Integrand size = 15, antiderivative size = 55 \[ \int \frac {\sqrt [4]{-x+x^4}}{x^2} \, dx=-\frac {4 \sqrt [4]{-x+x^4}}{3 x}-\frac {2}{3} \arctan \left (\frac {x}{\sqrt [4]{-x+x^4}}\right )+\frac {2}{3} \text {arctanh}\left (\frac {x}{\sqrt [4]{-x+x^4}}\right ) \]
[Out]
Time = 0.05 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.98, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {2045, 2057, 335, 281, 338, 304, 209, 212} \[ \int \frac {\sqrt [4]{-x+x^4}}{x^2} \, dx=-\frac {2 x^{3/4} \left (x^3-1\right )^{3/4} \arctan \left (\frac {x^{3/4}}{\sqrt [4]{x^3-1}}\right )}{3 \left (x^4-x\right )^{3/4}}+\frac {2 x^{3/4} \left (x^3-1\right )^{3/4} \text {arctanh}\left (\frac {x^{3/4}}{\sqrt [4]{x^3-1}}\right )}{3 \left (x^4-x\right )^{3/4}}-\frac {4 \sqrt [4]{x^4-x}}{3 x} \]
[In]
[Out]
Rule 209
Rule 212
Rule 281
Rule 304
Rule 335
Rule 338
Rule 2045
Rule 2057
Rubi steps \begin{align*} \text {integral}& = -\frac {4 \sqrt [4]{-x+x^4}}{3 x}+\int \frac {x^2}{\left (-x+x^4\right )^{3/4}} \, dx \\ & = -\frac {4 \sqrt [4]{-x+x^4}}{3 x}+\frac {\left (x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \int \frac {x^{5/4}}{\left (-1+x^3\right )^{3/4}} \, dx}{\left (-x+x^4\right )^{3/4}} \\ & = -\frac {4 \sqrt [4]{-x+x^4}}{3 x}+\frac {\left (4 x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \text {Subst}\left (\int \frac {x^8}{\left (-1+x^{12}\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{\left (-x+x^4\right )^{3/4}} \\ & = -\frac {4 \sqrt [4]{-x+x^4}}{3 x}+\frac {\left (4 x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4}} \, dx,x,x^{3/4}\right )}{3 \left (-x+x^4\right )^{3/4}} \\ & = -\frac {4 \sqrt [4]{-x+x^4}}{3 x}+\frac {\left (4 x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \left (-x+x^4\right )^{3/4}} \\ & = -\frac {4 \sqrt [4]{-x+x^4}}{3 x}+\frac {\left (2 x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \left (-x+x^4\right )^{3/4}}-\frac {\left (2 x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \left (-x+x^4\right )^{3/4}} \\ & = -\frac {4 \sqrt [4]{-x+x^4}}{3 x}-\frac {2 x^{3/4} \left (-1+x^3\right )^{3/4} \arctan \left (\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \left (-x+x^4\right )^{3/4}}+\frac {2 x^{3/4} \left (-1+x^3\right )^{3/4} \text {arctanh}\left (\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \left (-x+x^4\right )^{3/4}} \\ \end{align*}
Time = 4.05 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.47 \[ \int \frac {\sqrt [4]{-x+x^4}}{x^2} \, dx=-\frac {2 \left (-1+x^3\right )^{3/4} \left (2 \sqrt [4]{-1+x^3}+x^{3/4} \arctan \left (\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )-x^{3/4} \text {arctanh}\left (\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )\right )}{3 \left (x \left (-1+x^3\right )\right )^{3/4}} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 7.16 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.60
method | result | size |
meijerg | \(-\frac {4 \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{4}} \operatorname {hypergeom}\left (\left [-\frac {1}{4}, -\frac {1}{4}\right ], \left [\frac {3}{4}\right ], x^{3}\right )}{3 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{4}} x^{\frac {3}{4}}}\) | \(33\) |
pseudoelliptic | \(\frac {-4 \left (x^{4}-x \right )^{\frac {1}{4}}+x \left (2 \arctan \left (\frac {\left (x^{4}-x \right )^{\frac {1}{4}}}{x}\right )-\ln \left (\frac {\left (x^{4}-x \right )^{\frac {1}{4}}-x}{x}\right )+\ln \left (\frac {\left (x^{4}-x \right )^{\frac {1}{4}}+x}{x}\right )\right )}{3 x}\) | \(73\) |
trager | \(-\frac {4 \left (x^{4}-x \right )^{\frac {1}{4}}}{3 x}+\frac {\ln \left (-2 \left (x^{4}-x \right )^{\frac {3}{4}}-2 x \sqrt {x^{4}-x}-2 x^{2} \left (x^{4}-x \right )^{\frac {1}{4}}-2 x^{3}+1\right )}{3}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (2 \sqrt {x^{4}-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \left (x^{4}-x \right )^{\frac {3}{4}}+2 x^{2} \left (x^{4}-x \right )^{\frac {1}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{3}\) | \(133\) |
risch | \(-\frac {4 {\left (x \left (x^{3}-1\right )\right )}^{\frac {1}{4}}}{3 x}+\frac {\left (-\frac {\ln \left (-\frac {-2 x^{9}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{6}+5 x^{6}-2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}\, x^{3}-4 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{3}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {3}{4}}-4 x^{3}+2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}}+1}{\left (x^{2}+x +1\right )^{2} \left (x -1\right )^{2}}\right )}{3}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 x^{9}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{6}-5 x^{6}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{3}-2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}\, x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {3}{4}}+4 x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}}+2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}-1}{\left (x^{2}+x +1\right )^{2} \left (x -1\right )^{2}}\right )}{3}\right ) {\left (x \left (x^{3}-1\right )\right )}^{\frac {1}{4}} \left (x^{3} \left (x^{3}-1\right )^{3}\right )^{\frac {1}{4}}}{x \left (x^{3}-1\right )}\) | \(444\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (43) = 86\).
Time = 0.77 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.69 \[ \int \frac {\sqrt [4]{-x+x^4}}{x^2} \, dx=\frac {x \arctan \left (2 \, {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{2} + 2 \, {\left (x^{4} - x\right )}^{\frac {3}{4}}\right ) + x \log \left (-2 \, x^{3} - 2 \, {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{2} - 2 \, \sqrt {x^{4} - x} x - 2 \, {\left (x^{4} - x\right )}^{\frac {3}{4}} + 1\right ) - 4 \, {\left (x^{4} - x\right )}^{\frac {1}{4}}}{3 \, x} \]
[In]
[Out]
\[ \int \frac {\sqrt [4]{-x+x^4}}{x^2} \, dx=\int \frac {\sqrt [4]{x \left (x - 1\right ) \left (x^{2} + x + 1\right )}}{x^{2}}\, dx \]
[In]
[Out]
\[ \int \frac {\sqrt [4]{-x+x^4}}{x^2} \, dx=\int { \frac {{\left (x^{4} - x\right )}^{\frac {1}{4}}}{x^{2}} \,d x } \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt [4]{-x+x^4}}{x^2} \, dx=-\frac {4}{3} \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}} + \frac {2}{3} \, \arctan \left ({\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{3} \, \log \left ({\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {1}{3} \, \log \left ({\left | {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt [4]{-x+x^4}}{x^2} \, dx=\int \frac {{\left (x^4-x\right )}^{1/4}}{x^2} \,d x \]
[In]
[Out]