Optimal. Leaf size=74 \[ \frac {1}{3} \left (1+x^4\right )^{3/4}-\frac {1}{2} \text {ArcTan}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{2} \text {ArcTan}\left (\sqrt [4]{1+x^4}\right )-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{2} \tanh ^{-1}\left (\sqrt [4]{1+x^4}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1847, 246,
218, 212, 209, 457, 81, 65, 304} \begin {gather*} -\frac {1}{2} \text {ArcTan}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )-\frac {1}{2} \text {ArcTan}\left (\sqrt [4]{x^4+1}\right )+\frac {1}{3} \left (x^4+1\right )^{3/4}-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \tanh ^{-1}\left (\sqrt [4]{x^4+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 81
Rule 209
Rule 212
Rule 218
Rule 246
Rule 304
Rule 457
Rule 1847
Rubi steps
\begin {align*} \int \frac {-1-x+x^4}{x \sqrt [4]{1+x^4}} \, dx &=\int \left (-\frac {1}{\sqrt [4]{1+x^4}}+\frac {-1+x^4}{x \sqrt [4]{1+x^4}}\right ) \, dx\\ &=-\int \frac {1}{\sqrt [4]{1+x^4}} \, dx+\int \frac {-1+x^4}{x \sqrt [4]{1+x^4}} \, dx\\ &=\frac {1}{4} \text {Subst}\left (\int \frac {-1+x}{x \sqrt [4]{1+x}} \, dx,x,x^4\right )-\text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\frac {1}{3} \left (1+x^4\right )^{3/4}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{x \sqrt [4]{1+x}} \, dx,x,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 {1}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\frac {1}{3} \left (1+x^4\right )^{3/4}-\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 )-\text {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt [4]{1+x^4}\right )\\ &=\frac {1}{3} \left (1+x^4\right )^{3/4}-\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 )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{1+x^4}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{1+x^4}\right )\\ &=\frac {1}{3} \left (1+x^4\right )^{3/4}-\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{2} \tan ^{-1}\left (\sqrt [4]{1+x^4}\right )-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{2} \tanh ^{-1}\left (\sqrt [4]{1+x^4}\right )\\ \end {align*}
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Mathematica [A]
time = 5.26, size = 68, normalized size = 0.92 \begin {gather*} \frac {1}{6} \left (2 \left (1+x^4\right )^{3/4}-3 \text {ArcTan}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-3 \text {ArcTan}\left (\sqrt [4]{1+x^4}\right )-3 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )+3 \tanh ^{-1}\left (\sqrt [4]{1+x^4}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
3.
time = 2.51, size = 90, normalized size = 1.22
method | result | size |
meijerg | \(-\frac {\sqrt {2}\, \Gamma \left (\frac {3}{4}\right ) \left (-\frac {\pi \sqrt {2}\, x^{4} \hypergeom \left (\left [1, 1, \frac {5}{4}\right ], \left [2, 2\right ], -x^{4}\right )}{4 \Gamma \left (\frac {3}{4}\right )}+\frac {\left (-3 \ln \left (2\right )-\frac {\pi }{2}+4 \ln \left (x \right )\right ) \pi \sqrt {2}}{\Gamma \left (\frac {3}{4}\right )}\right )}{8 \pi }+\frac {x^{4} \hypergeom \left (\left [\frac {1}{4}, 1\right ], \left [2\right ], -x^{4}\right )}{4}-x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{4}\right ], \left [\frac {5}{4}\right ], -x^{4}\right )\) | \(90\) |
trager | \(\frac {\left (x^{4}+1\right )^{\frac {3}{4}}}{3}+\frac {\ln \left (\frac {\left (x^{4}+1\right )^{\frac {3}{4}} x -\sqrt {x^{4}+1}\, x^{2}+x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}-x^{4}-\left (x^{4}+1\right )^{\frac {3}{4}}+\sqrt {x^{4}+1}\, x -x^{2} \left (x^{4}+1\right )^{\frac {1}{4}}+x^{3}-\sqrt {x^{4}+1}+x \left (x^{4}+1\right )^{\frac {1}{4}}-\left (x^{4}+1\right )^{\frac {1}{4}}+x -1}{x^{2}}\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}+1}\, x^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+\left (x^{4}+1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}+1}\, x -x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-\left (x^{4}+1\right )^{\frac {3}{4}}-\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}+1}+x^{2} \left (x^{4}+1\right )^{\frac {1}{4}}-x \left (x^{4}+1\right )^{\frac {1}{4}}-x \RootOf \left (\textit {\_Z}^{2}+1\right )+\left (x^{4}+1\right )^{\frac {1}{4}}+\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{2}}\right )}{2}\) | \(275\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 90, normalized size = 1.22 \begin {gather*} \frac {1}{3} \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} - \frac {1}{2} \, \arctan \left ({\left (x^{4} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{2} \, \arctan \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {1}{4} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{4}} - 1\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. 123 vs.
\(2 (54) = 108\).
time = 4.78, size = 123, normalized size = 1.66 \begin {gather*} \frac {1}{3} \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} - \frac {1}{2} \, \arctan \left (\frac {{\left (x^{4} + 1\right )}^{\frac {3}{4}} {\left (x - 1\right )} - {\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{2} - x\right )}}{x^{4} - x^{2} + 1}\right ) + \frac {1}{2} \, \log \left (-\frac {x^{4} - x^{3} - {\left (x^{4} + 1\right )}^{\frac {3}{4}} {\left (x - 1\right )} + \sqrt {x^{4} + 1} {\left (x^{2} - x + 1\right )} - {\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{3} - x^{2} + x - 1\right )} - x + 1}{x^{2}}\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 = 2.49, size = 66, normalized size = 0.89 \begin {gather*} - \frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {\left (x^{4} + 1\right )^{\frac {3}{4}}}{3} + \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^{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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.24, size = 43, normalized size = 0.58 \begin {gather*} \frac {\mathrm {atanh}\left ({\left (x^4+1\right )}^{1/4}\right )}{2}-\frac {\mathrm {atan}\left ({\left (x^4+1\right )}^{1/4}\right )}{2}-x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{4};\ \frac {5}{4};\ -x^4\right )+\frac {{\left (x^4+1\right )}^{3/4}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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