Optimal. Leaf size=55 \[ -\frac {4 \sqrt [4]{x^4-x}}{3 x}-\frac {2}{3} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x}}\right )+\frac {2}{3} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x}}\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 109, normalized size of antiderivative = 1.98, number of steps used = 8, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {2020, 2032, 329, 275, 331, 298, 203, 206} \begin {gather*} -\frac {4 \sqrt [4]{x^4-x}}{3 x}-\frac {2 x^{3/4} \left (x^3-1\right )^{3/4} \tan ^{-1}\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} \tanh ^{-1}\left (\frac {x^{3/4}}{\sqrt [4]{x^3-1}}\right )}{3 \left (x^4-x\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 275
Rule 298
Rule 329
Rule 331
Rule 2020
Rule 2032
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{-x+x^4}}{x^2} \, dx &=-\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} \tan ^{-1}\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} \tanh ^{-1}\left (\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \left (-x+x^4\right )^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 42, normalized size = 0.76 \begin {gather*} -\frac {4 \sqrt [4]{x \left (x^3-1\right )} \, _2F_1\left (-\frac {1}{4},-\frac {1}{4};\frac {3}{4};x^3\right )}{3 x \sqrt [4]{1-x^3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.21, size = 55, normalized size = 1.00 \begin {gather*} -\frac {4 \sqrt [4]{-x+x^4}}{3 x}-\frac {2}{3} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-x+x^4}}\right )+\frac {2}{3} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.33, size = 93, normalized size = 1.69 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 53, normalized size = 0.96 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 5.47, size = 33, normalized size = 0.60
method | result | size |
meijerg | \(-\frac {4 \mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{4}} \hypergeom \left (\left [-\frac {1}{4}, -\frac {1}{4}\right ], \left [\frac {3}{4}\right ], x^{3}\right )}{3 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{4}} x^{\frac {3}{4}}}\) | \(33\) |
trager | \(-\frac {4 \left (x^{4}-x \right )^{\frac {1}{4}}}{3 x}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (2 \sqrt {x^{4}-x}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x -2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \left (x^{4}-x \right )^{\frac {3}{4}}+2 x^{2} \left (x^{4}-x \right )^{\frac {1}{4}}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{3}-\frac {\ln \left (-2 \left (x^{4}-x \right )^{\frac {3}{4}}+2 x \sqrt {x^{4}-x}-2 x^{2} \left (x^{4}-x \right )^{\frac {1}{4}}+2 x^{3}-1\right )}{3}\) | \(133\) |
risch | \(-\frac {4 \left (x \left (x^{3}-1\right )\right )^{\frac {1}{4}}}{3 x}+\frac {\left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 x^{9}-2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}-5 x^{6}-2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}\, x^{3}+4 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {3}{4}}+4 x^{3}+2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}}-1}{\left (-1+x \right )^{2} \left (x^{2}+x +1\right )^{2}}\right )}{3}-\frac {\ln \left (-\frac {-2 x^{9}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{6}+5 x^{6}-2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}\, x^{3}-4 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{3}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {3}{4}}-4 x^{3}+2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}}+1}{\left (-1+x \right )^{2} \left (x^{2}+x +1\right )^{2}}\right )}{3}\right ) \left (x \left (x^{3}-1\right )\right )^{\frac {1}{4}} \left (x^{3} \left (x^{3}-1\right )^{3}\right )^{\frac {1}{4}}}{x \left (x^{3}-1\right )}\) | \(444\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} - x\right )}^{\frac {1}{4}}}{x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (x^4-x\right )}^{1/4}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{x \left (x - 1\right ) \left (x^{2} + x + 1\right )}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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