Integrand size = 23, antiderivative size = 17 \[ \int \frac {1}{\sqrt {8 x-8 x^2+4 x^3-x^4}} \, dx=-\frac {\operatorname {EllipticF}\left (\arcsin (1-x),-\frac {1}{3}\right )}{\sqrt {3}} \]
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Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1120, 1109, 430} \[ \int \frac {1}{\sqrt {8 x-8 x^2+4 x^3-x^4}} \, dx=-\frac {\operatorname {EllipticF}\left (\arcsin (1-x),-\frac {1}{3}\right )}{\sqrt {3}} \]
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Rule 430
Rule 1109
Rule 1120
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{\sqrt {3-2 x^2-x^4}} \, dx,x,-1+x\right ) \\ & = 2 \text {Subst}\left (\int \frac {1}{\sqrt {2-2 x^2} \sqrt {6+2 x^2}} \, dx,x,-1+x\right ) \\ & = -\frac {F\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )}{\sqrt {3}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 32.15 (sec) , antiderivative size = 156, normalized size of antiderivative = 9.18 \[ \int \frac {1}{\sqrt {8 x-8 x^2+4 x^3-x^4}} \, dx=\frac {\sqrt {-i+\sqrt {3}+\frac {4 i}{x}} \sqrt {-\frac {i (-2+x)}{\left (-i+\sqrt {3}\right ) x}} x \left (-4+x-i \sqrt {3} x\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}-\frac {4 i}{x}}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{-i+\sqrt {3}}\right )}{\sqrt {2} \sqrt {i+\sqrt {3}-\frac {4 i}{x}} \sqrt {-x \left (-8+8 x-4 x^2+x^3\right )}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (15 ) = 30\).
Time = 1.36 (sec) , antiderivative size = 200, normalized size of antiderivative = 11.76
method | result | size |
default | \(\frac {2 \left (-1-i \sqrt {3}\right ) \sqrt {\frac {\left (-1+i \sqrt {3}\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \left (x -2\right )^{2} \sqrt {\frac {x -1+i \sqrt {3}}{\left (1-i \sqrt {3}\right ) \left (x -2\right )}}\, \sqrt {\frac {x -1-i \sqrt {3}}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, F\left (\sqrt {\frac {\left (-1+i \sqrt {3}\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}, \sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-1-i \sqrt {3}\right )}{\left (-1+i \sqrt {3}\right ) \left (1-i \sqrt {3}\right )}}\right )}{\left (-1+i \sqrt {3}\right ) \sqrt {-x \left (x -2\right ) \left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right )}}\) | \(200\) |
elliptic | \(\frac {2 \left (-1-i \sqrt {3}\right ) \sqrt {\frac {\left (-1+i \sqrt {3}\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \left (x -2\right )^{2} \sqrt {\frac {x -1+i \sqrt {3}}{\left (1-i \sqrt {3}\right ) \left (x -2\right )}}\, \sqrt {\frac {x -1-i \sqrt {3}}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, F\left (\sqrt {\frac {\left (-1+i \sqrt {3}\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}, \sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-1-i \sqrt {3}\right )}{\left (-1+i \sqrt {3}\right ) \left (1-i \sqrt {3}\right )}}\right )}{\left (-1+i \sqrt {3}\right ) \sqrt {-x \left (x -2\right ) \left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right )}}\) | \(200\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\sqrt {8 x-8 x^2+4 x^3-x^4}} \, dx=-\frac {1}{2} \, \sqrt {2} {\rm weierstrassPInverse}\left (-\frac {2}{3}, \frac {7}{54}, -\frac {x - 3}{3 \, x}\right ) \]
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\[ \int \frac {1}{\sqrt {8 x-8 x^2+4 x^3-x^4}} \, dx=\int \frac {1}{\sqrt {- x^{4} + 4 x^{3} - 8 x^{2} + 8 x}}\, dx \]
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\[ \int \frac {1}{\sqrt {8 x-8 x^2+4 x^3-x^4}} \, dx=\int { \frac {1}{\sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x}} \,d x } \]
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\[ \int \frac {1}{\sqrt {8 x-8 x^2+4 x^3-x^4}} \, dx=\int { \frac {1}{\sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {8 x-8 x^2+4 x^3-x^4}} \, dx=\int \frac {1}{\sqrt {-x^4+4\,x^3-8\,x^2+8\,x}} \,d x \]
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