Integrand size = 28, antiderivative size = 35 \[ \int \frac {1-2 x}{\sqrt {5+5 x-4 x^2-2 x^3+x^4}} \, dx=\log \left (5+2 x-2 x^2+2 \sqrt {5+5 x-4 x^2-2 x^3+x^4}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1694, 12, 1121, 635, 212} \[ \int \frac {1-2 x}{\sqrt {5+5 x-4 x^2-2 x^3+x^4}} \, dx=\text {arctanh}\left (\frac {11-4 \left (x-\frac {1}{2}\right )^2}{\sqrt {16 \left (x-\frac {1}{2}\right )^4-88 \left (x-\frac {1}{2}\right )^2+101}}\right ) \]
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Rule 12
Rule 212
Rule 635
Rule 1121
Rule 1694
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int -\frac {8 x}{\sqrt {101-88 x^2+16 x^4}} \, dx,x,-\frac {1}{2}+x\right ) \\ & = -\left (8 \text {Subst}\left (\int \frac {x}{\sqrt {101-88 x^2+16 x^4}} \, dx,x,-\frac {1}{2}+x\right )\right ) \\ & = -\left (4 \text {Subst}\left (\int \frac {1}{\sqrt {101-88 x+16 x^2}} \, dx,x,\left (-\frac {1}{2}+x\right )^2\right )\right ) \\ & = -\left (8 \text {Subst}\left (\int \frac {1}{64-x^2} \, dx,x,\frac {8 \left (-11+4 \left (-\frac {1}{2}+x\right )^2\right )}{\sqrt {101+(1-2 x)^4-88 \left (-\frac {1}{2}+x\right )^2}}\right )\right ) \\ & = \text {arctanh}\left (\frac {11-(-1+2 x)^2}{\sqrt {101-22 (1-2 x)^2+(1-2 x)^4}}\right ) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {1-2 x}{\sqrt {5+5 x-4 x^2-2 x^3+x^4}} \, dx=\log \left (5+2 x-2 x^2+2 \sqrt {5+5 x-4 x^2-2 x^3+x^4}\right ) \]
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Time = 2.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.03
method | result | size |
default | \(-\ln \left (2 x^{2}-2 x -5+2 \sqrt {x^{4}-2 x^{3}-4 x^{2}+5 x +5}\right )\) | \(36\) |
trager | \(-\ln \left (2 x^{2}-2 x -5+2 \sqrt {x^{4}-2 x^{3}-4 x^{2}+5 x +5}\right )\) | \(36\) |
pseudoelliptic | \(-\ln \left (2 x^{2}-2 x -5+2 \sqrt {x^{4}-2 x^{3}-4 x^{2}+5 x +5}\right )\) | \(36\) |
elliptic | \(\text {Expression too large to display}\) | \(1182\) |
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Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int \frac {1-2 x}{\sqrt {5+5 x-4 x^2-2 x^3+x^4}} \, dx=\log \left (-2 \, x^{2} + 2 \, x + 2 \, \sqrt {x^{4} - 2 \, x^{3} - 4 \, x^{2} + 5 \, x + 5} + 5\right ) \]
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\[ \int \frac {1-2 x}{\sqrt {5+5 x-4 x^2-2 x^3+x^4}} \, dx=- \int \frac {2 x}{\sqrt {x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 5}}\, dx - \int \left (- \frac {1}{\sqrt {x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 5}}\right )\, dx \]
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\[ \int \frac {1-2 x}{\sqrt {5+5 x-4 x^2-2 x^3+x^4}} \, dx=\int { -\frac {2 \, x - 1}{\sqrt {x^{4} - 2 \, x^{3} - 4 \, x^{2} + 5 \, x + 5}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (33) = 66\).
Time = 0.30 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.03 \[ \int \frac {1-2 x}{\sqrt {5+5 x-4 x^2-2 x^3+x^4}} \, dx=-\frac {1}{4} \, \sqrt {{\left (x^{2} - x\right )}^{2} - 5 \, x^{2} + 5 \, x + 5} {\left (2 \, x^{2} - 2 \, x - 5\right )} - \frac {5}{8} \, \log \left ({\left | -2 \, x^{2} + 2 \, x + 2 \, \sqrt {{\left (x^{2} - x\right )}^{2} - 5 \, x^{2} + 5 \, x + 5} + 5 \right |}\right ) \]
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Timed out. \[ \int \frac {1-2 x}{\sqrt {5+5 x-4 x^2-2 x^3+x^4}} \, dx=\int -\frac {2\,x-1}{\sqrt {x^4-2\,x^3-4\,x^2+5\,x+5}} \,d x \]
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