Integrand size = 73, antiderivative size = 45 \[ \int \frac {(-1+x)^2 \left (-10-8 x+5 x^2+5 x^3\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right ) \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx=2 \arctan \left (\frac {-1+x}{\sqrt [4]{\frac {1+x}{-2+x^2}}}\right )-2 \text {arctanh}\left (\frac {-1+x}{\sqrt [4]{\frac {1+x}{-2+x^2}}}\right ) \]
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\[ \int \frac {(-1+x)^2 \left (-10-8 x+5 x^2+5 x^3\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right ) \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx=\int \frac {(-1+x)^2 \left (-10-8 x+5 x^2+5 x^3\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right ) \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {(1+x)^{3/4} \int \frac {(-1+x)^2 \left (-10-8 x+5 x^2+5 x^3\right )}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}} \\ & = \frac {(1+x)^{3/4} \int \left (-\frac {10}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )}+\frac {12 x}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )}+\frac {11 x^2}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )}-\frac {13 x^3}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )}-\frac {5 x^4}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )}+\frac {5 x^5}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )}\right ) \, dx}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}} \\ & = -\frac {\left (5 (1+x)^{3/4}\right ) \int \frac {x^4}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (5 (1+x)^{3/4}\right ) \int \frac {x^5}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (10 (1+x)^{3/4}\right ) \int \frac {1}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (11 (1+x)^{3/4}\right ) \int \frac {x^2}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (12 (1+x)^{3/4}\right ) \int \frac {x}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (13 (1+x)^{3/4}\right ) \int \frac {x^3}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}} \\ & = -\frac {\left (20 (1+x)^{3/4}\right ) \text {Subst}\left (\int \frac {\left (-1+x^4\right )^4}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (20 (1+x)^{3/4}\right ) \text {Subst}\left (\int \frac {\left (-1+x^4\right )^5}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (40 (1+x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (44 (1+x)^{3/4}\right ) \text {Subst}\left (\int \frac {\left (-1+x^4\right )^2}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (48 (1+x)^{3/4}\right ) \text {Subst}\left (\int \frac {-1+x^4}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (52 (1+x)^{3/4}\right ) \text {Subst}\left (\int \frac {\left (-1+x^4\right )^3}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}} \\ & = \text {Too large to display} \\ \end{align*}
Time = 10.90 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \frac {(-1+x)^2 \left (-10-8 x+5 x^2+5 x^3\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right ) \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx=2 \arctan \left (\frac {-1+x}{\sqrt [4]{\frac {1+x}{-2+x^2}}}\right )-2 \text {arctanh}\left (\frac {-1+x}{\sqrt [4]{\frac {1+x}{-2+x^2}}}\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.46 (sec) , antiderivative size = 807, normalized size of antiderivative = 17.93
method | result | size |
trager | \(\ln \left (-\frac {2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}} x^{3}-2 \sqrt {-\frac {-1-x}{x^{2}-2}}\, x^{4}+2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{5}-x^{6}-2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}} x^{2}+4 \sqrt {-\frac {-1-x}{x^{2}-2}}\, x^{3}-6 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{4}+4 x^{5}-4 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}} x +2 \sqrt {-\frac {-1-x}{x^{2}-2}}\, x^{2}+2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{3}-4 x^{4}+4 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}}-8 \sqrt {-\frac {-1-x}{x^{2}-2}}\, x +10 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{2}-4 x^{3}+4 \sqrt {-\frac {-1-x}{x^{2}-2}}-12 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x +11 x^{2}+4 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}}-9 x +1}{x^{6}-4 x^{5}+4 x^{4}+4 x^{3}-11 x^{2}+7 x -3}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-\frac {-1-x}{x^{2}-2}}\, x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}+2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}} x^{3}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-\frac {-1-x}{x^{2}-2}}\, x^{3}-2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{5}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{5}-2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}} x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-\frac {-1-x}{x^{2}-2}}\, x^{2}+6 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{4}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}-4 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}} x -8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-\frac {-1-x}{x^{2}-2}}\, x -2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{3}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+4 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-\frac {-1-x}{x^{2}-2}}-10 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{2}-11 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+12 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x +9 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -4 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{6}-4 x^{5}+4 x^{4}+4 x^{3}-11 x^{2}+7 x -3}\right )\) | \(807\) |
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Leaf count of result is larger than twice the leaf count of optimal. 256 vs. \(2 (41) = 82\).
