Integrand size = 22, antiderivative size = 37 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{(-1+x) x} \, dx=-2 \arctan \left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )+2 \text {arctanh}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right ) \]
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Leaf count is larger than twice the leaf count of optimal. \(83\) vs. \(2(37)=74\).
Time = 0.07 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.24, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2081, 65, 246, 218, 212, 209} \[ \int \frac {\sqrt [4]{-x^3+x^4}}{(-1+x) x} \, dx=\frac {2 \sqrt [4]{x^4-x^3} \arctan \left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{\sqrt [4]{x-1} x^{3/4}}+\frac {2 \sqrt [4]{x^4-x^3} \text {arctanh}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{\sqrt [4]{x-1} x^{3/4}} \]
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Rule 65
Rule 209
Rule 212
Rule 218
Rule 246
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{-x^3+x^4} \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{\sqrt [4]{-1+x} x^{3/4}} \\ & = \frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4}} \, dx,x,\sqrt [4]{-1+x}\right )}{\sqrt [4]{-1+x} x^{3/4}} \\ & = \frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{\sqrt [4]{-1+x} x^{3/4}} \\ & = \frac {\left (2 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{\sqrt [4]{-1+x} x^{3/4}} \\ & = \frac {2 \sqrt [4]{-x^3+x^4} \arctan \left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {2 \sqrt [4]{-x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{\sqrt [4]{-1+x} x^{3/4}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.41 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{(-1+x) x} \, dx=\frac {2 (-1+x)^{3/4} x^{9/4} \left (-\arctan \left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )+\text {arctanh}\left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )\right )}{\left ((-1+x) x^3\right )^{3/4}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.83 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.73
method | result | size |
meijerg | \(-\frac {4 \operatorname {signum}\left (x -1\right )^{\frac {1}{4}} x^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [\frac {3}{4}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], x\right )}{3 \left (-\operatorname {signum}\left (x -1\right )\right )^{\frac {1}{4}}}\) | \(27\) |
pseudoelliptic | \(2 \,\operatorname {arctanh}\left (\frac {\left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}}{x}\right )+2 \arctan \left (\frac {\left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}}{x}\right )\) | \(34\) |
trager | \(\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}-x^{3}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}-2 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}}{x^{2}}\right )+\ln \left (\frac {2 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}+2 \sqrt {x^{4}-x^{3}}\, x +2 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}+2 x^{3}-x^{2}}{x^{2}}\right )\) | \(144\) |
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Time = 0.24 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.62 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{(-1+x) x} \, dx=2 \, \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \log \left (-\frac {x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]
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\[ \int \frac {\sqrt [4]{-x^3+x^4}}{(-1+x) x} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x - 1\right )}}{x \left (x - 1\right )}\, dx \]
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\[ \int \frac {\sqrt [4]{-x^3+x^4}}{(-1+x) x} \, dx=\int { \frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{{\left (x - 1\right )} x} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{(-1+x) x} \, dx=2 \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) - \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]
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Timed out. \[ \int \frac {\sqrt [4]{-x^3+x^4}}{(-1+x) x} \, dx=\int \frac {{\left (x^4-x^3\right )}^{1/4}}{x\,\left (x-1\right )} \,d x \]
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