Optimal. Leaf size=33 \[ \frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {240, 212, 206, 203} \begin {gather*} \frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\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 203
Rule 206
Rule 212
Rule 240
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [4]{1+x^4}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\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 = 0.00, size = 54, normalized size = 1.64 \begin {gather*} -\frac {1}{4} \log \left (1-\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{4} \log \left (\frac {x}{\sqrt [4]{x^4+1}}+1\right )+\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.15, size = 33, normalized size = 1.00 \begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 50, normalized size = 1.52 \begin {gather*} -\frac {1}{2} \, \arctan \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, \log \left (\frac {x + {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} \, \log \left (-\frac {x - {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 47, normalized size = 1.42 \begin {gather*} -\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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maple [C] time = 2.25, size = 14, normalized size = 0.42
method | result | size |
meijerg | \(\hypergeom \left (\left [\frac {1}{4}, \frac {1}{4}\right ], \left [\frac {5}{4}\right ], -x^{4}\right ) x\) | \(14\) |
trager | \(-\frac {\ln \left (2 \left (x^{4}+1\right )^{\frac {3}{4}} x -2 x^{2} \sqrt {x^{4}+1}+2 x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}-2 x^{4}-1\right )}{4}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}+1}\, x^{2}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (x^{4}+1\right )^{\frac {3}{4}} x -2 x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{4}\) | \(113\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 47, normalized size = 1.42 \begin {gather*} -\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 = 0.20, size = 12, normalized size = 0.36 \begin {gather*} x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{4};\ \frac {5}{4};\ -x^4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.72, size = 27, normalized size = 0.82 \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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