Integrand size = 34, antiderivative size = 67 \[ \int \frac {3-3 x^2+2 x^4}{\sqrt [4]{-1+x^2} \left (2-3 x^2+x^4\right )} \, dx=\frac {4 x}{\sqrt [4]{-1+x^2}}+\frac {5 \arctan \left (\frac {\sqrt {2} \sqrt [4]{-1+x^2}}{x}\right )}{2 \sqrt {2}}-\frac {5 \text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{2 \sqrt {2}} \]
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Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.97, number of steps used = 20, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {1702, 6857, 205, 236, 311, 226, 1210, 294, 412, 407} \[ \int \frac {3-3 x^2+2 x^4}{\sqrt [4]{-1+x^2} \left (2-3 x^2+x^4\right )} \, dx=-\frac {5 \arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}}-\frac {5 \text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}}+\frac {4 x}{\sqrt [4]{x^2-1}} \]
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Rule 205
Rule 226
Rule 236
Rule 294
Rule 311
Rule 407
Rule 412
Rule 1210
Rule 1702
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \frac {3-3 x^2+2 x^4}{\left (-2+x^2\right ) \left (-1+x^2\right )^{5/4}} \, dx \\ & = \int \left (\frac {1}{\left (-1+x^2\right )^{5/4}}+\frac {2 x^2}{\left (-1+x^2\right )^{5/4}}+\frac {5}{\left (-2+x^2\right ) \left (-1+x^2\right )^{5/4}}\right ) \, dx \\ & = 2 \int \frac {x^2}{\left (-1+x^2\right )^{5/4}} \, dx+5 \int \frac {1}{\left (-2+x^2\right ) \left (-1+x^2\right )^{5/4}} \, dx+\int \frac {1}{\left (-1+x^2\right )^{5/4}} \, dx \\ & = -\frac {6 x}{\sqrt [4]{-1+x^2}}+4 \int \frac {1}{\sqrt [4]{-1+x^2}} \, dx-5 \int \frac {1}{\left (-1+x^2\right )^{5/4}} \, dx+5 \int \frac {1}{\left (-2+x^2\right ) \sqrt [4]{-1+x^2}} \, dx+\int \frac {1}{\sqrt [4]{-1+x^2}} \, dx \\ & = \frac {4 x}{\sqrt [4]{-1+x^2}}-\frac {5 \arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{2 \sqrt {2}}-\frac {5 \text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{2 \sqrt {2}}-5 \int \frac {1}{\sqrt [4]{-1+x^2}} \, dx+\frac {\left (2 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+x^2}\right )}{x}+\frac {\left (8 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+x^2}\right )}{x} \\ & = \frac {4 x}{\sqrt [4]{-1+x^2}}-\frac {5 \arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{2 \sqrt {2}}-\frac {5 \text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{2 \sqrt {2}}+\frac {\left (2 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+x^2}\right )}{x}-\frac {\left (2 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+x^2}\right )}{x}+\frac {\left (8 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+x^2}\right )}{x}-\frac {\left (8 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+x^2}\right )}{x}-\frac {\left (10 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+x^2}\right )}{x} \\ & = \frac {4 x}{\sqrt [4]{-1+x^2}}+\frac {10 x \sqrt [4]{-1+x^2}}{1+\sqrt {-1+x^2}}-\frac {5 \arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{2 \sqrt {2}}-\frac {5 \text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{2 \sqrt {2}}-\frac {10 \sqrt {\frac {x^2}{\left (1+\sqrt {-1+x^2}\right )^2}} \left (1+\sqrt {-1+x^2}\right ) E\left (2 \arctan \left (\sqrt [4]{-1+x^2}\right )|\frac {1}{2}\right )}{x}+\frac {5 \sqrt {\frac {x^2}{\left (1+\sqrt {-1+x^2}\right )^2}} \left (1+\sqrt {-1+x^2}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{-1+x^2}\right ),\frac {1}{2}\right )}{x}-\frac {\left (10 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+x^2}\right )}{x}+\frac {\left (10 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+x^2}\right )}{x} \\ & = \frac {4 x}{\sqrt [4]{-1+x^2}}-\frac {5 \arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{2 \sqrt {2}}-\frac {5 \text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{2 \sqrt {2}} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00 \[ \int \frac {3-3 x^2+2 x^4}{\sqrt [4]{-1+x^2} \left (2-3 x^2+x^4\right )} \, dx=\frac {4 x}{\sqrt [4]{-1+x^2}}+\frac {5 \arctan \left (\frac {\sqrt {2} \sqrt [4]{-1+x^2}}{x}\right )}{2 \sqrt {2}}-\frac {5 \text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{2 \sqrt {2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.02 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.90
