Integrand size = 22, antiderivative size = 217 \[ \int \frac {-1+x}{(1+x) \sqrt [3]{-x^2+x^3}} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x^2+x^3}}\right )-2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-x^2+x^3}}\right )-\log \left (-x+\sqrt [3]{-x^2+x^3}\right )+2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{-x^2+x^3}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{-x^2+x^3}+\left (-x^2+x^3\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-x^2+x^3}+\sqrt [3]{2} \left (-x^2+x^3\right )^{2/3}\right )}{\sqrt [3]{2}} \]
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Time = 0.09 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.17, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {2081, 129, 399, 245, 384} \[ \int \frac {-1+x}{(1+x) \sqrt [3]{-x^2+x^3}} \, dx=\frac {\sqrt {3} \sqrt [3]{x-1} x^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{\sqrt [3]{x^3-x^2}}-\frac {2^{2/3} \sqrt {3} \sqrt [3]{x-1} x^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{\sqrt [3]{x^3-x^2}}-\frac {3 \sqrt [3]{x-1} x^{2/3} \log \left (\sqrt [3]{x-1}-\sqrt [3]{x}\right )}{2 \sqrt [3]{x^3-x^2}}+\frac {3 \sqrt [3]{x-1} x^{2/3} \log \left (\sqrt [3]{x-1}-\sqrt [3]{2} \sqrt [3]{x}\right )}{\sqrt [3]{2} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log (x+1)}{\sqrt [3]{2} \sqrt [3]{x^3-x^2}} \]
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Rule 129
Rule 245
Rule 384
Rule 399
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {(-1+x)^{2/3}}{x^{2/3} (1+x)} \, dx}{\sqrt [3]{-x^2+x^3}} \\ & = \frac {\left (3 \sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {\left (-1+x^3\right )^{2/3}}{1+x^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^3}} \\ & = \frac {\left (3 \sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^3}}-\frac {\left (6 \sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^3}} \\ & = \frac {\sqrt {3} \sqrt [3]{-1+x} x^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{x}}{\sqrt [3]{-1+x}}}{\sqrt {3}}\right )}{\sqrt [3]{-x^2+x^3}}-\frac {2^{2/3} \sqrt {3} \sqrt [3]{-1+x} x^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{-1+x}}}{\sqrt {3}}\right )}{\sqrt [3]{-x^2+x^3}}-\frac {3 \sqrt [3]{-1+x} x^{2/3} \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^3}}+\frac {3 \sqrt [3]{-1+x} x^{2/3} \log \left (\sqrt [3]{-1+x}-\sqrt [3]{2} \sqrt [3]{x}\right )}{\sqrt [3]{2} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log (1+x)}{\sqrt [3]{2} \sqrt [3]{-x^2+x^3}} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.06 \[ \int \frac {-1+x}{(1+x) \sqrt [3]{-x^2+x^3}} \, dx=\frac {\sqrt [3]{-1+x} x^{2/3} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{2 \sqrt [3]{-1+x}+\sqrt [3]{x}}\right )-2\ 2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{2^{2/3} \sqrt [3]{-1+x}+\sqrt [3]{x}}\right )+2\ 2^{2/3} \log \left (2^{2/3} \sqrt [3]{-1+x}-2 \sqrt [3]{x}\right )-2 \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x}\right )+\log \left ((-1+x)^{2/3}+\sqrt [3]{-1+x} \sqrt [3]{x}+x^{2/3}\right )-2^{2/3} \log \left (\sqrt [3]{2} (-1+x)^{2/3}+2^{2/3} \sqrt [3]{-1+x} \sqrt [3]{x}+2 x^{2/3}\right )\right )}{2 \sqrt [3]{(-1+x) x^2}} \]
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Time = 2.41 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.82
method | result | size |
pseudoelliptic | \(2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )-\frac {2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right )}{2}+\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+x \right )}{3 x}\right )-\ln \left (\frac {-x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )+\frac {\ln \left (\frac {\left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}}+x \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+x^{2}}{x^{2}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\left (2 \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )\) | \(177\) |
trager | \(\text {Expression too large to display}\) | \(1437\) |
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Time = 0.27 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.95 \[ \int \frac {-1+x}{(1+x) \sqrt [3]{-x^2+x^3}} \, dx=4^{\frac {1}{3}} \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 4^{\frac {1}{3}} \sqrt {3} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) - \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) + 4^{\frac {1}{3}} \log \left (-\frac {4^{\frac {2}{3}} x - 2 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{2} \cdot 4^{\frac {1}{3}} \log \left (\frac {2 \cdot 4^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x + 2 \, {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - \log \left (-\frac {x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{2} \, \log \left (\frac {x^{2} + {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]
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\[ \int \frac {-1+x}{(1+x) \sqrt [3]{-x^2+x^3}} \, dx=\int \frac {x - 1}{\sqrt [3]{x^{2} \left (x - 1\right )} \left (x + 1\right )}\, dx \]
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\[ \int \frac {-1+x}{(1+x) \sqrt [3]{-x^2+x^3}} \, dx=\int { \frac {x - 1}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} {\left (x + 1\right )}} \,d x } \]
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Time = 0.31 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.68 \[ \int \frac {-1+x}{(1+x) \sqrt [3]{-x^2+x^3}} \, dx=\sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}}\right )}\right ) - \frac {1}{2} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}}\right ) + 2^{\frac {2}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} \right |}\right ) - \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{2} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) - \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
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Timed out. \[ \int \frac {-1+x}{(1+x) \sqrt [3]{-x^2+x^3}} \, dx=\int \frac {x-1}{{\left (x^3-x^2\right )}^{1/3}\,\left (x+1\right )} \,d x \]
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