Time = 29.11 (sec) , antiderivative size = 256, normalized size of antiderivative = 5.69 \[ \int \frac {(-1+x)^2 \left (-10-8 x+5 x^2+5 x^3\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right ) \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx=-\arctan \left (\frac {2 \, {\left ({\left (x^{3} - x^{2} - 2 \, x + 2\right )} \left (\frac {x + 1}{x^{2} - 2}\right )^{\frac {3}{4}} + {\left (x^{5} - 3 \, x^{4} + x^{3} + 5 \, x^{2} - 6 \, x + 2\right )} \left (\frac {x + 1}{x^{2} - 2}\right )^{\frac {1}{4}}\right )}}{x^{6} - 4 \, x^{5} + 4 \, x^{4} + 4 \, x^{3} - 11 \, x^{2} + 7 \, x - 3}\right ) + \log \left (\frac {x^{6} - 4 \, x^{5} + 4 \, x^{4} + 4 \, x^{3} - 11 \, x^{2} - 2 \, {\left (x^{3} - x^{2} - 2 \, x + 2\right )} \left (\frac {x + 1}{x^{2} - 2}\right )^{\frac {3}{4}} + 2 \, {\left (x^{4} - 2 \, x^{3} - x^{2} + 4 \, x - 2\right )} \sqrt {\frac {x + 1}{x^{2} - 2}} - 2 \, {\left (x^{5} - 3 \, x^{4} + x^{3} + 5 \, x^{2} - 6 \, x + 2\right )} \left (\frac {x + 1}{x^{2} - 2}\right )^{\frac {1}{4}} + 9 \, x - 1}{x^{6} - 4 \, x^{5} + 4 \, x^{4} + 4 \, x^{3} - 11 \, x^{2} + 7 \, x - 3}\right ) \]
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Timed out. \[ \int \frac {(-1+x)^2 \left (-10-8 x+5 x^2+5 x^3\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right ) \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {(-1+x)^2 \left (-10-8 x+5 x^2+5 x^3\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right ) \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx=\int { \frac {{\left (5 \, x^{3} + 5 \, x^{2} - 8 \, x - 10\right )} {\left (x - 1\right )}^{2}}{{\left (x^{6} - 4 \, x^{5} + 4 \, x^{4} + 4 \, x^{3} - 11 \, x^{2} + 7 \, x - 3\right )} {\left (x^{2} - 2\right )} \left (\frac {x + 1}{x^{2} - 2}\right )^{\frac {3}{4}}} \,d x } \]
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\[ \int \frac {(-1+x)^2 \left (-10-8 x+5 x^2+5 x^3\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right ) \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx=\int { \frac {{\left (5 \, x^{3} + 5 \, x^{2} - 8 \, x - 10\right )} {\left (x - 1\right )}^{2}}{{\left (x^{6} - 4 \, x^{5} + 4 \, x^{4} + 4 \, x^{3} - 11 \, x^{2} + 7 \, x - 3\right )} {\left (x^{2} - 2\right )} \left (\frac {x + 1}{x^{2} - 2}\right )^{\frac {3}{4}}} \,d x } \]
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Timed out. \[ \int \frac {(-1+x)^2 \left (-10-8 x+5 x^2+5 x^3\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right ) \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx=\int -\frac {{\left (x-1\right )}^2\,\left (-5\,x^3-5\,x^2+8\,x+10\right )}{\left (x^2-2\right )\,{\left (\frac {x+1}{x^2-2}\right )}^{3/4}\,\left (x^6-4\,x^5+4\,x^4+4\,x^3-11\,x^2+7\,x-3\right )} \,d x \]
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