method | result | size |
trager | \(\frac {4 x}{\left (x^{2}-1\right )^{\frac {1}{4}}}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {\left (x^{2}-1\right )^{\frac {3}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )-x \sqrt {x^{2}-1}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \left (x^{2}-1\right )^{\frac {1}{4}}+x}{x^{2}-2}\right )}{4}-\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {\left (x^{2}-1\right )^{\frac {3}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )+x \sqrt {x^{2}-1}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \left (x^{2}-1\right )^{\frac {1}{4}}+x}{x^{2}-2}\right )}{4}\) | \(127\) |
risch | \(\frac {4 x}{\left (x^{2}-1\right )^{\frac {1}{4}}}-\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\left (x^{2}-1\right )^{\frac {3}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )+x \sqrt {x^{2}-1}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \left (x^{2}-1\right )^{\frac {1}{4}}-x}{x^{2}-2}\right )}{4}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {\left (x^{2}-1\right )^{\frac {3}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )-x \sqrt {x^{2}-1}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \left (x^{2}-1\right )^{\frac {1}{4}}-x}{x^{2}-2}\right )}{4}\) | \(131\) |
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Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (50) = 100\).
Time = 0.27 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.79 \[ \int \frac {3-3 x^2+2 x^4}{\sqrt [4]{-1+x^2} \left (2-3 x^2+x^4\right )} \, dx=\frac {10 \, \sqrt {2} {\left (x^{2} - 1\right )} \arctan \left (\frac {\sqrt {2} {\left (x^{2} - 1\right )}^{\frac {1}{4}}}{x}\right ) + 5 \, \sqrt {2} {\left (x^{2} - 1\right )} \log \left (-\frac {x^{4} - 2 \, \sqrt {2} {\left (x^{2} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{2} - 1} x^{2} - 4 \, \sqrt {2} {\left (x^{2} - 1\right )}^{\frac {3}{4}} x + 4 \, x^{2} - 4}{x^{4} - 4 \, x^{2} + 4}\right ) + 32 \, {\left (x^{2} - 1\right )}^{\frac {3}{4}} x}{8 \, {\left (x^{2} - 1\right )}} \]
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\[ \int \frac {3-3 x^2+2 x^4}{\sqrt [4]{-1+x^2} \left (2-3 x^2+x^4\right )} \, dx=\int \frac {2 x^{4} - 3 x^{2} + 3}{\sqrt [4]{\left (x - 1\right ) \left (x + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - 2\right )}\, dx \]
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\[ \int \frac {3-3 x^2+2 x^4}{\sqrt [4]{-1+x^2} \left (2-3 x^2+x^4\right )} \, dx=\int { \frac {2 \, x^{4} - 3 \, x^{2} + 3}{{\left (x^{4} - 3 \, x^{2} + 2\right )} {\left (x^{2} - 1\right )}^{\frac {1}{4}}} \,d x } \]
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\[ \int \frac {3-3 x^2+2 x^4}{\sqrt [4]{-1+x^2} \left (2-3 x^2+x^4\right )} \, dx=\int { \frac {2 \, x^{4} - 3 \, x^{2} + 3}{{\left (x^{4} - 3 \, x^{2} + 2\right )} {\left (x^{2} - 1\right )}^{\frac {1}{4}}} \,d x } \]
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Timed out. \[ \int \frac {3-3 x^2+2 x^4}{\sqrt [4]{-1+x^2} \left (2-3 x^2+x^4\right )} \, dx=\int \frac {2\,x^4-3\,x^2+3}{{\left (x^2-1\right )}^{1/4}\,\left (x^4-3\,x^2+2\right )} \,d x \]
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