Optimal. Leaf size=47 \[ -\frac {\sqrt [4]{-1+x^4}}{x}-\frac {1}{2} \text {ArcTan}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {283, 338, 304,
209, 212} \begin {gather*} -\frac {1}{2} \text {ArcTan}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\frac {\sqrt [4]{x^4-1}}{x}+\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 283
Rule 304
Rule 338
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{-1+x^4}}{x^2} \, dx &=-\frac {\sqrt [4]{-1+x^4}}{x}+\int \frac {x^2}{\left (-1+x^4\right )^{3/4}} \, dx\\ &=-\frac {\sqrt [4]{-1+x^4}}{x}+\text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=-\frac {\sqrt [4]{-1+x^4}}{x}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=-\frac {\sqrt [4]{-1+x^4}}{x}-\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 47, normalized size = 1.00 \begin {gather*} -\frac {\sqrt [4]{-1+x^4}}{x}-\frac {1}{2} \text {ArcTan}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right ) \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 = 1.97, size = 33, normalized size = 0.70
method | result | size |
meijerg | \(-\frac {\mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{4}} \hypergeom \left (\left [-\frac {1}{4}, -\frac {1}{4}\right ], \left [\frac {3}{4}\right ], x^{4}\right )}{\left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{4}} x}\) | \(33\) |
risch | \(-\frac {\left (x^{4}-1\right )^{\frac {1}{4}}}{x}+\frac {\left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {3}{4}} x^{3} \hypergeom \left (\left [\frac {3}{4}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], x^{4}\right )}{3 \mathrm {signum}\left (x^{4}-1\right )^{\frac {3}{4}}}\) | \(46\) |
trager | \(-\frac {\left (x^{4}-1\right )^{\frac {1}{4}}}{x}-\frac {\ln \left (2 \left (x^{4}-1\right )^{\frac {3}{4}} x -2 x^{2} \sqrt {x^{4}-1}+2 x^{3} \left (x^{4}-1\right )^{\frac {1}{4}}-2 x^{4}+1\right )}{4}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (2 \sqrt {x^{4}-1}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (x^{4}-1\right )^{\frac {3}{4}} x -2 x^{3} \left (x^{4}-1\right )^{\frac {1}{4}}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{4}\) | \(125\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 59, normalized size = 1.26 \begin {gather*} -\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} + \frac {1}{2} \, \arctan \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, \log \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} + 1\right ) - \frac {1}{4} \, \log \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 85 vs.
\(2 (37) = 74\).
time = 1.17, size = 85, normalized size = 1.81 \begin {gather*} \frac {x \arctan \left (2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x\right ) + x \log \left (2 \, x^{4} + 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {x^{4} - 1} x^{2} + 2 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x - 1\right ) - 4 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{4 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.54, size = 34, normalized size = 0.72 \begin {gather*} \frac {e^{\frac {i \pi }{4}} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {x^{4}} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 59, normalized size = 1.26 \begin {gather*} \frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} - \frac {1}{2} \, \arctan \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} \, \log \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} + 1\right ) + \frac {1}{4} \, \log \left (-\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.68, size = 29, normalized size = 0.62 \begin {gather*} -\frac {{\left (x^4-1\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},-\frac {1}{4};\ \frac {3}{4};\ x^4\right )}{x\,{\left (1-x^4\right )}^{1/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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