Optimal. Leaf size=77 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{-1+x^4}}{-1+\sqrt {-1+x^4}}\right )}{2 \sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-1+x^4}}{1+\sqrt {-1+x^4}}\right )}{2 \sqrt {2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 129, normalized size of antiderivative = 1.68, number of steps
used = 11, number of rules used = 8, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {272, 65, 303,
1176, 631, 210, 1179, 642} \begin {gather*} -\frac {\text {ArcTan}\left (1-\sqrt {2} \sqrt [4]{x^4-1}\right )}{2 \sqrt {2}}+\frac {\text {ArcTan}\left (\sqrt {2} \sqrt [4]{x^4-1}+1\right )}{2 \sqrt {2}}+\frac {\log \left (\sqrt {x^4-1}-\sqrt {2} \sqrt [4]{x^4-1}+1\right )}{4 \sqrt {2}}-\frac {\log \left (\sqrt {x^4-1}+\sqrt {2} \sqrt [4]{x^4-1}+1\right )}{4 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 210
Rule 272
Rule 303
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt [4]{-1+x^4}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x} x} \, dx,x,x^4\right )\\ &=\text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt [4]{-1+x^4}\right )\\ &=-\left (\frac {1}{2} \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt [4]{-1+x^4}\right )\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt [4]{-1+x^4}\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+x^4}\right )+\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+x^4}\right )+\frac {\text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt [4]{-1+x^4}\right )}{4 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt [4]{-1+x^4}\right )}{4 \sqrt {2}}\\ &=\frac {\log \left (1-\sqrt {2} \sqrt [4]{-1+x^4}+\sqrt {-1+x^4}\right )}{4 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} \sqrt [4]{-1+x^4}+\sqrt {-1+x^4}\right )}{4 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt [4]{-1+x^4}\right )}{2 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt [4]{-1+x^4}\right )}{2 \sqrt {2}}\\ &=-\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt [4]{-1+x^4}\right )}{2 \sqrt {2}}+\frac {\tan ^{-1}\left (1+\sqrt {2} \sqrt [4]{-1+x^4}\right )}{2 \sqrt {2}}+\frac {\log \left (1-\sqrt {2} \sqrt [4]{-1+x^4}+\sqrt {-1+x^4}\right )}{4 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} \sqrt [4]{-1+x^4}+\sqrt {-1+x^4}\right )}{4 \sqrt {2}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 68, normalized size = 0.88 \begin {gather*} \frac {\text {ArcTan}\left (\frac {-1+\sqrt {-1+x^4}}{\sqrt {2} \sqrt [4]{-1+x^4}}\right )-\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-1+x^4}}{1+\sqrt {-1+x^4}}\right )}{2 \sqrt {2}} \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 = 5.27, size = 79, normalized size = 1.03
method | result | size |
meijerg | \(\frac {\sqrt {2}\, \Gamma \left (\frac {3}{4}\right ) \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{4}} \left (\frac {\pi \sqrt {2}\, x^{4} \hypergeom \left (\left [1, 1, \frac {5}{4}\right ], \left [2, 2\right ], x^{4}\right )}{4 \Gamma \left (\frac {3}{4}\right )}+\frac {\left (-3 \ln \left (2\right )-\frac {\pi }{2}+4 \ln \left (x \right )+i \pi \right ) \pi \sqrt {2}}{\Gamma \left (\frac {3}{4}\right )}\right )}{8 \pi \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{4}}}\) | \(79\) |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}-\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{4}+2 \left (x^{4}-1\right )^{\frac {3}{4}}-2 \left (x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )}{x^{4}}\right )}{4}-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+2 \left (x^{4}-1\right )^{\frac {3}{4}}-2 \sqrt {x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )+2 \left (x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{x^{4}}\right )}{4}\) | \(157\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 102, normalized size = 1.32 \begin {gather*} \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (\sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{4} - 1} + 1\right ) + \frac {1}{8} \, \sqrt {2} \log \left (-\sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{4} - 1} + 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. 155 vs.
\(2 (57) = 114\).
time = 0.36, size = 155, normalized size = 2.01 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \arctan \left (\sqrt {2} \sqrt {\sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{4} - 1} + 1} - \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} - 1\right ) - \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-4 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{4} - 1} + 4} - \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {1}{8} \, \sqrt {2} \log \left (4 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{4} - 1} + 4\right ) + \frac {1}{8} \, \sqrt {2} \log \left (-4 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{4} - 1} + 4\right ) \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 = 0.45, size = 31, normalized size = 0.40 \begin {gather*} - \frac {\Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{4}}} \right )}}{4 x \Gamma \left (\frac {5}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 102, normalized size = 1.32 \begin {gather*} \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (\sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{4} - 1} + 1\right ) + \frac {1}{8} \, \sqrt {2} \log \left (-\sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{4} - 1} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.89, size = 45, normalized size = 0.58 \begin {gather*} \sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (x^4-1\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (x^4-1\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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