Integrand size = 23, antiderivative size = 25 \[ \int \frac {-1+2 x}{\sqrt {4+x^2-2 x^3+x^4}} \, dx=\log \left (-x+x^2+\sqrt {4+x^2-2 x^3+x^4}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.36, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1694, 12, 1121, 633, 221} \[ \int \frac {-1+2 x}{\sqrt {4+x^2-2 x^3+x^4}} \, dx=\text {arcsinh}\left (\frac {1}{2} (x-1) x\right ) \]
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Rule 12
Rule 221
Rule 633
Rule 1121
Rule 1694
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {8 x}{\sqrt {65-8 x^2+16 x^4}} \, dx,x,-\frac {1}{2}+x\right ) \\ & = 8 \text {Subst}\left (\int \frac {x}{\sqrt {65-8 x^2+16 x^4}} \, dx,x,-\frac {1}{2}+x\right ) \\ & = 4 \text {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} \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{4096}}} \, dx,x,32 (-1+x) x\right ) \\ & = -\text {arcsinh}\left (\frac {1}{2} (1-x) x\right ) \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {-1+2 x}{\sqrt {4+x^2-2 x^3+x^4}} \, dx=\log \left (-x+x^2+\sqrt {4+x^2-2 x^3+x^4}\right ) \]
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Time = 1.43 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.32
method | result | size |
default | \(\operatorname {arcsinh}\left (\frac {x \left (x -1\right )}{2}\right )\) | \(8\) |
pseudoelliptic | \(\operatorname {arcsinh}\left (\frac {x \left (x -1\right )}{2}\right )\) | \(8\) |
trager | \(\ln \left (-x +x^{2}+\sqrt {x^{4}-2 x^{3}+x^{2}+4}\right )\) | \(24\) |
elliptic | \(\text {Expression too large to display}\) | \(888\) |
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Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {-1+2 x}{\sqrt {4+x^2-2 x^3+x^4}} \, dx=\log \left (x^{2} - x + \sqrt {x^{4} - 2 \, x^{3} + x^{2} + 4}\right ) \]
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\[ \int \frac {-1+2 x}{\sqrt {4+x^2-2 x^3+x^4}} \, dx=\int \frac {2 x - 1}{\sqrt {x^{4} - 2 x^{3} + x^{2} + 4}}\, dx \]
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\[ \int \frac {-1+2 x}{\sqrt {4+x^2-2 x^3+x^4}} \, dx=\int { \frac {2 \, x - 1}{\sqrt {x^{4} - 2 \, x^{3} + x^{2} + 4}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.84 \[ \int \frac {-1+2 x}{\sqrt {4+x^2-2 x^3+x^4}} \, dx=\frac {1}{2} \, \sqrt {{\left (x^{2} - x\right )}^{2} + 4} {\left (x^{2} - x\right )} - 2 \, \log \left (-x^{2} + x + \sqrt {{\left (x^{2} - x\right )}^{2} + 4}\right ) \]
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Timed out. \[ \int \frac {-1+2 x}{\sqrt {4+x^2-2 x^3+x^4}} \, dx=\int \frac {2\,x-1}{\sqrt {x^4-2\,x^3+x^2+4}} \,d x \]
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