Optimal. Leaf size=30 \[ -\log \left (-2 x^2+2 \sqrt {x^4-2 x^3+x+1}+2 x+1\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 23, normalized size of antiderivative = 0.77, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1680, 12, 1107, 619, 215} \begin {gather*} -\sinh ^{-1}\left (\frac {3-4 \left (x-\frac {1}{2}\right )^2}{2 \sqrt {3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 215
Rule 619
Rule 1107
Rule 1680
Rubi steps
\begin {align*} \int \frac {-1+2 x}{\sqrt {1+x-2 x^3+x^4}} \, dx &=\operatorname {Subst}\left (\int \frac {8 x}{\sqrt {21-24 x^2+16 x^4}} \, dx,x,-\frac {1}{2}+x\right )\\ &=8 \operatorname {Subst}\left (\int \frac {x}{\sqrt {21-24 x^2+16 x^4}} \, dx,x,-\frac {1}{2}+x\right )\\ &=4 \operatorname {Subst}\left (\int \frac {1}{\sqrt {21-24 x+16 x^2}} \, dx,x,\left (-\frac {1}{2}+x\right )^2\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{768}}} \, dx,x,8 \left (-3+4 \left (-\frac {1}{2}+x\right )^2\right )\right )}{16 \sqrt {3}}\\ &=-\sinh ^{-1}\left (\frac {3-(-1+2 x)^2}{2 \sqrt {3}}\right )\\ \end {align*}
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Mathematica [C] time = 3.37, size = 717, normalized size = 23.90 \begin {gather*} \frac {\left (-2 x+\sqrt {1+4 \sqrt [3]{-1}}+1\right ) \sqrt {\frac {\sqrt {1+4 \sqrt [3]{-1}} \left (-2 x+\sqrt {1-4 (-1)^{2/3}}+1\right )}{\left (\sqrt {1+4 \sqrt [3]{-1}}+\sqrt {1-4 (-1)^{2/3}}\right ) \left (-2 x+\sqrt {1+4 \sqrt [3]{-1}}+1\right )}} \left (2 x+\sqrt {1+4 \sqrt [3]{-1}}-1\right ) \sqrt {-\frac {\sqrt {1+4 \sqrt [3]{-1}} \left (2 x+\sqrt {1-4 (-1)^{2/3}}-1\right )}{\left (\sqrt {1+4 \sqrt [3]{-1}}-\sqrt {1-4 (-1)^{2/3}}\right ) \left (-2 x+\sqrt {1+4 \sqrt [3]{-1}}+1\right )}} \left (F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {1+4 \sqrt [3]{-1}}-\sqrt {1-4 (-1)^{2/3}}\right ) \left (2 x+\sqrt {1+4 \sqrt [3]{-1}}-1\right )}{\left (\sqrt {1+4 \sqrt [3]{-1}}+\sqrt {1-4 (-1)^{2/3}}\right ) \left (-2 x+\sqrt {1+4 \sqrt [3]{-1}}+1\right )}}\right )|\frac {\left (\sqrt {1+4 \sqrt [3]{-1}}+\sqrt {1-4 (-1)^{2/3}}\right )^2}{\left (\sqrt {1+4 \sqrt [3]{-1}}-\sqrt {1-4 (-1)^{2/3}}\right )^2}\right )-2 \Pi \left (-\frac {\sqrt {1+4 \sqrt [3]{-1}}+\sqrt {1-4 (-1)^{2/3}}}{\sqrt {1+4 \sqrt [3]{-1}}-\sqrt {1-4 (-1)^{2/3}}};\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {1+4 \sqrt [3]{-1}}-\sqrt {1-4 (-1)^{2/3}}\right ) \left (2 x+\sqrt {1+4 \sqrt [3]{-1}}-1\right )}{\left (\sqrt {1+4 \sqrt [3]{-1}}+\sqrt {1-4 (-1)^{2/3}}\right ) \left (-2 x+\sqrt {1+4 \sqrt [3]{-1}}+1\right )}}\right )|\frac {\left (\sqrt {1+4 \sqrt [3]{-1}}+\sqrt {1-4 (-1)^{2/3}}\right )^2}{\left (\sqrt {1+4 \sqrt [3]{-1}}-\sqrt {1-4 (-1)^{2/3}}\right )^2}\right )\right )}{\sqrt {\frac {\left (\sqrt {1+4 \sqrt [3]{-1}}-\sqrt {1-4 (-1)^{2/3}}\right ) \left (2 x+\sqrt {1+4 \sqrt [3]{-1}}-1\right )}{\left (\sqrt {1+4 \sqrt [3]{-1}}+\sqrt {1-4 (-1)^{2/3}}\right ) \left (-2 x+\sqrt {1+4 \sqrt [3]{-1}}+1\right )}} \sqrt {x^4-2 x^3+x+1}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.37, size = 30, normalized size = 1.00 \begin {gather*} -\log \left (1+2 x-2 x^2+2 \sqrt {1+x-2 x^3+x^4}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 26, normalized size = 0.87 \begin {gather*} \log \left (2 \, x^{2} - 2 \, x + 2 \, \sqrt {x^{4} - 2 \, x^{3} + x + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 34, normalized size = 1.13 \begin {gather*} -\log \left (-2 \, x^{2} + 2 \, x + 2 \, \sqrt {{\left (x^{2} - x\right )}^{2} - x^{2} + x + 1} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.61, size = 29, normalized size = 0.97
method | result | size |
trager | \(-\ln \left (1+2 x -2 x^{2}+2 \sqrt {x^{4}-2 x^{3}+x +1}\right )\) | \(29\) |
