Optimal. Leaf size=54 \[ \sqrt [4]{x^4-x^3}+\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right )-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 100, normalized size of antiderivative = 1.85, 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 = {2021, 2032, 63, 240, 212, 206, 203} \begin {gather*} \sqrt [4]{x^4-x^3}-\frac {(x-1)^{3/4} x^{9/4} \tan ^{-1}\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} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{2 \left (x^4-x^3\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 206
Rule 212
Rule 240
Rule 2021
Rule 2032
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{-x^3+x^4}}{x} \, dx &=\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} \tan ^{-1}\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} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{2 \left (-x^3+x^4\right )^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 35, normalized size = 0.65 \begin {gather*} \frac {4 \left ((x-1) x^3\right )^{5/4} \, _2F_1\left (\frac {1}{4},\frac {5}{4};\frac {9}{4};1-x\right )}{5 x^{15/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.22, size = 54, normalized size = 1.00 \begin {gather*} \sqrt [4]{-x^3+x^4}+\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 73, normalized size = 1.35 \begin {gather*} {\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 54, normalized size = 1.00 \begin {gather*} -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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.80, size = 27, normalized size = 0.50
method | result | size |
meijerg | \(\frac {4 \mathrm {signum}\left (-1+x \right )^{\frac {1}{4}} x^{\frac {3}{4}} \hypergeom \left (\left [-\frac {1}{4}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], x\right )}{3 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{4}}}\) | \(27\) |
trager | \(\left (x^{4}-x^{3}\right )^{\frac {1}{4}}+\frac {\ln \left (\frac {2 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}-2 \sqrt {x^{4}-x^{3}}\, x +2 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}-2 x^{3}+x^{2}}{x^{2}}\right )}{4}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-2 \sqrt {x^{4}-x^{3}}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x +2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}-2 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}}{x^{2}}\right )}{4}\) | \(157\) |
risch | \(\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}+\frac {\left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x -2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \RootOf \left (\textit {\_Z}^{2}+1\right )+5 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}-2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}-4 \RootOf \left (\textit {\_Z}^{2}+1\right ) x +4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x +\RootOf \left (\textit {\_Z}^{2}+1\right )-2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}}}{\left (-1+x \right )^{2}}\right )}{4}+\frac {\ln \left (\frac {2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}-2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, x +2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}-2 x^{3}+2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}-4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x +5 x^{2}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}}-4 x +1}{\left (-1+x \right )^{2}}\right )}{4}\right ) \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \left (x \left (-1+x \right )^{3}\right )^{\frac {1}{4}}}{x \left (-1+x \right )}\) | \(390\) |
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^{3}\right )}^{\frac {1}{4}}}{x}\,{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^3\right )}^{1/4}}{x} \,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^{3} \left (x - 1\right )}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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