Optimal. Leaf size=68 \[ -\frac {\sqrt [4]{4 x^4+3}}{x}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{\sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.02, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {277, 331, 298, 203, 206} \[ -\frac {\sqrt [4]{4 x^4+3}}{x}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{\sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 277
Rule 298
Rule 331
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{3+4 x^4}}{x^2} \, dx &=-\frac {\sqrt [4]{3+4 x^4}}{x}+4 \int \frac {x^2}{\left (3+4 x^4\right )^{3/4}} \, dx\\ &=-\frac {\sqrt [4]{3+4 x^4}}{x}+4 \operatorname {Subst}\left (\int \frac {x^2}{1-4 x^4} \, dx,x,\frac {x}{\sqrt [4]{3+4 x^4}}\right )\\ &=-\frac {\sqrt [4]{3+4 x^4}}{x}+\operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x}{\sqrt [4]{3+4 x^4}}\right )-\operatorname {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {x}{\sqrt [4]{3+4 x^4}}\right )\\ &=-\frac {\sqrt [4]{3+4 x^4}}{x}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{\sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 27, normalized size = 0.40 \[ -\frac {\sqrt [4]{3} \, _2F_1\left (-\frac {1}{4},-\frac {1}{4};\frac {3}{4};-\frac {4 x^4}{3}\right )}{x} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.16, size = 68, normalized size = 1.00 \[ -\frac {\sqrt [4]{4 x^4+3}}{x}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{\sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 5.60, size = 146, normalized size = 2.15 \[ -\frac {2 \, \sqrt {2} x \arctan \left (\frac {4}{3} \, \sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}} x^{3} + \frac {2}{3} \, \sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {3}{4}} x\right ) - \sqrt {2} x \log \left (-256 \, x^{8} - 192 \, x^{4} - 4 \, \sqrt {2} {\left (16 \, x^{5} + 3 \, x\right )} {\left (4 \, x^{4} + 3\right )}^{\frac {3}{4}} - 8 \, \sqrt {2} {\left (16 \, x^{7} + 9 \, x^{3}\right )} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}} - 16 \, {\left (8 \, x^{6} + 3 \, x^{2}\right )} \sqrt {4 \, x^{4} + 3} - 9\right ) + 8 \, {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{8 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.67, size = 83, normalized size = 1.22 \[ \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{2 \, x}\right ) - \frac {1}{4} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}{\sqrt {2} + \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}\right ) - \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.74, size = 20, normalized size = 0.29
method | result | size |
meijerg | \(-\frac {3^{\frac {1}{4}} \hypergeom \left (\left [-\frac {1}{4}, -\frac {1}{4}\right ], \left [\frac {3}{4}\right ], -\frac {4 x^{4}}{3}\right )}{x}\) | \(20\) |
risch | \(-\frac {\left (4 x^{4}+3\right )^{\frac {1}{4}}}{x}+\frac {4 \,3^{\frac {1}{4}} x^{3} \hypergeom \left (\left [\frac {3}{4}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], -\frac {4 x^{4}}{3}\right )}{9}\) | \(35\) |
trager | \(-\frac {\left (4 x^{4}+3\right )^{\frac {1}{4}}}{x}-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (4 \RootOf \left (\textit {\_Z}^{2}-2\right ) \sqrt {4 x^{4}+3}\, x^{2}+8 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{4}-4 \left (4 x^{4}+3\right )^{\frac {3}{4}} x -8 x^{3} \left (4 x^{4}+3\right )^{\frac {1}{4}}+3 \RootOf \left (\textit {\_Z}^{2}-2\right )\right )}{4}-\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (4 \RootOf \left (\textit {\_Z}^{2}+2\right ) \sqrt {4 x^{4}+3}\, x^{2}-8 \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{4}-4 \left (4 x^{4}+3\right )^{\frac {3}{4}} x +8 x^{3} \left (4 x^{4}+3\right )^{\frac {1}{4}}-3 \RootOf \left (\textit {\_Z}^{2}+2\right )\right )}{4}\) | \(166\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 83, normalized size = 1.22 \[ \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{2 \, x}\right ) - \frac {1}{4} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}{\sqrt {2} + \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}\right ) - \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 18, normalized size = 0.26 \[ -\frac {3^{1/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},-\frac {1}{4};\ \frac {3}{4};\ -\frac {4\,x^4}{3}\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.03, size = 41, normalized size = 0.60 \[ \frac {\sqrt [4]{3} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {4 x^{4} e^{i \pi }}{3}} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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