Optimal. Leaf size=32 \[ \sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {-x+x^4}}{1+x+x^2}\right ) \]
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Rubi [F]
time = 1.69, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {-1+2 x+2 x^2}{\left (1-x+3 x^2\right ) \sqrt {-x+x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-1+2 x+2 x^2}{\left (1-x+3 x^2\right ) \sqrt {-x+x^4}} \, dx &=\frac {\left (\sqrt {x} \sqrt {-1+x^3}\right ) \int \frac {-1+2 x+2 x^2}{\sqrt {x} \left (1-x+3 x^2\right ) \sqrt {-1+x^3}} \, dx}{\sqrt {-x+x^4}}\\ &=\frac {\left (\sqrt {x} \sqrt {-1+x^3}\right ) \int \left (\frac {2}{3 \sqrt {x} \sqrt {-1+x^3}}-\frac {5-8 x}{3 \sqrt {x} \left (1-x+3 x^2\right ) \sqrt {-1+x^3}}\right ) \, dx}{\sqrt {-x+x^4}}\\ &=-\frac {\left (\sqrt {x} \sqrt {-1+x^3}\right ) \int \frac {5-8 x}{\sqrt {x} \left (1-x+3 x^2\right ) \sqrt {-1+x^3}} \, dx}{3 \sqrt {-x+x^4}}+\frac {\left (2 \sqrt {x} \sqrt {-1+x^3}\right ) \int \frac {1}{\sqrt {x} \sqrt {-1+x^3}} \, dx}{3 \sqrt {-x+x^4}}\\ &=-\frac {\left (\sqrt {x} \sqrt {-1+x^3}\right ) \int \left (\frac {-8-2 i \sqrt {11}}{\sqrt {x} \left (-1-i \sqrt {11}+6 x\right ) \sqrt {-1+x^3}}+\frac {-8+2 i \sqrt {11}}{\sqrt {x} \left (-1+i \sqrt {11}+6 x\right ) \sqrt {-1+x^3}}\right ) \, dx}{3 \sqrt {-x+x^4}}+\frac {\left (4 \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-x+x^4}}\\ &=\frac {2 (1-x) x \sqrt {\frac {1+x+x^2}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} F\left (\cos ^{-1}\left (\frac {1-\left (1-\sqrt {3}\right ) x}{1-\left (1+\sqrt {3}\right ) x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{3 \sqrt [4]{3} \sqrt {-\frac {(1-x) x}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {-x+x^4}}+\frac {\left (2 \left (4-i \sqrt {11}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \int \frac {1}{\sqrt {x} \left (-1+i \sqrt {11}+6 x\right ) \sqrt {-1+x^3}} \, dx}{3 \sqrt {-x+x^4}}+\frac {\left (2 \left (4+i \sqrt {11}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \int \frac {1}{\sqrt {x} \left (-1-i \sqrt {11}+6 x\right ) \sqrt {-1+x^3}} \, dx}{3 \sqrt {-x+x^4}}\\ &=\frac {2 (1-x) x \sqrt {\frac {1+x+x^2}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} F\left (\cos ^{-1}\left (\frac {1-\left (1-\sqrt {3}\right ) x}{1-\left (1+\sqrt {3}\right ) x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{3 \sqrt [4]{3} \sqrt {-\frac {(1-x) x}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {-x+x^4}}+\frac {\left (4 \left (4-i \sqrt {11}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+i \sqrt {11}+6 x^2\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-x+x^4}}+\frac {\left (4 \left (4+i \sqrt {11}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (-1-i \sqrt {11}+6 x^2\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-x+x^4}}\\ &=\frac {2 (1-x) x \sqrt {\frac {1+x+x^2}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} F\left (\cos ^{-1}\left (\frac {1-\left (1-\sqrt {3}\right ) x}{1-\left (1+\sqrt {3}\right ) x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{3 \sqrt [4]{3} \sqrt {-\frac {(1-x) x}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {-x+x^4}}+\frac {\left (4 \left (4-i \sqrt {11}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \left (\frac {\sqrt {1-i \sqrt {11}}}{2 \left (-1+i \sqrt {11}\right ) \left (\sqrt {1-i \sqrt {11}}-\sqrt {6} x\right ) \sqrt {-1+x^6}}+\frac {\sqrt {1-i \sqrt {11}}}{2 \left (-1+i \sqrt {11}\right ) \left (\sqrt {1-i \sqrt {11}}+\sqrt {6} x\right ) \sqrt {-1+x^6}}\right ) \, dx,x,\sqrt {x}\right )}{3 \sqrt {-x+x^4}}+\frac {\left (4 \left (4+i \sqrt {11}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \left (\frac {\sqrt {1+i \sqrt {11}}}{2 \left (-1-i \sqrt {11}\right ) \left (\sqrt {1+i \sqrt {11}}-\sqrt {6} x\right ) \sqrt {-1+x^6}}+\frac {\sqrt {1+i \sqrt {11}}}{2 \left (-1-i \sqrt {11}\right ) \left (\sqrt {1+i \sqrt {11}}+\sqrt {6} x\right ) \sqrt {-1+x^6}}\right ) \, dx,x,\sqrt {x}\right )}{3 \sqrt {-x+x^4}}\\ &=\frac {2 (1-x) x \sqrt {\frac {1+x+x^2}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} F\left (\cos ^{-1}\left (\frac {1-\left (1-\sqrt {3}\right ) x}{1-\left (1+\sqrt {3}\right ) x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{3 \sqrt [4]{3} \sqrt {-\frac {(1-x) x}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {-x+x^4}}-\frac {\left (2 \left (4-i \sqrt {11}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {1-i \sqrt {11}}-\sqrt {6} x\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {1-i \sqrt {11}} \sqrt {-x+x^4}}-\frac {\left (2 \left (4-i \sqrt {11}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {1-i \sqrt {11}}+\sqrt {6} x\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {1-i \sqrt {11}} \sqrt {-x+x^4}}-\frac {\left (2 \left (4+i \sqrt {11}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {1+i \sqrt {11}}-\sqrt {6} x\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {1+i \sqrt {11}} \sqrt {-x+x^4}}-\frac {\left (2 \left (4+i \sqrt {11}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {1+i \sqrt {11}}+\sqrt {6} x\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {1+i \sqrt {11}} \sqrt {-x+x^4}}\\ \end {align*}
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Mathematica [A]
time = 20.01, size = 56, normalized size = 1.75 \begin {gather*} -\frac {\sqrt {2} \sqrt {-1+\frac {1}{x^3}} x^2 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {-1+\frac {1}{x^3}}}{1+\frac {1}{x^2}+\frac {1}{x}}\right )}{\sqrt {x \left (-1+x^3\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 2.49, size = 828, normalized size = 25.88
method | result | size |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) x^{2}-3 \RootOf \left (\textit {\_Z}^{2}+2\right ) x -\RootOf \left (\textit {\_Z}^{2}+2\right )+4 \sqrt {x^{4}-x}}{3 x^{2}-x +1}\right )}{2}\) | \(62\) |
default | \(\text {Expression too large to display}\) | \(828\) |
elliptic | \(\text {Expression too large to display}\) | \(828\) |
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 = 0.40, size = 28, normalized size = 0.88 \begin {gather*} \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x^{2} - 3 \, x - 1\right )}}{4 \, \sqrt {x^{4} - 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 {2 x^{2} + 2 x - 1}{\sqrt {x \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (3 x^{2} - x + 1\right )}\, 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 = -0.03 \begin {gather*} \int \frac {2\,x^2+2\,x-1}{\sqrt {x^4-x}\,\left (3\,x^2-x+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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