Integrand size = 42, antiderivative size = 146 \[ \int \frac {\left (-1-x^4+2 x^6\right ) \sqrt [3]{x-x^5+x^7}}{\left (1+x^2-x^4+x^6\right )^2} \, dx=-\frac {x \sqrt [3]{x-x^5+x^7}}{2 \left (1+x^2-x^4+x^6\right )}-\frac {\arctan \left (\frac {\sqrt {3} \sqrt [3]{x-x^5+x^7}}{-2 x+\sqrt [3]{x-x^5+x^7}}\right )}{2 \sqrt {3}}+\frac {1}{6} \log \left (x+\sqrt [3]{x-x^5+x^7}\right )-\frac {1}{12} \log \left (x^2-x \sqrt [3]{x-x^5+x^7}+\left (x-x^5+x^7\right )^{2/3}\right ) \]
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\[ \int \frac {\left (-1-x^4+2 x^6\right ) \sqrt [3]{x-x^5+x^7}}{\left (1+x^2-x^4+x^6\right )^2} \, dx=\int \frac {\left (-1-x^4+2 x^6\right ) \sqrt [3]{x-x^5+x^7}}{\left (1+x^2-x^4+x^6\right )^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{x-x^5+x^7} \int \frac {\sqrt [3]{x} \sqrt [3]{1-x^4+x^6} \left (-1-x^4+2 x^6\right )}{\left (1+x^2-x^4+x^6\right )^2} \, dx}{\sqrt [3]{x} \sqrt [3]{1-x^4+x^6}} \\ & = \frac {\left (3 \sqrt [3]{x-x^5+x^7}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{1-x^{12}+x^{18}} \left (-1-x^{12}+2 x^{18}\right )}{\left (1+x^6-x^{12}+x^{18}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^4+x^6}} \\ & = \frac {\left (3 \sqrt [3]{x-x^5+x^7}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1-x^6+x^9} \left (-1-x^6+2 x^9\right )}{\left (1+x^3-x^6+x^9\right )^2} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1-x^4+x^6}} \\ & = \frac {\left (3 \sqrt [3]{x-x^5+x^7}\right ) \text {Subst}\left (\int \left (\frac {x \left (-3-2 x^3+x^6\right ) \sqrt [3]{1-x^6+x^9}}{\left (1+x^3-x^6+x^9\right )^2}+\frac {2 x \sqrt [3]{1-x^6+x^9}}{1+x^3-x^6+x^9}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1-x^4+x^6}} \\ & = \frac {\left (3 \sqrt [3]{x-x^5+x^7}\right ) \text {Subst}\left (\int \frac {x \left (-3-2 x^3+x^6\right ) \sqrt [3]{1-x^6+x^9}}{\left (1+x^3-x^6+x^9\right )^2} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1-x^4+x^6}}+\frac {\left (3 \sqrt [3]{x-x^5+x^7}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1-x^6+x^9}}{1+x^3-x^6+x^9} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^4+x^6}} \\ & = \frac {\left (3 \sqrt [3]{x-x^5+x^7}\right ) \text {Subst}\left (\int \left (-\frac {3 x \sqrt [3]{1-x^6+x^9}}{\left (1+x^3-x^6+x^9\right )^2}-\frac {2 x^4 \sqrt [3]{1-x^6+x^9}}{\left (1+x^3-x^6+x^9\right )^2}+\frac {x^7 \sqrt [3]{1-x^6+x^9}}{\left (1+x^3-x^6+x^9\right )^2}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1-x^4+x^6}}+\frac {\left (3 \sqrt [3]{x-x^5+x^7}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1-x^6+x^9}}{1+x^3-x^6+x^9} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^4+x^6}} \\ & = \frac {\left (3 \sqrt [3]{x-x^5+x^7}\right ) \text {Subst}\left (\int \frac {x^7 \sqrt [3]{1-x^6+x^9}}{\left (1+x^3-x^6+x^9\right )^2} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1-x^4+x^6}}-\frac {\left (3 \sqrt [3]{x-x^5+x^7}\right ) \text {Subst}\left (\int \frac {x^4 \sqrt [3]{1-x^6+x^9}}{\left (1+x^3-x^6+x^9\right )^2} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^4+x^6}}+\frac {\left (3 \sqrt [3]{x-x^5+x^7}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1-x^6+x^9}}{1+x^3-x^6+x^9} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^4+x^6}}-\frac {\left (9 \sqrt [3]{x-x^5+x^7}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1-x^6+x^9}}{\left (1+x^3-x^6+x^9\right )^2} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1-x^4+x^6}} \\ \end{align*}
