Integrand size = 24, antiderivative size = 79 \[ \int \frac {1+x^4}{\sqrt {-x+x^3} \left (-1+x^4\right )} \, dx=\frac {\sqrt {-x+x^3}}{1-x^2}-\frac {1}{4} \arctan \left (\frac {2 \sqrt {-x+x^3}}{-1-2 x+x^2}\right )-\frac {1}{4} \text {arctanh}\left (\frac {-\frac {1}{2}+x+\frac {x^2}{2}}{\sqrt {-x+x^3}}\right ) \]
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Result contains complex when optimal does not.
Time = 0.67 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.51, number of steps used = 18, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2081, 6847, 6857, 228, 1418, 425, 537, 418, 1225, 1713, 212, 209} \[ \int \frac {1+x^4}{\sqrt {-x+x^3} \left (-1+x^4\right )} \, dx=-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \sqrt {x^2-1} \sqrt {x} \arctan \left (\frac {(1+i) \sqrt {x}}{\sqrt {x^2-1}}\right )}{\sqrt {x^3-x}}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \sqrt {x^2-1} \sqrt {x} \text {arctanh}\left (\frac {(1+i) \sqrt {x}}{\sqrt {x^2-1}}\right )}{\sqrt {x^3-x}}-\frac {x}{\sqrt {x^3-x}} \]
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Rule 209
Rule 212
Rule 228
Rule 418
Rule 425
Rule 537
Rule 1225
Rule 1418
Rule 1713
Rule 2081
Rule 6847
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \int \frac {1+x^4}{\sqrt {x} \sqrt {-1+x^2} \left (-1+x^4\right )} \, dx}{\sqrt {-x+x^3}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {-1+x^2}\right ) \text {Subst}\left (\int \frac {1+x^8}{\sqrt {-1+x^4} \left (-1+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {-1+x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{\sqrt {-1+x^4}}+\frac {2}{\sqrt {-1+x^4} \left (-1+x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}}+\frac {\left (4 \sqrt {x} \sqrt {-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^4} \left (-1+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}} \\ & = \frac {\sqrt {2} \sqrt {-1+x} \sqrt {x} \sqrt {1+x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right ),\frac {1}{2}\right )}{\sqrt {-x+x^3}}+\frac {\left (4 \sqrt {x} \sqrt {-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+x^4\right )^{3/2} \left (1+x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}} \\ & = -\frac {x}{\sqrt {-x+x^3}}+\frac {\sqrt {2} \sqrt {-1+x} \sqrt {x} \sqrt {1+x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right ),\frac {1}{2}\right )}{\sqrt {-x+x^3}}+\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \text {Subst}\left (\int \frac {-3-x^4}{\sqrt {-1+x^4} \left (1+x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}} \\ & = -\frac {x}{\sqrt {-x+x^3}}+\frac {\sqrt {2} \sqrt {-1+x} \sqrt {x} \sqrt {1+x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right ),\frac {1}{2}\right )}{\sqrt {-x+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}}-\frac {\left (2 \sqrt {x} \sqrt {-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^4} \left (1+x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}} \\ & = -\frac {x}{\sqrt {-x+x^3}}-\frac {\sqrt {-1+x} \sqrt {x} \sqrt {1+x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right ),\frac {1}{2}\right )}{\sqrt {2} \sqrt {-x+x^3}}+\frac {\sqrt {2} \sqrt {-1+x} \sqrt {x} \sqrt {1+x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right ),\frac {1}{2}\right )}{\sqrt {-x+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-i x^2\right ) \sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+i x^2\right ) \sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}} \\ & = -\frac {x}{\sqrt {-x+x^3}}-\frac {\sqrt {-1+x} \sqrt {x} \sqrt {1+x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right ),\frac {1}{2}\right )}{\sqrt {2} \sqrt {-x+x^3}}+\frac {\sqrt {2} \sqrt {-1+x} \sqrt {x} \sqrt {1+x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right ),\frac {1}{2}\right )}{\sqrt {-x+x^3}}-2 \frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \text {Subst}\left (\int \frac {1-i x^2}{\left (1+i x^2\right ) \sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \text {Subst}\left (\int \frac {1+i x^2}{\left (1-i x^2\right ) \sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x+x^3}} \\ & = -\frac {x}{\sqrt {-x+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1-2 i x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {-1+x^2}}\right )}{2 \sqrt {-x+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1+2 i x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {-1+x^2}}\right )}{2 \sqrt {-x+x^3}} \\ & = -\frac {x}{\sqrt {-x+x^3}}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \sqrt {x} \sqrt {-1+x^2} \arctan \left (\frac {(1+i) \sqrt {x}}{\sqrt {-1+x^2}}\right )}{\sqrt {-x+x^3}}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \sqrt {x} \sqrt {-1+x^2} \text {arctanh}\left (\frac {(1+i) \sqrt {x}}{\sqrt {-1+x^2}}\right )}{\sqrt {-x+x^3}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.23 \[ \int \frac {1+x^4}{\sqrt {-x+x^3} \left (-1+x^4\right )} \, dx=\frac {-4 x-(1-i) \sqrt {x} \sqrt {-1+x^2} \arctan \left (\frac {(1+i) \sqrt {x}}{\sqrt {-1+x^2}}\right )+(1+i) \sqrt {x} \sqrt {-1+x^2} \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-1+x^2}}{\sqrt {x}}\right )}{4 \sqrt {x \left (-1+x^2\right )}} \]
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Time = 7.87 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.32
method | result | size |
risch | \(-\frac {x}{\sqrt {x \left (x^{2}-1\right )}}-\frac {\ln \left (\frac {x^{2}+2 \sqrt {x^{3}-x}+2 x -1}{x}\right )}{8}+\frac {\arctan \left (\frac {\sqrt {x^{3}-x}+x}{x}\right )}{4}+\frac {\ln \left (\frac {x^{2}-2 \sqrt {x^{3}-x}+2 x -1}{x}\right )}{8}+\frac {\arctan \left (\frac {\sqrt {x^{3}-x}-x}{x}\right )}{4}\) | \(104\) |
default | \(\frac {\left (\arctan \left (\frac {\sqrt {x^{3}-x}+x}{x}\right )-\frac {\ln \left (\frac {x^{2}+2 \sqrt {x^{3}-x}+2 x -1}{x}\right )}{2}+\frac {\ln \left (\frac {x^{2}-2 \sqrt {x^{3}-x}+2 x -1}{x}\right )}{2}+\arctan \left (\frac {\sqrt {x^{3}-x}-x}{x}\right )\right ) \sqrt {x^{3}-x}-4 x}{4 \sqrt {x^{3}-x}}\) | \(113\) |