default | \(-\frac {2 \left (-\frac {\sqrt {3-2 i \sqrt {3}}}{2}-\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )^{2} \sqrt {\frac {\sqrt {3-2 i \sqrt {3}}\, \left (x -\frac {1}{2}+\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right )}{\left (-\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\, \sqrt {\frac {\sqrt {3-2 i \sqrt {3}}\, \left (x -\frac {1}{2}-\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}, \sqrt {-\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )^{2}}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (-\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \sqrt {3-2 i \sqrt {3}}\, \sqrt {\left (x -\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right )}}+\frac {4 \left (-\frac {\sqrt {3-2 i \sqrt {3}}}{2}-\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )^{2} \sqrt {\frac {\sqrt {3-2 i \sqrt {3}}\, \left (x -\frac {1}{2}+\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right )}{\left (-\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\, \sqrt {\frac {\sqrt {3-2 i \sqrt {3}}\, \left (x -\frac {1}{2}-\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\, \left (\left (\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}, \sqrt {-\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )^{2}}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (-\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\right )-\sqrt {3-2 i \sqrt {3}}\, \EllipticPi \left (\sqrt {\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}, \frac {\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}}{\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}}, \sqrt {-\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )^{2}}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (-\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\right )\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \sqrt {3-2 i \sqrt {3}}\, \sqrt {\left (x -\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right )}}\) | \(1352\) |
elliptic | \(-\frac {2 \left (-\frac {\sqrt {3-2 i \sqrt {3}}}{2}-\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )^{2} \sqrt {\frac {\sqrt {3-2 i \sqrt {3}}\, \left (x -\frac {1}{2}+\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right )}{\left (-\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\, \sqrt {\frac {\sqrt {3-2 i \sqrt {3}}\, \left (x -\frac {1}{2}-\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}, \sqrt {-\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )^{2}}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (-\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \sqrt {3-2 i \sqrt {3}}\, \sqrt {\left (x -\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right )}}+\frac {4 \left (-\frac {\sqrt {3-2 i \sqrt {3}}}{2}-\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )^{2} \sqrt {\frac {\sqrt {3-2 i \sqrt {3}}\, \left (x -\frac {1}{2}+\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right )}{\left (-\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\, \sqrt {\frac {\sqrt {3-2 i \sqrt {3}}\, \left (x -\frac {1}{2}-\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\, \left (\left (\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}, \sqrt {-\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )^{2}}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (-\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\right )-\sqrt {3-2 i \sqrt {3}}\, \EllipticPi \left (\sqrt {\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}, \frac {\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}}{\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}}, \sqrt {-\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )^{2}}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (-\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\right )\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \sqrt {3-2 i \sqrt {3}}\, \sqrt {\left (x -\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right )}}\) | \(1352\) |
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*} \int \frac {2 \, x - 1}{\sqrt {x^{4} - 2 \, x^{3} + x + 1}}\,{d x} \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-1}{\sqrt {x^4-2\,x^3+x+1}} \,d x \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 - 1}{\sqrt {x^{4} - 2 x^{3} + x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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