Optimal. Leaf size=49 \[ -\frac {1}{8} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{8} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{4} \sqrt [4]{x^4+1} x^3 \]
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Rubi [A] time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {279, 331, 298, 203, 206} \begin {gather*} -\frac {1}{8} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{8} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{4} \sqrt [4]{x^4+1} x^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 279
Rule 298
Rule 331
Rubi steps
\begin {align*} \int x^2 \sqrt [4]{1+x^4} \, dx &=\frac {1}{4} x^3 \sqrt [4]{1+x^4}+\frac {1}{4} \int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx\\ &=\frac {1}{4} x^3 \sqrt [4]{1+x^4}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\frac {1}{4} x^3 \sqrt [4]{1+x^4}+\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\frac {1}{4} x^3 \sqrt [4]{1+x^4}-\frac {1}{8} \tan ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{8} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )\\ \end {align*}
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Mathematica [C] time = 0.00, size = 22, normalized size = 0.45 \begin {gather*} \frac {1}{3} x^3 \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};-x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.15, size = 49, normalized size = 1.00 \begin {gather*} \frac {1}{4} x^3 \sqrt [4]{1+x^4}-\frac {1}{8} \tan ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{8} \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 = 62, normalized size = 1.27 \begin {gather*} \frac {1}{4} \, {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + \frac {1}{8} \, \arctan \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{16} \, \log \left (\frac {x + {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{16} \, \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 = 0.51, size = 59, normalized size = 1.20 \begin {gather*} \frac {1}{4} \, {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + \frac {1}{8} \, \arctan \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{16} \, \log \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x} + 1\right ) - \frac {1}{16} \, \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 = 1.78, size = 17, normalized size = 0.35
method | result | size |
meijerg | \(\frac {\hypergeom \left (\left [-\frac {1}{4}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], -x^{4}\right ) x^{3}}{3}\) | \(17\) |
risch | \(\frac {x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}}{4}+\frac {\hypergeom \left (\left [\frac {3}{4}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], -x^{4}\right ) x^{3}}{12}\) | \(30\) |
trager | \(\frac {x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}}{4}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}+1}\, x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (x^{4}+1\right )^{\frac {3}{4}} x -2 x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{16}-\frac {\ln \left (2 \left (x^{4}+1\right )^{\frac {3}{4}} x -2 x^{2} \sqrt {x^{4}+1}+2 x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}-2 x^{4}-1\right )}{16}\) | \(127\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 72, normalized size = 1.47 \begin {gather*} \frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{4 \, x {\left (\frac {x^{4} + 1}{x^{4}} - 1\right )}} + \frac {1}{8} \, \arctan \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{16} \, \log \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x} + 1\right ) - \frac {1}{16} \, \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 [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x^2\,{\left (x^4+1\right )}^{1/4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.82, size = 31, normalized size = 0.63 \begin {gather*} \frac {x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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