Optimal. Leaf size=25 \[ \log \left (x^2+\sqrt {x^4-2 x^3+x^2+13}-x\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 15, normalized size of antiderivative = 0.60, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1680, 12, 1107, 619, 215} \begin {gather*} -\sinh ^{-1}\left (\frac {(1-x) x}{\sqrt {13}}\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 {13+x^2-2 x^3+x^4}} \, dx &=\operatorname {Subst}\left (\int \frac {8 x}{\sqrt {209-8 x^2+16 x^4}} \, dx,x,-\frac {1}{2}+x\right )\\ &=8 \operatorname {Subst}\left (\int \frac {x}{\sqrt {209-8 x^2+16 x^4}} \, dx,x,-\frac {1}{2}+x\right )\\ &=4 \operatorname {Subst}\left (\int \frac {1}{\sqrt {209-8 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}{13312}}} \, dx,x,32 (-1+x) x\right )}{32 \sqrt {13}}\\ &=-\sinh ^{-1}\left (\frac {(1-x) x}{\sqrt {13}}\right )\\ \end {align*}
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Mathematica [C] time = 3.04, size = 803, normalized size = 32.12 \begin {gather*} \frac {\left (-2 x+\sqrt {1-4 i \sqrt {13}}+1\right ) \sqrt {\frac {\sqrt {1-4 i \sqrt {13}} \left (-2 x+\sqrt {1+4 i \sqrt {13}}+1\right )}{\left (\sqrt {1-4 i \sqrt {13}}+\sqrt {1+4 i \sqrt {13}}\right ) \left (-2 x+\sqrt {1-4 i \sqrt {13}}+1\right )}} \left (2 x+\sqrt {1-4 i \sqrt {13}}-1\right ) \sqrt {-\frac {\sqrt {1-4 i \sqrt {13}} \left (2 x+\sqrt {1+4 i \sqrt {13}}-1\right )}{\left (\sqrt {1-4 i \sqrt {13}}-\sqrt {1+4 i \sqrt {13}}\right ) \left (-2 x+\sqrt {1-4 i \sqrt {13}}+1\right )}} \left (F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {1-4 i \sqrt {13}}-\sqrt {1+4 i \sqrt {13}}\right ) \left (2 x+\sqrt {1-4 i \sqrt {13}}-1\right )}{\left (\sqrt {1-4 i \sqrt {13}}+\sqrt {1+4 i \sqrt {13}}\right ) \left (-2 x+\sqrt {1-4 i \sqrt {13}}+1\right )}}\right )|\frac {\left (\sqrt {1-4 i \sqrt {13}}+\sqrt {1+4 i \sqrt {13}}\right )^2}{\left (\sqrt {1-4 i \sqrt {13}}-\sqrt {1+4 i \sqrt {13}}\right )^2}\right )-2 \Pi \left (-\frac {\sqrt {1-4 i \sqrt {13}}+\sqrt {1+4 i \sqrt {13}}}{\sqrt {1-4 i \sqrt {13}}-\sqrt {1+4 i \sqrt {13}}};\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {1-4 i \sqrt {13}}-\sqrt {1+4 i \sqrt {13}}\right ) \left (2 x+\sqrt {1-4 i \sqrt {13}}-1\right )}{\left (\sqrt {1-4 i \sqrt {13}}+\sqrt {1+4 i \sqrt {13}}\right ) \left (-2 x+\sqrt {1-4 i \sqrt {13}}+1\right )}}\right )|\frac {\left (\sqrt {1-4 i \sqrt {13}}+\sqrt {1+4 i \sqrt {13}}\right )^2}{\left (\sqrt {1-4 i \sqrt {13}}-\sqrt {1+4 i \sqrt {13}}\right )^2}\right )\right )}{\sqrt {\frac {\left (\sqrt {1-4 i \sqrt {13}}-\sqrt {1+4 i \sqrt {13}}\right ) \left (2 x+\sqrt {1-4 i \sqrt {13}}-1\right )}{\left (\sqrt {1-4 i \sqrt {13}}+\sqrt {1+4 i \sqrt {13}}\right ) \left (-2 x+\sqrt {1-4 i \sqrt {13}}+1\right )}} \sqrt {x^4-2 x^3+x^2+13}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.22, size = 25, normalized size = 1.00 \begin {gather*} \log \left (-x+x^2+\sqrt {13+x^2-2 x^3+x^4}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 23, normalized size = 0.92 \begin {gather*} \log \left (x^{2} - x + \sqrt {x^{4} - 2 \, x^{3} + x^{2} + 13}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 23, normalized size = 0.92 \begin {gather*} -\log \left (-x^{2} + x + \sqrt {{\left (x^{2} - x\right )}^{2} + 13}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.61, size = 24, normalized size = 0.96
method | result | size |
trager | \(\ln \left (-x +x^{2}+\sqrt {x^{4}-2 x^{3}+x^{2}+13}\right )\) | \(24\) |
default | \(-\frac {2 \left (-\frac {\sqrt {1+4 i \sqrt {13}}}{2}-\frac {\sqrt {1-4 i \sqrt {13}}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}{\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )^{2} \sqrt {\frac {\sqrt {1+4 i \sqrt {13}}\, \left (x -\frac {1}{2}+\frac {\sqrt {1-4 i \sqrt {13}}}{2}\right )}{\left (-\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}}\, \sqrt {\frac {\sqrt {1+4 i \sqrt {13}}\, \left (x -\frac {1}{2}-\frac {\sqrt {1-4 i \sqrt {13}}}{2}\right )}{\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}{\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}}, \sqrt {-\frac {\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )^{2}}{\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (-\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}}\right )}{\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \sqrt {1+4 i \sqrt {13}}\, \sqrt {\left (x -\frac {1}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1-4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1-4 i \sqrt {13}}}{2}\right )}}+\frac {4 \left (-\frac {\sqrt {1+4 i \sqrt {13}}}{2}-\frac {\sqrt {1-4 i \sqrt {13}}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}{\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )^{2} \sqrt {\frac {\sqrt {1+4 i \sqrt {13}}\, \left (x -\frac {1}{2}+\frac {\sqrt {1-4 i \sqrt {13}}}{2}\right )}{\left (-\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}}\, \sqrt {\frac {\sqrt {1+4 i \sqrt {13}}\, \left (x -\frac {1}{2}-\frac {\sqrt {1-4 i \sqrt {13}}}{2}\right )}{\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}}\, \left (\left (\frac {1}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}{\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}}, \sqrt {-\frac {\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )^{2}}{\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (-\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}}\right )-\sqrt {1+4 i \sqrt {13}}\, \EllipticPi \left (\sqrt {\frac {\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}{\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}}, \frac {\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}}{\frac {\sqrt {1-4 i \sqrt {13}}}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}}, \sqrt {-\frac {\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )^{2}}{\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (-\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}}\right )\right )}{\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \sqrt {1+4 i \sqrt {13}}\, \sqrt {\left (x -\frac {1}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1-4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1-4 i \sqrt {13}}}{2}\right )}}\) | \(1352\) |
elliptic | \(-\frac {2 \left (-\frac {\sqrt {1+4 i \sqrt {13}}}{2}-\frac {\sqrt {1-4 i \sqrt {13}}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}{\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )^{2} \sqrt {\frac {\sqrt {1+4 i \sqrt {13}}\, \left (x -\frac {1}{2}+\frac {\sqrt {1-4 i \sqrt {13}}}{2}\right )}{\left (-\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}}\, \sqrt {\frac {\sqrt {1+4 i \sqrt {13}}\, \left (x -\frac {1}{2}-\frac {\sqrt {1-4 i \sqrt {13}}}{2}\right )}{\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}{\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}}, \sqrt {-\frac {\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )^{2}}{\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (-\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}}\right )}{\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \sqrt {1+4 i \sqrt {13}}\, \sqrt {\left (x -\frac {1}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1-4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1-4 i \sqrt {13}}}{2}\right )}}+\frac {4 \left (-\frac {\sqrt {1+4 i \sqrt {13}}}{2}-\frac {\sqrt {1-4 i \sqrt {13}}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}{\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )^{2} \sqrt {\frac {\sqrt {1+4 i \sqrt {13}}\, \left (x -\frac {1}{2}+\frac {\sqrt {1-4 i \sqrt {13}}}{2}\right )}{\left (-\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}}\, \sqrt {\frac {\sqrt {1+4 i \sqrt {13}}\, \left (x -\frac {1}{2}-\frac {\sqrt {1-4 i \sqrt {13}}}{2}\right )}{\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}}\, \left (\left (\frac {1}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}{\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}}, \sqrt {-\frac {\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )^{2}}{\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (-\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}}\right )-\sqrt {1+4 i \sqrt {13}}\, \EllipticPi \left (\sqrt {\frac {\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}{\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}}, \frac {\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}}{\frac {\sqrt {1-4 i \sqrt {13}}}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}}, \sqrt {-\frac {\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )^{2}}{\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (-\frac {\sqrt {1-4 i \sqrt {13}}}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right )}}\right )\right )}{\left (\frac {\sqrt {1-4 i \sqrt {13}}}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \sqrt {1+4 i \sqrt {13}}\, \sqrt {\left (x -\frac {1}{2}+\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1-4 i \sqrt {13}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1-4 i \sqrt {13}}}{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^{2} + 13}}\,{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 = -0.04 \begin {gather*} \int \frac {2\,x-1}{\sqrt {x^4-2\,x^3+x^2+13}} \,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^{2} + 13}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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