Time = 3.27 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.36 \[ \int \frac {\left (-1-x^4+2 x^6\right ) \sqrt [3]{x-x^5+x^7}}{\left (1+x^2-x^4+x^6\right )^2} \, dx=\frac {\sqrt [3]{x-x^5+x^7} \left (-\frac {6 x^{4/3}}{1+x^2-x^4+x^6}+\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}-2 \sqrt [3]{1-x^4+x^6}}\right )}{\sqrt [3]{1-x^4+x^6}}+\frac {2 \log \left (x^{2/3}+\sqrt [3]{1-x^4+x^6}\right )}{\sqrt [3]{1-x^4+x^6}}-\frac {\log \left (x^{4/3}-x^{2/3} \sqrt [3]{1-x^4+x^6}+\left (1-x^4+x^6\right )^{2/3}\right )}{\sqrt [3]{1-x^4+x^6}}\right )}{12 \sqrt [3]{x}} \]
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Time = 12.06 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.39
method | result | size |
pseudoelliptic | \(-\frac {\left (\left (-2 x^{6}+2 x^{4}-2 x^{2}-2\right ) \ln \left (\frac {x +{\left (x \left (x^{6}-x^{4}+1\right )\right )}^{\frac {1}{3}}}{x}\right )+6 {\left (x \left (x^{6}-x^{4}+1\right )\right )}^{\frac {1}{3}} x +\left (x^{6}-x^{4}+x^{2}+1\right ) \left (2 \arctan \left (\frac {\left (-2 {\left (x \left (x^{6}-x^{4}+1\right )\right )}^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right ) \sqrt {3}+\ln \left (\frac {{\left (x \left (x^{6}-x^{4}+1\right )\right )}^{\frac {2}{3}}-{\left (x \left (x^{6}-x^{4}+1\right )\right )}^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )\right )\right ) x}{12 \left (x +{\left (x \left (x^{6}-x^{4}+1\right )\right )}^{\frac {1}{3}}\right ) \left ({\left (x \left (x^{6}-x^{4}+1\right )\right )}^{\frac {2}{3}}+x \left (x -{\left (x \left (x^{6}-x^{4}+1\right )\right )}^{\frac {1}{3}}\right )\right )}\) | \(203\) |
trager | \(\text {Expression too large to display}\) | \(739\) |
risch | \(\text {Expression too large to display}\) | \(1528\) |
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Time = 1.80 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.36 \[ \int \frac {\left (-1-x^4+2 x^6\right ) \sqrt [3]{x-x^5+x^7}}{\left (1+x^2-x^4+x^6\right )^2} \, dx=-\frac {2 \, \sqrt {3} {\left (x^{6} - x^{4} + x^{2} + 1\right )} \arctan \left (-\frac {2 \, \sqrt {3} {\left (x^{7} - x^{5} + x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{6} - x^{4} - x^{2} + 1\right )} - 2 \, \sqrt {3} {\left (x^{7} - x^{5} + x\right )}^{\frac {2}{3}}}{x^{6} - x^{4} + x^{2} + 1}\right ) - {\left (x^{6} - x^{4} + x^{2} + 1\right )} \log \left (\frac {x^{6} - x^{4} + x^{2} + 3 \, {\left (x^{7} - x^{5} + x\right )}^{\frac {1}{3}} x + 3 \, {\left (x^{7} - x^{5} + x\right )}^{\frac {2}{3}} + 1}{x^{6} - x^{4} + x^{2} + 1}\right ) + 6 \, {\left (x^{7} - x^{5} + x\right )}^{\frac {1}{3}} x}{12 \, {\left (x^{6} - x^{4} + x^{2} + 1\right )}} \]
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Timed out. \[ \int \frac {\left (-1-x^4+2 x^6\right ) \sqrt [3]{x-x^5+x^7}}{\left (1+x^2-x^4+x^6\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (-1-x^4+2 x^6\right ) \sqrt [3]{x-x^5+x^7}}{\left (1+x^2-x^4+x^6\right )^2} \, dx=\int { \frac {{\left (x^{7} - x^{5} + x\right )}^{\frac {1}{3}} {\left (2 \, x^{6} - x^{4} - 1\right )}}{{\left (x^{6} - x^{4} + x^{2} + 1\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (-1-x^4+2 x^6\right ) \sqrt [3]{x-x^5+x^7}}{\left (1+x^2-x^4+x^6\right )^2} \, dx=\int { \frac {{\left (x^{7} - x^{5} + x\right )}^{\frac {1}{3}} {\left (2 \, x^{6} - x^{4} - 1\right )}}{{\left (x^{6} - x^{4} + x^{2} + 1\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (-1-x^4+2 x^6\right ) \sqrt [3]{x-x^5+x^7}}{\left (1+x^2-x^4+x^6\right )^2} \, dx=\int -\frac {\left (-2\,x^6+x^4+1\right )\,{\left (x^7-x^5+x\right )}^{1/3}}{{\left (x^6-x^4+x^2+1\right )}^2} \,d x \]
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