Optimal. Leaf size=47 \[ \frac {1}{5} \left (-4+x^4\right ) \sqrt [4]{1+x^4}+\frac {1}{2} \text {ArcTan}\left (\sqrt [4]{1+x^4}\right )+\frac {1}{2} \tanh ^{-1}\left (\sqrt [4]{1+x^4}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.13, number of steps
used = 7, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {457, 81, 52, 65,
218, 212, 209} \begin {gather*} \frac {1}{2} \text {ArcTan}\left (\sqrt [4]{x^4+1}\right )+\frac {1}{5} \left (x^4+1\right )^{5/4}-\sqrt [4]{x^4+1}+\frac {1}{2} \tanh ^{-1}\left (\sqrt [4]{x^4+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 81
Rule 209
Rule 212
Rule 218
Rule 457
Rubi steps
\begin {align*} \int \frac {\left (-1+x^4\right ) \sqrt [4]{1+x^4}}{x} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {(-1+x) \sqrt [4]{1+x}}{x} \, dx,x,x^4\right )\\ &=\frac {1}{5} \left (1+x^4\right )^{5/4}-\frac {1}{4} \text {Subst}\left (\int \frac {\sqrt [4]{1+x}}{x} \, dx,x,x^4\right )\\ &=-\sqrt [4]{1+x^4}+\frac {1}{5} \left (1+x^4\right )^{5/4}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{x (1+x)^{3/4}} \, dx,x,x^4\right )\\ &=-\sqrt [4]{1+x^4}+\frac {1}{5} \left (1+x^4\right )^{5/4}-\text {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt [4]{1+x^4}\right )\\ &=-\sqrt [4]{1+x^4}+\frac {1}{5} \left (1+x^4\right )^{5/4}+\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 )\\ &=-\sqrt [4]{1+x^4}+\frac {1}{5} \left (1+x^4\right )^{5/4}+\frac {1}{2} \tan ^{-1}\left (\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 = 0.04, size = 47, normalized size = 1.00 \begin {gather*} \frac {1}{5} \left (-4+x^4\right ) \sqrt [4]{1+x^4}+\frac {1}{2} \text {ArcTan}\left (\sqrt [4]{1+x^4}\right )+\frac {1}{2} \tanh ^{-1}\left (\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 5 vs. order
3.
time = 3.07, size = 62, normalized size = 1.32
method | result | size |
meijerg | \(\frac {-\Gamma \left (\frac {3}{4}\right ) x^{4} \hypergeom \left (\left [\frac {3}{4}, 1, 1\right ], \left [2, 2\right ], -x^{4}\right )-4 \left (4-3 \ln \left (2\right )+\frac {\pi }{2}+4 \ln \left (x \right )\right ) \Gamma \left (\frac {3}{4}\right )}{16 \Gamma \left (\frac {3}{4}\right )}+\frac {x^{4} \hypergeom \left (\left [-\frac {1}{4}, 1\right ], \left [2\right ], -x^{4}\right )}{4}\) | \(62\) |
trager | \(\left (\frac {x^{4}}{5}-\frac {4}{5}\right ) \left (x^{4}+1\right )^{\frac {1}{4}}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (x^{4}+1\right )^{\frac {3}{4}}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}+1}-2 \left (x^{4}+1\right )^{\frac {1}{4}}-2 \RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{4}}\right )}{4}-\frac {\ln \left (\frac {-x^{4}+2 \left (x^{4}+1\right )^{\frac {3}{4}}-2 \sqrt {x^{4}+1}+2 \left (x^{4}+1\right )^{\frac {1}{4}}-2}{x^{4}}\right )}{4}\) | \(125\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 53, normalized size = 1.13 \begin {gather*} \frac {1}{5} \, {\left (x^{4} + 1\right )}^{\frac {5}{4}} - {\left (x^{4} + 1\right )}^{\frac {1}{4}} + \frac {1}{2} \, \arctan \left ({\left (x^{4} + 1\right )}^{\frac {1}{4}}\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 49, normalized size = 1.04 \begin {gather*} \frac {1}{5} \, {\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{4} - 4\right )} + \frac {1}{2} \, \arctan \left ({\left (x^{4} + 1\right )}^{\frac {1}{4}}\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 22.88, size = 56, normalized size = 1.19 \begin {gather*} \frac {\left (x^{4} + 1\right )^{\frac {5}{4}}}{5} - \sqrt [4]{x^{4} + 1} - \frac {\log {\left (\sqrt [4]{x^{4} + 1} - 1 \right )}}{4} + \frac {\log {\left (\sqrt [4]{x^{4} + 1} + 1 \right )}}{4} + \frac {\operatorname {atan}{\left (\sqrt [4]{x^{4} + 1} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 53, normalized size = 1.13 \begin {gather*} \frac {1}{5} \, {\left (x^{4} + 1\right )}^{\frac {5}{4}} - {\left (x^{4} + 1\right )}^{\frac {1}{4}} + \frac {1}{2} \, \arctan \left ({\left (x^{4} + 1\right )}^{\frac {1}{4}}\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.73, size = 39, normalized size = 0.83 \begin {gather*} \frac {\mathrm {atan}\left ({\left (x^4+1\right )}^{1/4}\right )}{2}+\frac {\mathrm {atanh}\left ({\left (x^4+1\right )}^{1/4}\right )}{2}-{\left (x^4+1\right )}^{1/4}+\frac {{\left (x^4+1\right )}^{5/4}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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