pseudoelliptic | \(\frac {\left (\arctan \left (\frac {\sqrt {x^{3}-x}+x}{x}\right )-\frac {\ln \left (\frac {x^{2}+2 \sqrt {x^{3}-x}+2 x -1}{x}\right )}{2}+\frac {\ln \left (\frac {x^{2}-2 \sqrt {x^{3}-x}+2 x -1}{x}\right )}{2}+\arctan \left (\frac {\sqrt {x^{3}-x}-x}{x}\right )\right ) \sqrt {x^{3}-x}-4 x}{4 \sqrt {x^{3}-x}}\) | \(113\) |
elliptic | \(-\frac {x}{\sqrt {x \left (x^{2}-1\right )}}+\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticF}\left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {x^{3}-x}}-\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticPi}\left (\sqrt {1+x}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}-\frac {i \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticPi}\left (\sqrt {1+x}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}-\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticPi}\left (\sqrt {1+x}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}+\frac {i \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticPi}\left (\sqrt {1+x}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}\) | \(223\) |
trager | \(-\frac {\sqrt {x^{3}-x}}{x^{2}-1}+\frac {\ln \left (-\frac {300672 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2} x^{2}-451008 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2} x -11052 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) x^{2}+14400 \sqrt {x^{3}-x}\, \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )-300672 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2}+3528 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) x +91 x^{2}-150 \sqrt {x^{3}-x}+11052 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )+26 x -91}{{\left (72 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) x -144 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )+x +3\right )}^{2}}\right )}{4}-9 \ln \left (-\frac {300672 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2} x^{2}-451008 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2} x -11052 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) x^{2}+14400 \sqrt {x^{3}-x}\, \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )-300672 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2}+3528 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) x +91 x^{2}-150 \sqrt {x^{3}-x}+11052 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )+26 x -91}{{\left (72 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) x -144 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )+x +3\right )}^{2}}\right ) \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )+9 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) \ln \left (\frac {-150336 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2} x^{2}+225504 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2} x +2826 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) x^{2}+7200 \sqrt {x^{3}-x}\, \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )+150336 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2}-10764 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) x -8 x^{2}-125 \sqrt {x^{3}-x}-2826 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )+112 x +8}{{\left (72 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) x -144 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )-3 x +1\right )}^{2}}\right )\) | \(559\) |
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Time = 0.28 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.33 \[ \int \frac {1+x^4}{\sqrt {-x+x^3} \left (-1+x^4\right )} \, dx=\frac {2 \, {\left (x^{2} - 1\right )} \arctan \left (\frac {x^{2} - 2 \, x - 1}{2 \, \sqrt {x^{3} - x}}\right ) + {\left (x^{2} - 1\right )} \log \left (\frac {x^{4} + 8 \, x^{3} + 2 \, x^{2} - 4 \, \sqrt {x^{3} - x} {\left (x^{2} + 2 \, x - 1\right )} - 8 \, x + 1}{x^{4} + 2 \, x^{2} + 1}\right ) - 8 \, \sqrt {x^{3} - x}}{8 \, {\left (x^{2} - 1\right )}} \]
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\[ \int \frac {1+x^4}{\sqrt {-x+x^3} \left (-1+x^4\right )} \, dx=\int \frac {x^{4} + 1}{\sqrt {x \left (x - 1\right ) \left (x + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \]
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\[ \int \frac {1+x^4}{\sqrt {-x+x^3} \left (-1+x^4\right )} \, dx=\int { \frac {x^{4} + 1}{{\left (x^{4} - 1\right )} \sqrt {x^{3} - x}} \,d x } \]
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\[ \int \frac {1+x^4}{\sqrt {-x+x^3} \left (-1+x^4\right )} \, dx=\int { \frac {x^{4} + 1}{{\left (x^{4} - 1\right )} \sqrt {x^{3} - x}} \,d x } \]
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Time = 0.05 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.94 \[ \int \frac {1+x^4}{\sqrt {-x+x^3} \left (-1+x^4\right )} \, dx=\frac {\sqrt {-x}\,\left (\frac {\sin \left (2\,\mathrm {asin}\left (\sqrt {-x}\right )\right )}{4\,\sqrt {1-x}}+\frac {\mathrm {E}\left (\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{2}\right )\,\sqrt {1-x}\,\sqrt {x+1}}{\sqrt {x^3-x}}+\frac {\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (-\mathrm {i};\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{\sqrt {x^3-x}}+\frac {\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (1{}\mathrm {i};\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{\sqrt {x^3-x}}-\frac {2\,\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{\sqrt {x^3-x}}+\frac {\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\left (\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )-\frac {\mathrm {E}\left (\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{2}+\frac {\sqrt {-x}\,\sqrt {1-x}}{2\,\sqrt {x+1}}\right )}{\sqrt {x^3-x}} \]
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