Optimal. Leaf size=25 \[ \log \left (x^2+\sqrt {x^4-2 x^3+x^2+4}-x\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 9, normalized size of antiderivative = 0.36, 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}{2} (x-1) x\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 {4+x^2-2 x^3+x^4}} \, dx &=\operatorname {Subst}\left (\int \frac {8 x}{\sqrt {65-8 x^2+16 x^4}} \, dx,x,-\frac {1}{2}+x\right )\\ &=8 \operatorname {Subst}\left (\int \frac {x}{\sqrt {65-8 x^2+16 x^4}} \, dx,x,-\frac {1}{2}+x\right )\\ &=4 \operatorname {Subst}\left (\int \frac {1}{\sqrt {65-8 x+16 x^2}} \, dx,x,\left (-\frac {1}{2}+x\right )^2\right )\\ &=\frac {1}{64} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{4096}}} \, dx,x,32 (-1+x) x\right )\\ &=-\sinh ^{-1}\left (\frac {1}{2} (1-x) x\right )\\ \end {align*}
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Mathematica [C] time = 2.36, size = 613, normalized size = 24.52 \begin {gather*} \frac {\left (-2 x+\sqrt {1-8 i}+1\right ) \sqrt {\frac {\sqrt {1-8 i} \left (-2 x+\sqrt {1+8 i}+1\right )}{\left (\sqrt {1-8 i}+\sqrt {1+8 i}\right ) \left (-2 x+\sqrt {1-8 i}+1\right )}} \left (2 x+\sqrt {1-8 i}-1\right ) \sqrt {\frac {\sqrt {1-8 i} \left (2 x+\sqrt {1+8 i}-1\right )}{\left (\sqrt {1+8 i}-\sqrt {1-8 i}\right ) \left (-2 x+\sqrt {1-8 i}+1\right )}} \left (\left (1+\sqrt {1-8 i}\right ) F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {1-8 i}-\sqrt {1+8 i}\right ) \left (2 x+\sqrt {1-8 i}-1\right )}{\left (\sqrt {1-8 i}+\sqrt {1+8 i}\right ) \left (-2 x+\sqrt {1-8 i}+1\right )}}\right )|\frac {\left (\sqrt {1-8 i}+\sqrt {1+8 i}\right )^2}{\left (\sqrt {1-8 i}-\sqrt {1+8 i}\right )^2}\right )-F\left (\sin ^{-1}\left (\sqrt {-\frac {2 \sqrt {1+8 i} x-2 \sqrt {1-8 i} x+\sqrt {65}-\sqrt {1+8 i}+\sqrt {1-8 i}-(1-8 i)}{\left (\sqrt {1-8 i}+\sqrt {1+8 i}\right ) \left (-2 x+\sqrt {1-8 i}+1\right )}}\right )|\frac {\left (\sqrt {1-8 i}+\sqrt {1+8 i}\right )^2}{\left (\sqrt {1-8 i}-\sqrt {1+8 i}\right )^2}\right )-2 \sqrt {1-8 i} \Pi \left (-\frac {\sqrt {1-8 i}+\sqrt {1+8 i}}{\sqrt {1-8 i}-\sqrt {1+8 i}};\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {1-8 i}-\sqrt {1+8 i}\right ) \left (2 x+\sqrt {1-8 i}-1\right )}{\left (\sqrt {1-8 i}+\sqrt {1+8 i}\right ) \left (-2 x+\sqrt {1-8 i}+1\right )}}\right )|\frac {\left (\sqrt {1-8 i}+\sqrt {1+8 i}\right )^2}{\left (\sqrt {1-8 i}-\sqrt {1+8 i}\right )^2}\right )\right )}{\sqrt {1-8 i} \sqrt {\frac {\left (\sqrt {1-8 i}-\sqrt {1+8 i}\right ) \left (2 x+\sqrt {1-8 i}-1\right )}{\left (\sqrt {1-8 i}+\sqrt {1+8 i}\right ) \left (-2 x+\sqrt {1-8 i}+1\right )}} \sqrt {x^4-2 x^3+x^2+4}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.11, size = 25, normalized size = 1.00 \begin {gather*} \log \left (-x+x^2+\sqrt {4+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} + 4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.68, size = 23, normalized size = 0.92 \begin {gather*} -\log \left (-x^{2} + x + \sqrt {{\left (x^{2} - x\right )}^{2} + 4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.47, size = 26, normalized size = 1.04
method | result | size |
trager | \(-\ln \left (-x^{2}+\sqrt {x^{4}-2 x^{3}+x^{2}+4}+x \right )\) | \(26\) |
default | \(-\frac {2 \left (-\frac {\sqrt {1+8 i}}{2}-\frac {\sqrt {1-8 i}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right )}{\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )^{2} \sqrt {\frac {\sqrt {1+8 i}\, \left (x -\frac {1}{2}+\frac {\sqrt {1-8 i}}{2}\right )}{\left (-\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}\, \sqrt {\frac {\sqrt {1+8 i}\, \left (x -\frac {1}{2}-\frac {\sqrt {1-8 i}}{2}\right )}{\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right )}{\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}, \sqrt {-\frac {\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right )^{2}}{\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (-\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right )}}\right )}{\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \sqrt {1+8 i}\, \sqrt {\left (x -\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1-8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1-8 i}}{2}\right )}}+\frac {4 \left (-\frac {\sqrt {1+8 i}}{2}-\frac {\sqrt {1-8 i}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right )}{\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )^{2} \sqrt {\frac {\sqrt {1+8 i}\, \left (x -\frac {1}{2}+\frac {\sqrt {1-8 i}}{2}\right )}{\left (-\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}\, \sqrt {\frac {\sqrt {1+8 i}\, \left (x -\frac {1}{2}-\frac {\sqrt {1-8 i}}{2}\right )}{\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}\, \left (\left (\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right )}{\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}, \sqrt {-\frac {\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right )^{2}}{\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (-\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right )}}\right )-\sqrt {1+8 i}\, \EllipticPi \left (\sqrt {\frac {\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right )}{\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}, \frac {\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}}{\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}}, \sqrt {-\frac {\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right )^{2}}{\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (-\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right )}}\right )\right )}{\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \sqrt {1+8 i}\, \sqrt {\left (x -\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1-8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1-8 i}}{2}\right )}}\) | \(882\) |
elliptic | \(-\frac {2 \left (-\frac {\sqrt {1+8 i}}{2}-\frac {\sqrt {1-8 i}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right )}{\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )^{2} \sqrt {\frac {\sqrt {1+8 i}\, \left (x -\frac {1}{2}+\frac {\sqrt {1-8 i}}{2}\right )}{\left (-\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}\, \sqrt {\frac {\sqrt {1+8 i}\, \left (x -\frac {1}{2}-\frac {\sqrt {1-8 i}}{2}\right )}{\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right )}{\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}, \sqrt {-\frac {\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right )^{2}}{\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (-\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right )}}\right )}{\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \sqrt {1+8 i}\, \sqrt {\left (x -\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1-8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1-8 i}}{2}\right )}}+\frac {4 \left (-\frac {\sqrt {1+8 i}}{2}-\frac {\sqrt {1-8 i}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right )}{\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )^{2} \sqrt {\frac {\sqrt {1+8 i}\, \left (x -\frac {1}{2}+\frac {\sqrt {1-8 i}}{2}\right )}{\left (-\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}\, \sqrt {\frac {\sqrt {1+8 i}\, \left (x -\frac {1}{2}-\frac {\sqrt {1-8 i}}{2}\right )}{\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}\, \left (\left (\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right )}{\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}, \sqrt {-\frac {\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right )^{2}}{\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (-\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right )}}\right )-\sqrt {1+8 i}\, \EllipticPi \left (\sqrt {\frac {\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right )}{\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}, \frac {\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}}{\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}}, \sqrt {-\frac {\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right )^{2}}{\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (-\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right )}}\right )\right )}{\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \sqrt {1+8 i}\, \sqrt {\left (x -\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1-8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1-8 i}}{2}\right )}}\) | \(882\) |
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} + 4}}\,{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+4}} \,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} + 4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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