Integrand size = 27, antiderivative size = 133 \[ \int \frac {x^2}{\sqrt {-x-x^2+x^3} \left (-1+x^4\right )} \, dx=-\frac {1}{2} \arctan \left (\frac {\sqrt {-x-x^2+x^3}}{-1-x+x^2}\right )+\frac {1}{4} \sqrt {\frac {1}{5}+\frac {2 i}{5}} \arctan \left (\frac {\sqrt {1-2 i} \sqrt {-x-x^2+x^3}}{-1-x+x^2}\right )+\frac {1}{4} \sqrt {\frac {1}{5}-\frac {2 i}{5}} \arctan \left (\frac {\sqrt {1+2 i} \sqrt {-x-x^2+x^3}}{-1-x+x^2}\right ) \]
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Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 2.02 (sec) , antiderivative size = 467, normalized size of antiderivative = 3.51, number of steps used = 55, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {2081, 6857, 971, 730, 1112, 948, 174, 552, 551, 1148, 1198} \[ \int \frac {x^2}{\sqrt {-x-x^2+x^3} \left (-1+x^4\right )} \, dx=-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \operatorname {EllipticPi}\left (\frac {1}{2} \left (-1-\sqrt {5}\right ),\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{4 \sqrt {x^3-x^2-x}}+\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \operatorname {EllipticPi}\left (-\frac {1}{2} i \left (1+\sqrt {5}\right ),\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{4 \sqrt {x^3-x^2-x}}+\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \operatorname {EllipticPi}\left (\frac {1}{2} i \left (1+\sqrt {5}\right ),\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{4 \sqrt {x^3-x^2-x}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \operatorname {EllipticPi}\left (\frac {1}{2} \left (1+\sqrt {5}\right ),\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{4 \sqrt {x^3-x^2-x}} \]
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Rule 174
Rule 551
Rule 552
Rule 730
Rule 948
Rule 971
Rule 1112
Rule 1148
Rule 1198
Rule 2081
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {x^{3/2}}{\sqrt {-1-x+x^2} \left (-1+x^4\right )} \, dx}{\sqrt {-x-x^2+x^3}} \\ & = \frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \left (-\frac {x^{3/2}}{2 \left (1-x^2\right ) \sqrt {-1-x+x^2}}-\frac {x^{3/2}}{2 \left (1+x^2\right ) \sqrt {-1-x+x^2}}\right ) \, dx}{\sqrt {-x-x^2+x^3}} \\ & = -\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {x^{3/2}}{\left (1-x^2\right ) \sqrt {-1-x+x^2}} \, dx}{2 \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {x^{3/2}}{\left (1+x^2\right ) \sqrt {-1-x+x^2}} \, dx}{2 \sqrt {-x-x^2+x^3}} \\ & = -\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \left (\frac {i x^{3/2}}{2 (i-x) \sqrt {-1-x+x^2}}+\frac {i x^{3/2}}{2 (i+x) \sqrt {-1-x+x^2}}\right ) \, dx}{2 \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \left (\frac {x^{3/2}}{2 (1-x) \sqrt {-1-x+x^2}}+\frac {x^{3/2}}{2 (1+x) \sqrt {-1-x+x^2}}\right ) \, dx}{2 \sqrt {-x-x^2+x^3}} \\ & = -\frac {\left (i \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {x^{3/2}}{(i-x) \sqrt {-1-x+x^2}} \, dx}{4 \sqrt {-x-x^2+x^3}}-\frac {\left (i \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {x^{3/2}}{(i+x) \sqrt {-1-x+x^2}} \, dx}{4 \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {x^{3/2}}{(1-x) \sqrt {-1-x+x^2}} \, dx}{4 \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {x^{3/2}}{(1+x) \sqrt {-1-x+x^2}} \, dx}{4 \sqrt {-x-x^2+x^3}} \\ & = -\frac {\left (i \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \left (-\frac {i}{\sqrt {x} \sqrt {-1-x+x^2}}+\frac {1}{(-i-x) \sqrt {x} \sqrt {-1-x+x^2}}+\frac {\sqrt {x}}{\sqrt {-1-x+x^2}}\right ) \, dx}{4 \sqrt {-x-x^2+x^3}}-\frac {\left (i \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \left (-\frac {i}{\sqrt {x} \sqrt {-1-x+x^2}}-\frac {\sqrt {x}}{\sqrt {-1-x+x^2}}+\frac {1}{\sqrt {x} (-i+x) \sqrt {-1-x+x^2}}\right ) \, dx}{4 \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \left (-\frac {1}{\sqrt {x} \sqrt {-1-x+x^2}}+\frac {1}{(1-x) \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\sqrt {x}}{\sqrt {-1-x+x^2}}\right ) \, dx}{4 \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \left (-\frac {1}{\sqrt {x} \sqrt {-1-x+x^2}}+\frac {\sqrt {x}}{\sqrt {-1-x+x^2}}+\frac {1}{\sqrt {x} (1+x) \sqrt {-1-x+x^2}}\right ) \, dx}{4 \sqrt {-x-x^2+x^3}} \\ & = -\frac {\left (i \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{(-i-x) \sqrt {x} \sqrt {-1-x+x^2}} \, dx}{4 \sqrt {-x-x^2+x^3}}-\frac {\left (i \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\sqrt {x} (-i+x) \sqrt {-1-x+x^2}} \, dx}{4 \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{(1-x) \sqrt {x} \sqrt {-1-x+x^2}} \, dx}{4 \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\sqrt {x} (1+x) \sqrt {-1-x+x^2}} \, dx}{4 \sqrt {-x-x^2+x^3}} \\ & = -\frac {\left (i \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \int \frac {1}{(-i-x) \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}} \, dx}{4 \sqrt {-x-x^2+x^3}}-\frac {\left (i \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \int \frac {1}{\sqrt {x} (-i+x) \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}} \, dx}{4 \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \int \frac {1}{(1-x) \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}} \, dx}{4 \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \int \frac {1}{\sqrt {x} (1+x) \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}} \, dx}{4 \sqrt {-x-x^2+x^3}} \\ & = \frac {\left (i \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \text {Subst}\left (\int \frac {1}{\left (i-x^2\right ) \sqrt {-1-\sqrt {5}+2 x^2} \sqrt {-1+\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x-x^2+x^3}}+\frac {\left (i \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \text {Subst}\left (\int \frac {1}{\left (i+x^2\right ) \sqrt {-1-\sqrt {5}+2 x^2} \sqrt {-1+\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x-x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \text {Subst}\left (\int \frac {1}{\left (-1-x^2\right ) \sqrt {-1-\sqrt {5}+2 x^2} \sqrt {-1+\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x-x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt {-1-\sqrt {5}+2 x^2} \sqrt {-1+\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x-x^2+x^3}} \\ & = \frac {\left (i \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}}\right ) \text {Subst}\left (\int \frac {1}{\left (i-x^2\right ) \sqrt {-1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x-x^2+x^3}}+\frac {\left (i \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}}\right ) \text {Subst}\left (\int \frac {1}{\left (i+x^2\right ) \sqrt {-1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x-x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}}\right ) \text {Subst}\left (\int \frac {1}{\left (-1-x^2\right ) \sqrt {-1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x-x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt {-1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x-x^2+x^3}} \\ & = -\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \operatorname {EllipticPi}\left (\frac {1}{2} \left (-1-\sqrt {5}\right ),\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{4 \sqrt {-x-x^2+x^3}}+\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \operatorname {EllipticPi}\left (-\frac {1}{2} i \left (1+\sqrt {5}\right ),\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{4 \sqrt {-x-x^2+x^3}}+\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \operatorname {EllipticPi}\left (\frac {1}{2} i \left (1+\sqrt {5}\right ),\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{4 \sqrt {-x-x^2+x^3}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \operatorname {EllipticPi}\left (\frac {1}{2} \left (1+\sqrt {5}\right ),\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{4 \sqrt {-x-x^2+x^3}} \\ \end{align*}
Time = 0.55 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.02 \[ \int \frac {x^2}{\sqrt {-x-x^2+x^3} \left (-1+x^4\right )} \, dx=\frac {\sqrt {x} \sqrt {-1-x+x^2} \left (-2 \sqrt {5} \arctan \left (\frac {\sqrt {x}}{\sqrt {-1-x+x^2}}\right )+\sqrt {1+2 i} \arctan \left (\frac {\sqrt {1-2 i} \sqrt {x}}{\sqrt {-1-x+x^2}}\right )+\sqrt {1-2 i} \arctan \left (\frac {\sqrt {1+2 i} \sqrt {x}}{\sqrt {-1-x+x^2}}\right )\right )}{4 \sqrt {5} \sqrt {x \left (-1-x+x^2\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(224\) vs. \(2(105)=210\).
