Integrand size = 25, antiderivative size = 73 \[ \int \frac {-1+x}{\left (-1-2 x+x^2\right ) \sqrt {-x+x^3}} \, dx=\frac {1}{4} \left (-2+\sqrt {2}\right ) \text {arctanh}\left (\frac {-1+x}{\left (-1+\sqrt {2}\right ) \sqrt {-x+x^3}}\right )+\frac {1}{4} \left (2+\sqrt {2}\right ) \text {arctanh}\left (\frac {-1+x}{\left (1+\sqrt {2}\right ) \sqrt {-x+x^3}}\right ) \]
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Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.51 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.41, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2081, 6860, 947, 174, 551} \[ \int \frac {-1+x}{\left (-1-2 x+x^2\right ) \sqrt {-x+x^3}} \, dx=\frac {\sqrt {x} \sqrt {1-x^2} \operatorname {EllipticPi}\left (-\frac {1}{\sqrt {2}},\arcsin \left (\sqrt {1-x}\right ),\frac {1}{2}\right )}{2 \sqrt {x^3-x}}-\frac {\sqrt {x} \sqrt {1-x^2} \operatorname {EllipticPi}\left (\frac {1}{\sqrt {2}},\arcsin \left (\sqrt {1-x}\right ),\frac {1}{2}\right )}{2 \sqrt {x^3-x}} \]
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Rule 174
Rule 551
Rule 947
Rule 2081
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \int \frac {-1+x}{\sqrt {x} \sqrt {-1+x^2} \left (-1-2 x+x^2\right )} \, dx}{\sqrt {-x+x^3}} \\ & = \frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \int \left (\frac {1}{\sqrt {x} \left (-2-2 \sqrt {2}+2 x\right ) \sqrt {-1+x^2}}+\frac {1}{\sqrt {x} \left (-2+2 \sqrt {2}+2 x\right ) \sqrt {-1+x^2}}\right ) \, dx}{\sqrt {-x+x^3}} \\ & = \frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \int \frac {1}{\sqrt {x} \left (-2-2 \sqrt {2}+2 x\right ) \sqrt {-1+x^2}} \, dx}{\sqrt {-x+x^3}}+\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \int \frac {1}{\sqrt {x} \left (-2+2 \sqrt {2}+2 x\right ) \sqrt {-1+x^2}} \, dx}{\sqrt {-x+x^3}} \\ & = \frac {\left (\sqrt {x} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1+x} \left (-2-2 \sqrt {2}+2 x\right )} \, dx}{\sqrt {-x+x^3}}+\frac {\left (\sqrt {x} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1+x} \left (-2+2 \sqrt {2}+2 x\right )} \, dx}{\sqrt {-x+x^3}} \\ & = -\frac {\left (2 \sqrt {x} \sqrt {1-x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-2 \sqrt {2}-2 x^2\right ) \sqrt {1-x^2} \sqrt {2-x^2}} \, dx,x,\sqrt {1-x}\right )}{\sqrt {-x+x^3}}-\frac {\left (2 \sqrt {x} \sqrt {1-x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (2 \sqrt {2}-2 x^2\right ) \sqrt {1-x^2} \sqrt {2-x^2}} \, dx,x,\sqrt {1-x}\right )}{\sqrt {-x+x^3}} \\ & = \frac {\sqrt {x} \sqrt {1-x^2} \operatorname {EllipticPi}\left (-\frac {1}{\sqrt {2}},\arcsin \left (\sqrt {1-x}\right ),\frac {1}{2}\right )}{2 \sqrt {-x+x^3}}-\frac {\sqrt {x} \sqrt {1-x^2} \operatorname {EllipticPi}\left (\frac {1}{\sqrt {2}},\arcsin \left (\sqrt {1-x}\right ),\frac {1}{2}\right )}{2 \sqrt {-x+x^3}} \\ \end{align*}
Time = 10.97 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.92 \[ \int \frac {-1+x}{\left (-1-2 x+x^2\right ) \sqrt {-x+x^3}} \, dx=\frac {1}{4} \left (\left (2+\sqrt {2}\right ) \text {arctanh}\left (\frac {\left (-1+\sqrt {2}\right ) (-1+x)}{\sqrt {x \left (-1+x^2\right )}}\right )+\left (-2+\sqrt {2}\right ) \text {arctanh}\left (\frac {\left (1+\sqrt {2}\right ) (-1+x)}{\sqrt {x \left (-1+x^2\right )}}\right )\right ) \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 2.35 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.59
method | result | size |
elliptic | \(\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticPi}\left (\sqrt {1+x}, -\frac {1}{-2-\sqrt {2}}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {x^{3}-x}\, \left (-2-\sqrt {2}\right )}+\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticPi}\left (\sqrt {1+x}, -\frac {1}{-2+\sqrt {2}}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {x^{3}-x}\, \left (-2+\sqrt {2}\right )}\) | \(116\) |
default | \(-\frac {\ln \left (\frac {278808 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2} x^{2}-1394040 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2} x -493895 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x^{2}+1672848 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2}+1038514 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) \sqrt {x^{3}-x}-74648 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x +66352 x^{2}-419247 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )-152085 \sqrt {x^{3}-x}+40832 x +25520}{{\left (4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x -8 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )-x +1\right )}^{2}}\right ) \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )}{2}+\frac {\ln \left (\frac {278808 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2} x^{2}-1394040 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2} x -493895 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x^{2}+1672848 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2}+1038514 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) \sqrt {x^{3}-x}-74648 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x +66352 x^{2}-419247 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )-152085 \sqrt {x^{3}-x}+40832 x +25520}{{\left (4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x -8 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )-x +1\right )}^{2}}\right )}{2}+\frac {\operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) \ln \left (\frac {278808 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2} x^{2}-1394040 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2} x -63721 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x^{2}+1672848 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2}-1038514 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) \sqrt {x^{3}-x}+2862728 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x -148735 x^{2}-2926449 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )+886429 \sqrt {x^{3}-x}-1427856 x +1279121}{{\left (4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x -8 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )-3 x +7\right )}^{2}}\right )}{2}\) | \(543\) |
trager | \(-\frac {\ln \left (\frac {278808 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2} x^{2}-1394040 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2} x -493895 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x^{2}+1672848 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2}+1038514 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) \sqrt {x^{3}-x}-74648 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x +66352 x^{2}-419247 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )-152085 \sqrt {x^{3}-x}+40832 x +25520}{{\left (4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x -8 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )-x +1\right )}^{2}}\right ) \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )}{2}+\frac {\ln \left (\frac {278808 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2} x^{2}-1394040 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2} x -493895 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x^{2}+1672848 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2}+1038514 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) \sqrt {x^{3}-x}-74648 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x +66352 x^{2}-419247 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )-152085 \sqrt {x^{3}-x}+40832 x +25520}{{\left (4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x -8 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )-x +1\right )}^{2}}\right )}{2}+\frac {\operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) \ln \left (\frac {278808 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2} x^{2}-1394040 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2} x -63721 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x^{2}+1672848 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2}-1038514 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) \sqrt {x^{3}-x}+2862728 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x -148735 x^{2}-2926449 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )+886429 \sqrt {x^{3}-x}-1427856 x +1279121}{{\left (4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x -8 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )-3 x +7\right )}^{2}}\right )}{2}\) | \(543\) |
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Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (57) = 114\).
Time = 0.30 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.73 \[ \int \frac {-1+x}{\left (-1-2 x+x^2\right ) \sqrt {-x+x^3}} \, dx=\frac {1}{8} \, \sqrt {2} \log \left (\frac {x^{4} + 12 \, x^{3} + 4 \, \sqrt {2} \sqrt {x^{3} - x} {\left (x^{2} + 2 \, x - 1\right )} + 2 \, x^{2} - 12 \, x + 1}{x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x + 1}\right ) + \frac {1}{4} \, \log \left (\frac {x^{4} + 4 \, x^{3} + 2 \, x^{2} - 4 \, \sqrt {x^{3} - x} {\left (x^{2} + 1\right )} - 4 \, x + 1}{x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x + 1}\right ) \]
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\[ \int \frac {-1+x}{\left (-1-2 x+x^2\right ) \sqrt {-x+x^3}} \, dx=\int \frac {x - 1}{\sqrt {x \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} - 2 x - 1\right )}\, dx \]
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\[ \int \frac {-1+x}{\left (-1-2 x+x^2\right ) \sqrt {-x+x^3}} \, dx=\int { \frac {x - 1}{\sqrt {x^{3} - x} {\left (x^{2} - 2 \, x - 1\right )}} \,d x } \]
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\[ \int \frac {-1+x}{\left (-1-2 x+x^2\right ) \sqrt {-x+x^3}} \, dx=\int { \frac {x - 1}{\sqrt {x^{3} - x} {\left (x^{2} - 2 \, x - 1\right )}} \,d x } \]
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Time = 6.33 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.40 \[ \int \frac {-1+x}{\left (-1-2 x+x^2\right ) \sqrt {-x+x^3}} \, dx=\frac {\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (-\frac {1}{\sqrt {2}+1};\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{\sqrt {x^3-x}\,\left (\sqrt {2}+1\right )}-\frac {\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (\frac {1}{\sqrt {2}-1};\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{\sqrt {x^3-x}\,\left (\sqrt {2}-1\right )} \]
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