Optimal. Leaf size=47 \[ \frac {1}{5} \sqrt [4]{x^4+1} \left (x^4-4\right )+\frac {1}{2} \tan ^{-1}\left (\sqrt [4]{x^4+1}\right )+\frac {1}{2} \tanh ^{-1}\left (\sqrt [4]{x^4+1}\right ) \]
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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 = {446, 80, 50, 63, 212, 206, 203} \begin {gather*} \frac {1}{5} \left (x^4+1\right )^{5/4}-\sqrt [4]{x^4+1}+\frac {1}{2} \tan ^{-1}\left (\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 50
Rule 63
Rule 80
Rule 203
Rule 206
Rule 212
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (-1+x^4\right ) \sqrt [4]{1+x^4}}{x} \, dx &=\frac {1}{4} \operatorname {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} \operatorname {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} \operatorname {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}-\operatorname {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} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{1+x^4}\right )+\frac {1}{2} \operatorname {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 = 0.02, size = 45, normalized size = 0.96 \begin {gather*} \frac {1}{10} \left (2 \sqrt [4]{x^4+1} \left (x^4-4\right )+5 \tan ^{-1}\left (\sqrt [4]{x^4+1}\right )+5 \tanh ^{-1}\left (\sqrt [4]{x^4+1}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [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} \tan ^{-1}\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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fricas [A] time = 0.47, 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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giac [A] time = 0.29, 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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maple [C] time = 4.36, size = 62, normalized size = 1.32
method | result | size |
meijerg | \(\frac {-4 \left (4-3 \ln \relax (2)+\frac {\pi }{2}+4 \ln \relax (x )\right ) \Gamma \left (\frac {3}{4}\right )-\hypergeom \left (\left [\frac {3}{4}, 1, 1\right ], \left [2, 2\right ], -x^{4}\right ) \Gamma \left (\frac {3}{4}\right ) x^{4}}{16 \Gamma \left (\frac {3}{4}\right )}+\frac {\hypergeom \left (\left [-\frac {1}{4}, 1\right ], \relax [2], -x^{4}\right ) x^{4}}{4}\) | \(62\) |
trager | \(\left (\frac {x^{4}}{5}-\frac {4}{5}\right ) \left (x^{4}+1\right )^{\frac {1}{4}}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}+1}+2 \left (x^{4}+1\right )^{\frac {3}{4}}-2 \RootOf \left (\textit {\_Z}^{2}+1\right )-2 \left (x^{4}+1\right )^{\frac {1}{4}}}{x^{4}}\right )}{4}-\frac {\ln \left (\frac {-x^{4}+2 \left (x^{4}+1\right )^{\frac {3}{4}}-2 \sqrt {x^{4}+1}+2 \left (x^{4}+1\right )^{\frac {1}{4}}-2}{x^{4}}\right )}{4}\) | \(124\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, 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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sympy [A] time = 37.10, 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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