Optimal. Leaf size=59 \[ \frac {1}{2} \sqrt [4]{x^4-x^2} x+\frac {1}{4} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^2}}\right )-\frac {1}{4} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^2}}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 113, normalized size of antiderivative = 1.92, number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {2004, 2032, 329, 331, 298, 203, 206} \begin {gather*} \frac {1}{2} \sqrt [4]{x^4-x^2} x+\frac {\left (x^2-1\right )^{3/4} x^{3/2} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{4 \left (x^4-x^2\right )^{3/4}}-\frac {\left (x^2-1\right )^{3/4} x^{3/2} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{4 \left (x^4-x^2\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 298
Rule 329
Rule 331
Rule 2004
Rule 2032
Rubi steps
\begin {align*} \int \sqrt [4]{-x^2+x^4} \, dx &=\frac {1}{2} x \sqrt [4]{-x^2+x^4}-\frac {1}{4} \int \frac {x^2}{\left (-x^2+x^4\right )^{3/4}} \, dx\\ &=\frac {1}{2} x \sqrt [4]{-x^2+x^4}-\frac {\left (x^{3/2} \left (-1+x^2\right )^{3/4}\right ) \int \frac {\sqrt {x}}{\left (-1+x^2\right )^{3/4}} \, dx}{4 \left (-x^2+x^4\right )^{3/4}}\\ &=\frac {1}{2} x \sqrt [4]{-x^2+x^4}-\frac {\left (x^{3/2} \left (-1+x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{2 \left (-x^2+x^4\right )^{3/4}}\\ &=\frac {1}{2} x \sqrt [4]{-x^2+x^4}-\frac {\left (x^{3/2} \left (-1+x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \left (-x^2+x^4\right )^{3/4}}\\ &=\frac {1}{2} x \sqrt [4]{-x^2+x^4}-\frac {\left (x^{3/2} \left (-1+x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \left (-x^2+x^4\right )^{3/4}}+\frac {\left (x^{3/2} \left (-1+x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \left (-x^2+x^4\right )^{3/4}}\\ &=\frac {1}{2} x \sqrt [4]{-x^2+x^4}+\frac {x^{3/2} \left (-1+x^2\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \left (-x^2+x^4\right )^{3/4}}-\frac {x^{3/2} \left (-1+x^2\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \left (-x^2+x^4\right )^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 42, normalized size = 0.71 \begin {gather*} \frac {2 x \sqrt [4]{x^2 \left (x^2-1\right )} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};x^2\right )}{3 \sqrt [4]{1-x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 59, normalized size = 1.00 \begin {gather*} \frac {1}{2} x \sqrt [4]{-x^2+x^4}+\frac {1}{4} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-x^2+x^4}}\right )-\frac {1}{4} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x^2+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 2.74, size = 110, normalized size = 1.86 \begin {gather*} \frac {1}{2} \, {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} x - \frac {1}{8} \, \arctan \left (\frac {2 \, {\left ({\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} x^{2} + {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}\right )}}{x}\right ) + \frac {1}{8} \, \log \left (-\frac {2 \, x^{3} - 2 \, {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{4} - x^{2}} x - x - 2 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 57, normalized size = 0.97 \begin {gather*} -\frac {1}{2} \, x^{2} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + \frac {1}{4} \, \arctan \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{8} \, \log \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {1}{8} \, \log \left (-{\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.40, size = 33, normalized size = 0.56
method | result | size |
meijerg | \(\frac {2 \mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{4}} x^{\frac {3}{2}} \hypergeom \left (\left [-\frac {1}{4}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], x^{2}\right )}{3 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{4}}}\) | \(33\) |
trager | \(\frac {x \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{2}-\frac {\ln \left (\frac {2 \left (x^{4}-x^{2}\right )^{\frac {3}{4}}+2 \sqrt {x^{4}-x^{2}}\, x +2 x^{2} \left (x^{4}-x^{2}\right )^{\frac {1}{4}}+2 x^{3}-x}{x}\right )}{8}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}-x^{2}}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x -2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{4}-x^{2}\right )^{\frac {3}{4}}-2 x^{2} \left (x^{4}-x^{2}\right )^{\frac {1}{4}}+\RootOf \left (\textit {\_Z}^{2}+1\right ) x}{x}\right )}{8}\) | \(157\) |
risch | \(\frac {x \left (x^{2} \left (x^{2}-1\right )\right )^{\frac {1}{4}}}{2}+\frac {\left (-\frac {\ln \left (\frac {2 x^{6}+2 \left (x^{8}-3 x^{6}+3 x^{4}-x^{2}\right )^{\frac {1}{4}} x^{4}+2 \sqrt {x^{8}-3 x^{6}+3 x^{4}-x^{2}}\, x^{2}-5 x^{4}+2 \left (x^{8}-3 x^{6}+3 x^{4}-x^{2}\right )^{\frac {3}{4}}-4 \left (x^{8}-3 x^{6}+3 x^{4}-x^{2}\right )^{\frac {1}{4}} x^{2}-2 \sqrt {x^{8}-3 x^{6}+3 x^{4}-x^{2}}+4 x^{2}+2 \left (x^{8}-3 x^{6}+3 x^{4}-x^{2}\right )^{\frac {1}{4}}-1}{\left (-1+x \right )^{2} \left (1+x \right )^{2}}\right )}{8}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-2 \left (x^{8}-3 x^{6}+3 x^{4}-x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-2 x^{6}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{8}-3 x^{6}+3 x^{4}-x^{2}\right )^{\frac {3}{4}}+2 \sqrt {x^{8}-3 x^{6}+3 x^{4}-x^{2}}\, x^{2}+4 \left (x^{8}-3 x^{6}+3 x^{4}-x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+5 x^{4}-2 \sqrt {x^{8}-3 x^{6}+3 x^{4}-x^{2}}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{8}-3 x^{6}+3 x^{4}-x^{2}\right )^{\frac {1}{4}}-4 x^{2}+1}{\left (-1+x \right )^{2} \left (1+x \right )^{2}}\right )}{8}\right ) \left (x^{2} \left (x^{2}-1\right )\right )^{\frac {1}{4}} \left (x^{2} \left (x^{2}-1\right )^{3}\right )^{\frac {1}{4}}}{x \left (x^{2}-1\right )}\) | \(438\) |
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 {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.77, size = 31, normalized size = 0.53 \begin {gather*} \frac {2\,x\,{\left (x^4-x^2\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {3}{4};\ \frac {7}{4};\ x^2\right )}{3\,{\left (1-x^2\right )}^{1/4}} \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 \sqrt [4]{x^{4} - x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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