Integrand size = 46, antiderivative size = 177 \[ \int \frac {1}{x^2 \sqrt [3]{-1-x+5 x^2+2 x^3-10 x^4+2 x^5+7 x^6-5 x^7+x^8}} \, dx=\frac {(-1+x)^{2/3} \left (-1-x+x^2\right ) \left (-\frac {\sqrt [3]{-1+x}}{x}-\frac {\arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (1+\sqrt [3]{-1+x}\right )-\frac {1}{6} \log \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right )-\text {RootSum}\left [-1+\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{-1+x}-\text {$\#$1}\right )+\log \left (\sqrt [3]{-1+x}-\text {$\#$1}\right ) \text {$\#$1}^3}{\text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ]\right )}{\sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3}} \]
[Out]
Leaf count is larger than twice the leaf count of optimal. \(892\) vs. \(2(177)=354\).
Time = 0.57 (sec) , antiderivative size = 892, normalized size of antiderivative = 5.04, number of steps used = 25, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.283, Rules used = {6820, 6851, 911, 1438, 652, 632, 210, 648, 642, 1436, 206, 31, 631} \[ \int \frac {1}{x^2 \sqrt [3]{-1-x+5 x^2+2 x^3-10 x^4+2 x^5+7 x^6-5 x^7+x^8}} \, dx=\frac {(x-1)^{2/3} \arctan \left (\frac {1-2 \sqrt [3]{x-1}}{\sqrt {3}}\right ) \left (-x^2+x+1\right )}{\sqrt {3} \sqrt [3]{-(1-x)^2 \left (-x^2+x+1\right )^3}}+\frac {\sqrt {\frac {3}{5}} \sqrt [3]{\frac {1}{2} \left (7-3 \sqrt {5}\right )} (x-1)^{2/3} \arctan \left (\frac {2 \sqrt [3]{\frac {2}{-1+\sqrt {5}}} \sqrt [3]{x-1}+1}{\sqrt {3}}\right ) \left (-x^2+x+1\right )}{\sqrt [3]{-(1-x)^2 \left (-x^2+x+1\right )^3}}-\frac {\sqrt {\frac {3}{5}} \sqrt [3]{\frac {1}{2} \left (7+3 \sqrt {5}\right )} (x-1)^{2/3} \arctan \left (\frac {1-2 \sqrt [3]{\frac {2}{1+\sqrt {5}}} \sqrt [3]{x-1}}{\sqrt {3}}\right ) \left (-x^2+x+1\right )}{\sqrt [3]{-(1-x)^2 \left (-x^2+x+1\right )^3}}-\frac {(x-1)^{2/3} \log \left (\sqrt [3]{x-1}+1\right ) \left (-x^2+x+1\right )}{3 \sqrt [3]{-(1-x)^2 \left (-x^2+x+1\right )^3}}-\frac {\sqrt [3]{\frac {1}{2} \left (7-3 \sqrt {5}\right )} (x-1)^{2/3} \log \left (\sqrt [3]{-1+\sqrt {5}}-\sqrt [3]{2} \sqrt [3]{x-1}\right ) \left (-x^2+x+1\right )}{\sqrt {5} \sqrt [3]{-(1-x)^2 \left (-x^2+x+1\right )^3}}+\frac {\sqrt [3]{\frac {1}{2} \left (7+3 \sqrt {5}\right )} (x-1)^{2/3} \log \left (\sqrt [3]{2} \sqrt [3]{x-1}+\sqrt [3]{1+\sqrt {5}}\right ) \left (-x^2+x+1\right )}{\sqrt {5} \sqrt [3]{-(1-x)^2 \left (-x^2+x+1\right )^3}}+\frac {(x-1)^{2/3} \log \left ((x-1)^{2/3}-\sqrt [3]{x-1}+1\right ) \left (-x^2+x+1\right )}{6 \sqrt [3]{-(1-x)^2 \left (-x^2+x+1\right )^3}}+\frac {\sqrt [3]{\frac {1}{2} \left (7-3 \sqrt {5}\right )} (x-1)^{2/3} \log \left (2^{2/3} (x-1)^{2/3}+\sqrt [3]{2 \left (-1+\sqrt {5}\right )} \sqrt [3]{x-1}+\left (-1+\sqrt {5}\right )^{2/3}\right ) \left (-x^2+x+1\right )}{2 \sqrt {5} \sqrt [3]{-(1-x)^2 \left (-x^2+x+1\right )^3}}-\frac {\sqrt [3]{\frac {1}{2} \left (7+3 \sqrt {5}\right )} (x-1)^{2/3} \log \left (2^{2/3} (x-1)^{2/3}-\sqrt [3]{2 \left (1+\sqrt {5}\right )} \sqrt [3]{x-1}+\left (1+\sqrt {5}\right )^{2/3}\right ) \left (-x^2+x+1\right )}{2 \sqrt {5} \sqrt [3]{-(1-x)^2 \left (-x^2+x+1\right )^3}}-\frac {(x-1)^{2/3} \left (-x^2+x+1\right )}{3 \left (\sqrt [3]{x-1}+1\right ) \sqrt [3]{-(1-x)^2 \left (-x^2+x+1\right )^3}}+\frac {\left (\sqrt [3]{x-1}+1\right ) (x-1)^{2/3} \left (-x^2+x+1\right )}{3 \left ((x-1)^{2/3}-\sqrt [3]{x-1}+1\right ) \sqrt [3]{-(1-x)^2 \left (-x^2+x+1\right )^3}} \]
[In]
[Out]
Rule 31
Rule 206
Rule 210
Rule 631
Rule 632
Rule 642
Rule 648
Rule 652
Rule 911
Rule 1436
Rule 1438
Rule 6820
Rule 6851
