Optimal. Leaf size=54 \[ \frac {\sqrt [4]{x^4+1} \left (x^4+4\right )}{4 x}+\frac {3}{8} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )-\frac {3}{8} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 62, normalized size of antiderivative = 1.15, number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {453, 279, 331, 298, 203, 206} \begin {gather*} \frac {\left (x^4+1\right )^{5/4}}{x}+\frac {3}{8} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )-\frac {3}{8} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )-\frac {3}{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
Rule 453
Rubi steps
\begin {align*} \int \frac {\left (-1+x^4\right ) \sqrt [4]{1+x^4}}{x^2} \, dx &=\frac {\left (1+x^4\right )^{5/4}}{x}-3 \int x^2 \sqrt [4]{1+x^4} \, dx\\ &=-\frac {3}{4} x^3 \sqrt [4]{1+x^4}+\frac {\left (1+x^4\right )^{5/4}}{x}-\frac {3}{4} \int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx\\ &=-\frac {3}{4} x^3 \sqrt [4]{1+x^4}+\frac {\left (1+x^4\right )^{5/4}}{x}-\frac {3}{4} \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=-\frac {3}{4} x^3 \sqrt [4]{1+x^4}+\frac {\left (1+x^4\right )^{5/4}}{x}-\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=-\frac {3}{4} x^3 \sqrt [4]{1+x^4}+\frac {\left (1+x^4\right )^{5/4}}{x}+\frac {3}{8} \tan ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {3}{8} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 34, normalized size = 0.63 \begin {gather*} \frac {\left (x^4+1\right )^{5/4}}{x}-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 = 0.17, size = 54, normalized size = 1.00 \begin {gather*} \frac {\sqrt [4]{1+x^4} \left (4+x^4\right )}{4 x}+\frac {3}{8} \tan ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {3}{8} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 2.58, size = 92, normalized size = 1.70 \begin {gather*} \frac {3 \, x \arctan \left (2 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 2 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x\right ) + 3 \, x \log \left (-2 \, x^{4} + 2 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 2 \, \sqrt {x^{4} + 1} x^{2} + 2 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x - 1\right ) + 4 \, {\left (x^{4} + 4\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{16 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 70, normalized size = 1.30 \begin {gather*} \frac {1}{4} \, {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + \frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x} - \frac {3}{8} \, \arctan \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {3}{16} \, \log \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x} + 1\right ) + \frac {3}{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 = 4.21, size = 33, normalized size = 0.61
method | result | size |
meijerg | \(\frac {\hypergeom \left (\left [-\frac {1}{4}, -\frac {1}{4}\right ], \left [\frac {3}{4}\right ], -x^{4}\right )}{x}+\frac {x^{3} \hypergeom \left (\left [-\frac {1}{4}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], -x^{4}\right )}{3}\) | \(33\) |
risch | \(\frac {x^{8}+5 x^{4}+4}{4 x \left (x^{4}+1\right )^{\frac {3}{4}}}-\frac {x^{3} \hypergeom \left (\left [\frac {3}{4}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], -x^{4}\right )}{4}\) | \(40\) |
trager | \(\frac {\left (x^{4}+1\right )^{\frac {1}{4}} \left (x^{4}+4\right )}{4 x}-\frac {3 \ln \left (-2 \left (x^{4}+1\right )^{\frac {3}{4}} x -2 x^{2} \sqrt {x^{4}+1}-2 x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}-2 x^{4}-1\right )}{16}+\frac {3 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}+1}\, x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-2 \left (x^{4}+1\right )^{\frac {3}{4}} x +2 x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{16}\) | \(132\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 83, normalized size = 1.54 \begin {gather*} \frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x} + \frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{4 \, x {\left (\frac {x^{4} + 1}{x^{4}} - 1\right )}} - \frac {3}{8} \, \arctan \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {3}{16} \, \log \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x} + 1\right ) + \frac {3}{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 \frac {\left (x^4-1\right )\,{\left (x^4+1\right )}^{1/4}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.46, size = 65, normalized size = 1.20 \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 )} - \frac {\Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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