Optimal. Leaf size=55 \[ -\frac {4 \sqrt [4]{-x^3+x^4}}{x}-2 \text {ArcTan}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )+2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right ) \]
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Rubi [A]
time = 0.05, antiderivative size = 101, normalized size of antiderivative = 1.84, number of steps
used = 7, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {2045, 2057, 65,
246, 218, 212, 209} \begin {gather*} \frac {2 (x-1)^{3/4} x^{9/4} \text {ArcTan}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{\left (x^4-x^3\right )^{3/4}}-\frac {4 \sqrt [4]{x^4-x^3}}{x}+\frac {2 (x-1)^{3/4} x^{9/4} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{\left (x^4-x^3\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 209
Rule 212
Rule 218
Rule 246
Rule 2045
Rule 2057
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{-x^3+x^4}}{x^2} \, dx &=-\frac {4 \sqrt [4]{-x^3+x^4}}{x}+\int \frac {x^2}{\left (-x^3+x^4\right )^{3/4}} \, dx\\ &=-\frac {4 \sqrt [4]{-x^3+x^4}}{x}+\frac {\left ((-1+x)^{3/4} x^{9/4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{\left (-x^3+x^4\right )^{3/4}}\\ &=-\frac {4 \sqrt [4]{-x^3+x^4}}{x}+\frac {\left (4 (-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}}\\ &=-\frac {4 \sqrt [4]{-x^3+x^4}}{x}+\frac {\left (4 (-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}}\\ &=-\frac {4 \sqrt [4]{-x^3+x^4}}{x}+\frac {\left (2 (-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 )}{\left (-x^3+x^4\right )^{3/4}}+\frac {\left (2 (-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 )}{\left (-x^3+x^4\right )^{3/4}}\\ &=-\frac {4 \sqrt [4]{-x^3+x^4}}{x}+\frac {2 (-1+x)^{3/4} x^{9/4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{\left (-x^3+x^4\right )^{3/4}}+\frac {2 (-1+x)^{3/4} x^{9/4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{\left (-x^3+x^4\right )^{3/4}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 70, normalized size = 1.27 \begin {gather*} -\frac {2 (-1+x)^{3/4} x^2 \left (2 \sqrt [4]{-1+x}+\sqrt [4]{x} \text {ArcTan}\left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )-\sqrt [4]{x} \tanh ^{-1}\left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )\right )}{\left ((-1+x) x^3\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.01, size = 27, normalized size = 0.49
method | result | size |
meijerg | \(-\frac {4 \mathrm {signum}\left (-1+x \right )^{\frac {1}{4}} \hypergeom \left (\left [-\frac {1}{4}, -\frac {1}{4}\right ], \left [\frac {3}{4}\right ], x\right )}{\left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{4}} x^{\frac {1}{4}}}\) | \(27\) |
trager | \(-\frac {4 \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}{x}+\ln \left (\frac {2 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}+2 \sqrt {x^{4}-x^{3}}\, x +2 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}+2 x^{3}-x^{2}}{x^{2}}\right )+\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}-x^{3}}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x -2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}-2 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}+\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}}{x^{2}}\right )\) | \(160\) |
risch | \(-\frac {4 \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}+\frac {\left (-\ln \left (\frac {2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}-2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, x +2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}-2 x^{3}+2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}-4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x +5 x^{2}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}}-4 x +1}{\left (-1+x \right )^{2}}\right )+\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x -2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \RootOf \left (\textit {\_Z}^{2}+1\right )+5 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}-4 x \RootOf \left (\textit {\_Z}^{2}+1\right )-4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x +\RootOf \left (\textit {\_Z}^{2}+1\right )+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}}}{\left (-1+x \right )^{2}}\right )\right ) \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \left (x \left (-1+x \right )^{3}\right )^{\frac {1}{4}}}{x \left (-1+x \right )}\) | \(395\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 81, normalized size = 1.47 \begin {gather*} \frac {2 \, x \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + x \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - x \log \left (-\frac {x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{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^{3} \left (x - 1\right )}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 51, normalized size = 0.93 \begin {gather*} 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.72, size = 29, normalized size = 0.53 \begin {gather*} -\frac {4\,{\left (x^4-x^3\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},-\frac {1}{4};\ \frac {3}{4};\ x\right )}{x\,{\left (1-x\right )}^{1/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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