Optimal. Leaf size=33 \[ \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^2}}\right )+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^2}}\right ) \]
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Rubi [B] time = 0.02, antiderivative size = 89, normalized size of antiderivative = 2.70, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2011, 329, 240, 212, 206, 203} \begin {gather*} \frac {\sqrt {x} \sqrt [4]{x^2-1} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{\sqrt [4]{x^4-x^2}}+\frac {\sqrt {x} \sqrt [4]{x^2-1} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{\sqrt [4]{x^4-x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 240
Rule 329
Rule 2011
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [4]{-x^2+x^4}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt [4]{-1+x^2}} \, dx}{\sqrt [4]{-x^2+x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-x^2+x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}\\ &=\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}+\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}\\ &=\frac {\sqrt {x} \sqrt [4]{-1+x^2} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}+\frac {\sqrt {x} \sqrt [4]{-1+x^2} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 61, normalized size = 1.85 \begin {gather*} \frac {\sqrt {x} \sqrt [4]{x^2-1} \left (\tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}\right )+\tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}\right )\right )}{\sqrt [4]{x^2 \left (x^2-1\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 33, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-x^2+x^4}}\right )+\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 = 3.03, size = 95, normalized size = 2.88 \begin {gather*} -\frac {1}{2} \, \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}{2} \, \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 = 0.35, size = 41, normalized size = 1.24 \begin {gather*} \arctan \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{2} \, \log \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{2} \, \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 = 1.30, size = 33, normalized size = 1.00
method | result | size |
meijerg | \(\frac {2 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{4}} \hypergeom \left (\left [\frac {1}{4}, \frac {1}{4}\right ], \left [\frac {5}{4}\right ], x^{2}\right ) \sqrt {x}}{\mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{4}}}\) | \(33\) |
trager | \(\frac {\ln \left (\frac {2 \left (x^{4}-x^{2}\right )^{\frac {3}{4}}+2 \sqrt {x^{4}-x^{2}}\, x +2 x^{2} \left (x^{4}-x^{2}\right )^{\frac {1}{4}}+2 x^{3}-x}{x}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}-x^{2}}\, x +2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{4}-x^{2}\right )^{\frac {3}{4}}-2 x^{2} \left (x^{4}-x^{2}\right )^{\frac {1}{4}}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x}{x}\right )}{2}\) | \(144\) |
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 {1}{{\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 = 0.40, size = 31, normalized size = 0.94 \begin {gather*} \frac {2\,x\,{\left (1-x^2\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{4};\ \frac {5}{4};\ x^2\right )}{{\left (x^4-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 \frac {1}{\sqrt [4]{x^{4} - x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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