {a +4}}\right )}} \]
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 \[ \int \frac {x}{\sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x}{\sqrt {-x^4+4\,x^3-8\,x^2+8\,x+a}} \,d x \]
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 \[ \int \frac {x}{\sqrt {a - x^{4} + 4 x^{3} - 8 x^{2} + 8 x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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