Integrand size = 17, antiderivative size = 54 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x} \, dx=\sqrt [4]{-x^3+x^4}+\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.85, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {2046, 2057, 65, 246, 218, 212, 209} \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x} \, dx=-\frac {(x-1)^{3/4} x^{9/4} \arctan \left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{2 \left (x^4-x^3\right )^{3/4}}-\frac {(x-1)^{3/4} x^{9/4} \text {arctanh}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{2 \left (x^4-x^3\right )^{3/4}}+\sqrt [4]{x^4-x^3} \]
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Rule 65
Rule 209
Rule 212
Rule 218
Rule 246
Rule 2046
Rule 2057
Rubi steps \begin{align*} \text {integral}& = \sqrt [4]{-x^3+x^4}-\frac {1}{4} \int \frac {x^2}{\left (-x^3+x^4\right )^{3/4}} \, dx \\ & = \sqrt [4]{-x^3+x^4}-\frac {\left ((-1+x)^{3/4} x^{9/4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{4 \left (-x^3+x^4\right )^{3/4}} \\ & = \sqrt [4]{-x^3+x^4}-\frac {\left ((-1+x)^{3/4} x^{9/4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4}} \, dx,x,\sqrt [4]{-1+x}\right )}{\left (-x^3+x^4\right )^{3/4}} \\ & = \sqrt [4]{-x^3+x^4}-\frac {\left ((-1+x)^{3/4} x^{9/4}\right ) \text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{\left (-x^3+x^4\right )^{3/4}} \\ & = \sqrt [4]{-x^3+x^4}-\frac {\left ((-1+x)^{3/4} x^{9/4}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{2 \left (-x^3+x^4\right )^{3/4}}-\frac {\left ((-1+x)^{3/4} x^{9/4}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{2 \left (-x^3+x^4\right )^{3/4}} \\ & = \sqrt [4]{-x^3+x^4}-\frac {(-1+x)^{3/4} x^{9/4} \arctan \left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{2 \left (-x^3+x^4\right )^{3/4}}-\frac {(-1+x)^{3/4} x^{9/4} \text {arctanh}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{2 \left (-x^3+x^4\right )^{3/4}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.26 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x} \, dx=\frac {(-1+x)^{3/4} x^{9/4} \left (2 \sqrt [4]{-1+x} x^{3/4}+\arctan \left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )-\text {arctanh}\left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )\right )}{2 \left ((-1+x) x^3\right )^{3/4}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.50
method | result | size |
meijerg | \(\frac {4 \operatorname {signum}\left (x -1\right )^{\frac {1}{4}} x^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [-\frac {1}{4}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], x\right )}{3 \left (-\operatorname {signum}\left (x -1\right )\right )^{\frac {1}{4}}}\) | \(27\) |
pseudoelliptic | \(\frac {\ln \left (\frac {\left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}-x}{x}\right )}{4}-\frac {\arctan \left (\frac {\left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}}{x}\right )}{2}-\frac {\ln \left (\frac {\left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}+x}{x}\right )}{4}+\left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}\) | \(65\) |
trager | \(\left (x^{4}-x^{3}\right )^{\frac {1}{4}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}-x^{3}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}-2 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}}{x^{2}}\right )}{4}-\frac {\ln \left (\frac {2 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}+2 \sqrt {x^{4}-x^{3}}\, x +2 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}+2 x^{3}-x^{2}}{x^{2}}\right )}{4}\) | \(158\) |
risch | \(\left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}+\frac {\left (\frac {\ln \left (\frac {2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}-2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, x +2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}-2 x^{3}+2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}-4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x +5 x^{2}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}}-4 x +1}{\left (x -1\right )^{2}}\right )}{4}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}-2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )-2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{\left (x -1\right )^{2}}\right )}{4}\right ) \left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}} \left (x \left (x -1\right )^{3}\right )^{\frac {1}{4}}}{x \left (x -1\right )}\) | \(390\) |
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Time = 0.28 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.35 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x} \, dx={\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} - \frac {1}{2} \, \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} \, \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, \log \left (-\frac {x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]
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\[ \int \frac {\sqrt [4]{-x^3+x^4}}{x} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x - 1\right )}}{x}\, dx \]
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\[ \int \frac {\sqrt [4]{-x^3+x^4}}{x} \, dx=\int { \frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x} \, dx=x {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - \frac {1}{2} \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{4} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{4} \, \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]
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Timed out. \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x} \, dx=\int \frac {{\left (x^4-x^3\right )}^{1/4}}{x} \,d x \]
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