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^2 \sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3}} \, dx \\ & = \frac {\left ((-1+x)^{2/3} \left (-1-x+x^2\right )\right ) \int \frac {1}{(-1+x)^{2/3} x^2 \left (-1-x+x^2\right )} \, dx}{\sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3}} \\ & = \frac {\left (3 (-1+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {1}{\left (1+x^3\right )^2 \left (-1+x^3+x^6\right )} \, dx,x,\sqrt [3]{-1+x}\right )}{\sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3}} \\ & = \frac {\left (3 (-1+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \left (-\frac {1}{9 (1+x)^2}+\frac {1}{9 (1+x)}+\frac {-1+x}{3 \left (1-x+x^2\right )^2}+\frac {3-x}{9 \left (1-x+x^2\right )}+\frac {1-x^3}{-1+x^3+x^6}\right ) \, dx,x,\sqrt [3]{-1+x}\right )}{\sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3}} \\ & = -\frac {(-1+x)^{2/3} \left (1+x-x^2\right )}{3 \left (1+\sqrt [3]{-1+x}\right ) \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}-\frac {(-1+x)^{2/3} \left (1+x-x^2\right ) \log \left (1+\sqrt [3]{-1+x}\right )}{3 \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}+\frac {\left ((-1+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {3-x}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x}\right )}{3 \sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3}}+\frac {\left ((-1+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {-1+x}{\left (1-x+x^2\right )^2} \, dx,x,\sqrt [3]{-1+x}\right )}{\sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3}}+\frac {\left (3 (-1+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {1-x^3}{-1+x^3+x^6} \, dx,x,\sqrt [3]{-1+x}\right )}{\sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3}} \\ & = -\frac {(-1+x)^{2/3} \left (1+x-x^2\right )}{3 \left (1+\sqrt [3]{-1+x}\right ) \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}+\frac {\left (1+\sqrt [3]{-1+x}\right ) (-1+x)^{2/3} \left (1+x-x^2\right )}{3 \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right ) \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}-\frac {(-1+x)^{2/3} \left (1+x-x^2\right ) \log \left (1+\sqrt [3]{-1+x}\right )}{3 \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}-\frac {\left ((-1+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x}\right )}{6 \sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3}}-\frac {\left ((-1+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x}\right )}{3 \sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3}}+\frac {\left (5 (-1+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x}\right )}{6 \sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3}}+\frac {\left (3 \left (-5+3 \sqrt {5}\right ) (-1+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2}-\frac {\sqrt {5}}{2}+x^3} \, dx,x,\sqrt [3]{-1+x}\right )}{10 \sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3}}-\frac {\left (3 \left (5+3 \sqrt {5}\right ) (-1+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2}+\frac {\sqrt {5}}{2}+x^3} \, dx,x,\sqrt [3]{-1+x}\right )}{10 \sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3}} \\ & = -\frac {(-1+x)^{2/3} \left (1+x-x^2\right )}{3 \left (1+\sqrt [3]{-1+x}\right ) \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}+\frac {\left (1+\sqrt [3]{-1+x}\right ) (-1+x)^{2/3} \left (1+x-x^2\right )}{3 \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right ) \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}-\frac {(-1+x)^{2/3} \left (1+x-x^2\right ) \log \left (1+\sqrt [3]{-1+x}\right )}{3 \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}+\frac {(-1+x)^{2/3} \left (1+x-x^2\right ) \log \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right )}{6 \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}+\frac {\left (2 (-1+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x}\right )}{3 \sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3}}-\frac {\left (5 (-1+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x}\right )}{3 \sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3}}+\frac {\left (\left (-5+3 \sqrt {5}\right ) (-1+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {1}{-\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {5}\right )}+x} \, dx,x,\sqrt [3]{-1+x}\right )}{5 \sqrt [3]{2} \left (-1+\sqrt {5}\right )^{2/3} \sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3}}+\frac {\left (\left (-5+3 \sqrt {5}\right ) (-1+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {-2^{2/3} \sqrt [3]{-1+\sqrt {5}}-x}{\left (\frac {1}{2} \left (-1+\sqrt {5}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {5}\right )} x+x^2} \, dx,x,\sqrt [3]{-1+x}\right )}{5 \sqrt [3]{2} \left (-1+\sqrt {5}\right )^{2/3} \sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3}}-\frac {\left (\left (5+3 \sqrt {5}\right ) (-1+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x} \, dx,x,\sqrt [3]{-1+x}\right )}{5 \sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3} \sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3}}-\frac {\left (\left (5+3 \sqrt {5}\right ) (-1+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {2^{2/3} \sqrt [3]{1+\sqrt {5}}-x}{\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )} x+x^2} \, dx,x,\sqrt [3]{-1+x}\right )}{5 \sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3} \sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3}} \\ & = -\frac {(-1+x)^{2/3} \left (1+x-x^2\right )}{3 \left (1+\sqrt [3]{-1+x}\right ) \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}+\frac {\left (1+\sqrt [3]{-1+x}\right ) (-1+x)^{2/3} \left (1+x-x^2\right )}{3 \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right ) \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}+\frac {(-1+x)^{2/3} \left (1+x-x^2\right ) \arctan \left (\frac {1-2 \sqrt [3]{-1+x}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}-\frac {(-1+x)^{2/3} \left (1+x-x^2\right ) \log \left (1+\sqrt [3]{-1+x}\right )}{3 \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}-\frac {\sqrt [3]{\frac {1}{2} \left (7-3 \sqrt {5}\right )} (-1+x)^{2/3} \left (1+x-x^2\right ) \log \left (\sqrt [3]{-1+\sqrt {5}}-\sqrt [3]{2} \sqrt [3]{-1+x}\right )}{\sqrt {5} \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}+\frac {\sqrt [3]{\frac {1}{2} \left (7+3 \sqrt {5}\right )} (-1+x)^{2/3} \left (1+x-x^2\right ) \log \left (\sqrt [3]{1+\sqrt {5}}+\sqrt [3]{2} \sqrt [3]{-1+x}\right )}{\sqrt {5} \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}+\frac {(-1+x)^{2/3} \left (1+x-x^2\right ) \log \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right )}{6 \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}-\frac {\left (\left (-5+3 \sqrt {5}\right ) (-1+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {5}\right )}+2 x}{\left (\frac {1}{2} \left (-1+\sqrt {5}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {5}\right )} x+x^2} \, dx,x,\sqrt [3]{-1+x}\right )}{10 \sqrt [3]{2} \left (-1+\sqrt {5}\right )^{2/3} \sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3}}-\frac {\left (3 \left (-5+3 \sqrt {5}\right ) (-1+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {1}{\left (\frac {1}{2} \left (-1+\sqrt {5}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {5}\right )} x+x^2} \, dx,x,\sqrt [3]{-1+x}\right )}{10\ 2^{2/3} \sqrt [3]{-1+\sqrt {5}} \sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3}}+\frac {\left (\left (5+3 \sqrt {5}\right ) (-1+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+2 x}{\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )} x+x^2} \, dx,x,\sqrt [3]{-1+x}\right )}{10 \sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3} \sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3}}-\frac {\left (3 \left (5+3 \sqrt {5}\right ) (-1+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {1}{\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )} x+x^2} \, dx,x,\sqrt [3]{-1+x}\right )}{10\ 2^{2/3} \sqrt [3]{1+\sqrt {5}} \sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3}} \\ & = -\frac {(-1+x)^{2/3} \left (1+x-x^2\right )}{3 \left (1+\sqrt [3]{-1+x}\right ) \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}+\frac {\left (1+\sqrt [3]{-1+x}\right ) (-1+x)^{2/3} \left (1+x-x^2\right )}{3 \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right ) \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}+\frac {(-1+x)^{2/3} \left (1+x-x^2\right ) \arctan \left (\frac {1-2 \sqrt [3]{-1+x}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}-\frac {(-1+x)^{2/3} \left (1+x-x^2\right ) \log \left (1+\sqrt [3]{-1+x}\right )}{3 \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}-\frac {\sqrt [3]{\frac {1}{2} \left (7-3 \sqrt {5}\right )} (-1+x)^{2/3} \left (1+x-x^2\right ) \log \left (\sqrt [3]{-1+\sqrt {5}}-\sqrt [3]{2} \sqrt [3]{-1+x}\right )}{\sqrt {5} \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}+\frac {\sqrt [3]{\frac {1}{2} \left (7+3 \sqrt {5}\right )} (-1+x)^{2/3} \left (1+x-x^2\right ) \log \left (\sqrt [3]{1+\sqrt {5}}+\sqrt [3]{2} \sqrt [3]{-1+x}\right )}{\sqrt {5} \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}+\frac {(-1+x)^{2/3} \left (1+x-x^2\right ) \log \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right )}{6 \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}+\frac {\sqrt [3]{\frac {1}{2} \left (7-3 \sqrt {5}\right )} (-1+x)^{2/3} \left (1+x-x^2\right ) \log \left (\left (-1+\sqrt {5}\right )^{2/3}+\sqrt [3]{2 \left (-1+\sqrt {5}\right )} \sqrt [3]{-1+x}+2^{2/3} (-1+x)^{2/3}\right )}{2 \sqrt {5} \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}-\frac {\sqrt [3]{\frac {1}{2} \left (7+3 \sqrt {5}\right )} (-1+x)^{2/3} \left (1+x-x^2\right ) \log \left (\left (1+\sqrt {5}\right )^{2/3}-\sqrt [3]{2 \left (1+\sqrt {5}\right )} \sqrt [3]{-1+x}+2^{2/3} (-1+x)^{2/3}\right )}{2 \sqrt {5} \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}+\frac {\left (3 \left (-5+3 \sqrt {5}\right ) (-1+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{\frac {2}{-1+\sqrt {5}}} \sqrt [3]{-1+x}\right )}{5 \sqrt [3]{2} \left (-1+\sqrt {5}\right )^{2/3} \sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3}}-\frac {\left (3 \left (5+3 \sqrt {5}\right ) (-1+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2 \sqrt [3]{\frac {2}{1+\sqrt {5}}} \sqrt [3]{-1+x}\right )}{5 \sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3} \sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3}} \\ & = -\frac {(-1+x)^{2/3} \left (1+x-x^2\right )}{3 \left (1+\sqrt [3]{-1+x}\right ) \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}+\frac {\left (1+\sqrt [3]{-1+x}\right ) (-1+x)^{2/3} \left (1+x-x^2\right )}{3 \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right ) \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}+\frac {(-1+x)^{2/3} \left (1+x-x^2\right ) \arctan \left (\frac {1-2 \sqrt [3]{-1+x}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}+\frac {\sqrt {\frac {3}{5}} \sqrt [3]{\frac {1}{2} \left (7-3 \sqrt {5}\right )} (-1+x)^{2/3} \left (1+x-x^2\right ) \arctan \left (\frac {1+2 \sqrt [3]{\frac {2}{-1+\sqrt {5}}} \sqrt [3]{-1+x}}{\sqrt {3}}\right )}{\sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}-\frac {\sqrt {\frac {3}{5}} \sqrt [3]{\frac {1}{2} \left (7+3 \sqrt {5}\right )} (-1+x)^{2/3} \left (1+x-x^2\right ) \arctan \left (\frac {1-2 \sqrt [3]{\frac {2}{1+\sqrt {5}}} \sqrt [3]{-1+x}}{\sqrt {3}}\right )}{\sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}-\frac {(-1+x)^{2/3} \left (1+x-x^2\right ) \log \left (1+\sqrt [3]{-1+x}\right )}{3 \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}-\frac {\sqrt [3]{\frac {1}{2} \left (7-3 \sqrt {5}\right )} (-1+x)^{2/3} \left (1+x-x^2\right ) \log \left (\sqrt [3]{-1+\sqrt {5}}-\sqrt [3]{2} \sqrt [3]{-1+x}\right )}{\sqrt {5} \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}+\frac {\sqrt [3]{\frac {1}{2} \left (7+3 \sqrt {5}\right )} (-1+x)^{2/3} \left (1+x-x^2\right ) \log \left (\sqrt [3]{1+\sqrt {5}}+\sqrt [3]{2} \sqrt [3]{-1+x}\right )}{\sqrt {5} \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}+\frac {(-1+x)^{2/3} \left (1+x-x^2\right ) \log \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right )}{6 \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}+\frac {\sqrt [3]{\frac {1}{2} \left (7-3 \sqrt {5}\right )} (-1+x)^{2/3} \left (1+x-x^2\right ) \log \left (\left (-1+\sqrt {5}\right )^{2/3}+\sqrt [3]{2 \left (-1+\sqrt {5}\right )} \sqrt [3]{-1+x}+2^{2/3} (-1+x)^{2/3}\right )}{2 \sqrt {5} \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}-\frac {\sqrt [3]{\frac {1}{2} \left (7+3 \sqrt {5}\right )} (-1+x)^{2/3} \left (1+x-x^2\right ) \log \left (\left (1+\sqrt {5}\right )^{2/3}-\sqrt [3]{2 \left (1+\sqrt {5}\right )} \sqrt [3]{-1+x}+2^{2/3} (-1+x)^{2/3}\right )}{2 \sqrt {5} \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^2 \sqrt [3]{-1-x+5 x^2+2 x^3-10 x^4+2 x^5+7 x^6-5 x^7+x^8}} \, dx=-\frac {\left (-1-x+x^2\right ) \left (-6+6 x+2 \sqrt {3} (-1+x)^{2/3} x \arctan \left (\frac {1-2 \sqrt [3]{-1+x}}{\sqrt {3}}\right )-2 (-1+x)^{2/3} x \log \left (1+\sqrt [3]{-1+x}\right )+(-1+x)^{2/3} x \log \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right )+6 (-1+x)^{2/3} x \text {RootSum}\left [-1+\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{-1+x}-\text {$\#$1}\right )+\log \left (\sqrt [3]{-1+x}-\text {$\#$1}\right ) \text {$\#$1}^3}{\text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ]\right )}{6 x \sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3}} \]
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Time = 56.36 (sec) , antiderivative size = 19808, normalized size of antiderivative = 111.91
method | result | size |
risch | \(\text {Expression too large to display}\) | \(19808\) |
trager | \(\text {Expression too large to display}\) | \(88405\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.28 (sec) , antiderivative size = 1492, normalized size of antiderivative = 8.43 \[ \int \frac {1}{x^2 \sqrt [3]{-1-x+5 x^2+2 x^3-10 x^4+2 x^5+7 x^6-5 x^7+x^8}} \, dx=\text {Too large to display} \]
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Not integrable
Time = 1.13 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.12 \[ \int \frac {1}{x^2 \sqrt [3]{-1-x+5 x^2+2 x^3-10 x^4+2 x^5+7 x^6-5 x^7+x^8}} \, dx=\int \frac {1}{x^{2} \sqrt [3]{\left (x - 1\right )^{2} \left (x^{2} - x - 1\right )^{3}}}\, dx \]
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Not integrable
Time = 0.20 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.26 \[ \int \frac {1}{x^2 \sqrt [3]{-1-x+5 x^2+2 x^3-10 x^4+2 x^5+7 x^6-5 x^7+x^8}} \, dx=\int { \frac {1}{{\left (x^{8} - 5 \, x^{7} + 7 \, x^{6} + 2 \, x^{5} - 10 \, x^{4} + 2 \, x^{3} + 5 \, x^{2} - x - 1\right )}^{\frac {1}{3}} x^{2}} \,d x } \]
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Not integrable
Time = 0.31 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.26 \[ \int \frac {1}{x^2 \sqrt [3]{-1-x+5 x^2+2 x^3-10 x^4+2 x^5+7 x^6-5 x^7+x^8}} \, dx=\int { \frac {1}{{\left (x^{8} - 5 \, x^{7} + 7 \, x^{6} + 2 \, x^{5} - 10 \, x^{4} + 2 \, x^{3} + 5 \, x^{2} - x - 1\right )}^{\frac {1}{3}} x^{2}} \,d x } \]
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Not integrable
Time = 6.23 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.26 \[ \int \frac {1}{x^2 \sqrt [3]{-1-x+5 x^2+2 x^3-10 x^4+2 x^5+7 x^6-5 x^7+x^8}} \, dx=\int \frac {1}{x^2\,{\left (x^8-5\,x^7+7\,x^6+2\,x^5-10\,x^4+2\,x^3+5\,x^2-x-1\right )}^{1/3}} \,d x \]
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