Time = 5.37 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.69
method | result | size |
default | \(\frac {-\sqrt {5}\, \ln \left (\frac {-\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {-2+2 \sqrt {5}}+x \sqrt {5}+x^{2}-x -1}{x}\right )+\sqrt {5}\, \ln \left (\frac {\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {-2+2 \sqrt {5}}+x \sqrt {5}+x^{2}-x -1}{x}\right )+\left (-5-\sqrt {5}\right ) \arctan \left (\frac {\sqrt {-2+2 \sqrt {5}}\, x +2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2+2 \sqrt {5}}}\right )+10 \arctan \left (\frac {\sqrt {x \left (x^{2}-x -1\right )}}{x}\right ) \sqrt {2+2 \sqrt {5}}+\left (5+\sqrt {5}\right ) \arctan \left (\frac {\sqrt {-2+2 \sqrt {5}}\, x -2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2+2 \sqrt {5}}}\right )}{20 \sqrt {2+2 \sqrt {5}}}\) | \(225\) |
pseudoelliptic | \(\frac {-\sqrt {5}\, \ln \left (\frac {-\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {-2+2 \sqrt {5}}+x \sqrt {5}+x^{2}-x -1}{x}\right )+\sqrt {5}\, \ln \left (\frac {\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {-2+2 \sqrt {5}}+x \sqrt {5}+x^{2}-x -1}{x}\right )+\left (-5-\sqrt {5}\right ) \arctan \left (\frac {\sqrt {-2+2 \sqrt {5}}\, x +2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2+2 \sqrt {5}}}\right )+10 \arctan \left (\frac {\sqrt {x \left (x^{2}-x -1\right )}}{x}\right ) \sqrt {2+2 \sqrt {5}}+\left (5+\sqrt {5}\right ) \arctan \left (\frac {\sqrt {-2+2 \sqrt {5}}\, x -2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2+2 \sqrt {5}}}\right )}{20 \sqrt {2+2 \sqrt {5}}}\) | \(225\) |
trager | \(\text {Expression too large to display}\) | \(911\) |
elliptic | \(\text {Expression too large to display}\) | \(1436\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 324 vs. \(2 (97) = 194\).
Time = 0.29 (sec) , antiderivative size = 324, normalized size of antiderivative = 2.44 \[ \int \frac {x^2}{\sqrt {-x-x^2+x^3} \left (-1+x^4\right )} \, dx=\frac {1}{80} \, \sqrt {5} \sqrt {-2 i - 1} \log \left (\frac {5 \, x^{4} + \left (20 i - 10\right ) \, x^{3} - 2 \, \sqrt {5} \sqrt {-2 i - 1} \sqrt {x^{3} - x^{2} - x} {\left (-\left (2 i - 1\right ) \, x^{2} + \left (2 i + 4\right ) \, x + 2 i - 1\right )} - \left (20 i + 30\right ) \, x^{2} - \left (20 i - 10\right ) \, x + 5}{x^{4} + 2 \, x^{2} + 1}\right ) - \frac {1}{80} \, \sqrt {5} \sqrt {-2 i - 1} \log \left (\frac {5 \, x^{4} + \left (20 i - 10\right ) \, x^{3} - 2 \, \sqrt {5} \sqrt {-2 i - 1} \sqrt {x^{3} - x^{2} - x} {\left (\left (2 i - 1\right ) \, x^{2} - \left (2 i + 4\right ) \, x - 2 i + 1\right )} - \left (20 i + 30\right ) \, x^{2} - \left (20 i - 10\right ) \, x + 5}{x^{4} + 2 \, x^{2} + 1}\right ) + \frac {1}{80} \, \sqrt {5} \sqrt {2 i - 1} \log \left (\frac {5 \, x^{4} - \left (20 i + 10\right ) \, x^{3} - 2 \, \sqrt {5} \sqrt {2 i - 1} \sqrt {x^{3} - x^{2} - x} {\left (\left (2 i + 1\right ) \, x^{2} - \left (2 i - 4\right ) \, x - 2 i - 1\right )} + \left (20 i - 30\right ) \, x^{2} + \left (20 i + 10\right ) \, x + 5}{x^{4} + 2 \, x^{2} + 1}\right ) - \frac {1}{80} \, \sqrt {5} \sqrt {2 i - 1} \log \left (\frac {5 \, x^{4} - \left (20 i + 10\right ) \, x^{3} - 2 \, \sqrt {5} \sqrt {2 i - 1} \sqrt {x^{3} - x^{2} - x} {\left (-\left (2 i + 1\right ) \, x^{2} + \left (2 i - 4\right ) \, x + 2 i + 1\right )} + \left (20 i - 30\right ) \, x^{2} + \left (20 i + 10\right ) \, x + 5}{x^{4} + 2 \, x^{2} + 1}\right ) + \frac {1}{4} \, \arctan \left (\frac {x^{2} - 2 \, x - 1}{2 \, \sqrt {x^{3} - x^{2} - x}}\right ) \]
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\[ \int \frac {x^2}{\sqrt {-x-x^2+x^3} \left (-1+x^4\right )} \, dx=\int \frac {x^{2}}{\sqrt {x \left (x^{2} - x - 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \]
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\[ \int \frac {x^2}{\sqrt {-x-x^2+x^3} \left (-1+x^4\right )} \, dx=\int { \frac {x^{2}}{{\left (x^{4} - 1\right )} \sqrt {x^{3} - x^{2} - x}} \,d x } \]
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\[ \int \frac {x^2}{\sqrt {-x-x^2+x^3} \left (-1+x^4\right )} \, dx=\int { \frac {x^{2}}{{\left (x^{4} - 1\right )} \sqrt {x^{3} - x^{2} - x}} \,d x } \]
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Time = 0.06 (sec) , antiderivative size = 533, normalized size of antiderivative = 4.01 \[ \int \frac {x^2}{\sqrt {-x-x^2+x^3} \left (-1+x^4\right )} \, dx=-\frac {\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\sqrt {\frac {x+\frac {\sqrt {5}}{2}-\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\,\sqrt {\frac {\frac {\sqrt {5}}{2}-x+\frac {1}{2}}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\Pi \left (-\frac {\sqrt {5}}{2}-\frac {1}{2};\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )}{2\,\sqrt {x^3-x^2-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,x}}-\frac {\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\sqrt {\frac {x+\frac {\sqrt {5}}{2}-\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\,\sqrt {\frac {\frac {\sqrt {5}}{2}-x+\frac {1}{2}}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\Pi \left (\frac {\sqrt {5}}{2}+\frac {1}{2};\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )}{2\,\sqrt {x^3-x^2-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,x}}+\frac {\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\sqrt {\frac {x+\frac {\sqrt {5}}{2}-\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\,\sqrt {\frac {\frac {\sqrt {5}}{2}-x+\frac {1}{2}}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\Pi \left (-\frac {\sqrt {5}\,1{}\mathrm {i}}{2}-\frac {1}{2}{}\mathrm {i};\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )}{2\,\sqrt {x^3-x^2-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,x}}+\frac {\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\sqrt {\frac {x+\frac {\sqrt {5}}{2}-\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\,\sqrt {\frac {\frac {\sqrt {5}}{2}-x+\frac {1}{2}}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\Pi \left (\frac {\sqrt {5}\,1{}\mathrm {i}}{2}+\frac {1}{2}{}\mathrm {i};\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )}{2\,\sqrt {x^3-x^2-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,x}} \]
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