Optimal. Leaf size=81 \[ \frac {\tan ^{-1}\left (\frac {\left (1-\sqrt [3]{3 x^2+1}\right )^2}{3 \sqrt {3} x}\right )}{4 \sqrt {3}}-\frac {1}{4} \tanh ^{-1}\left (\frac {1-\sqrt [3]{3 x^2+1}}{x}\right )+\frac {\tan ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{4 \sqrt {3}} \]
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Rubi [A] time = 0.01, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {394} \[ \frac {\tan ^{-1}\left (\frac {\left (1-\sqrt [3]{3 x^2+1}\right )^2}{3 \sqrt {3} x}\right )}{4 \sqrt {3}}-\frac {1}{4} \tanh ^{-1}\left (\frac {1-\sqrt [3]{3 x^2+1}}{x}\right )+\frac {\tan ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{4 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 394
Rubi steps
\begin {align*} \int \frac {1}{\left (3+x^2\right ) \sqrt [3]{1+3 x^2}} \, dx &=\frac {\tan ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\left (1-\sqrt [3]{1+3 x^2}\right )^2}{3 \sqrt {3} x}\right )}{4 \sqrt {3}}-\frac {1}{4} \tanh ^{-1}\left (\frac {1-\sqrt [3]{1+3 x^2}}{x}\right )\\ \end {align*}
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Mathematica [C] time = 0.10, size = 126, normalized size = 1.56 \[ -\frac {9 x F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};-3 x^2,-\frac {x^2}{3}\right )}{\left (x^2+3\right ) \sqrt [3]{3 x^2+1} \left (2 x^2 \left (F_1\left (\frac {3}{2};\frac {1}{3},2;\frac {5}{2};-3 x^2,-\frac {x^2}{3}\right )+3 F_1\left (\frac {3}{2};\frac {4}{3},1;\frac {5}{2};-3 x^2,-\frac {x^2}{3}\right )\right )-9 F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};-3 x^2,-\frac {x^2}{3}\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 2.76, size = 345, normalized size = 4.26 \[ \frac {1}{36} \, \sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} {\left (3 \, x^{4} - 10 \, x^{3} - 36 \, x^{2} + 18 \, x + 9\right )} {\left (3 \, x^{2} + 1\right )}^{\frac {2}{3}} - 4 \, \sqrt {3} {\left (x^{5} + 15 \, x^{4} - 26 \, x^{3} - 54 \, x^{2} + 9 \, x - 9\right )} {\left (3 \, x^{2} + 1\right )}^{\frac {1}{3}} + \sqrt {3} {\left (x^{6} - 2 \, x^{5} - 105 \, x^{4} - 28 \, x^{3} + 63 \, x^{2} + 126 \, x + 9\right )}}{x^{6} + 126 \, x^{5} - 225 \, x^{4} - 828 \, x^{3} - 81 \, x^{2} - 162 \, x + 81}\right ) - \frac {1}{36} \, \sqrt {3} \arctan \left (\frac {2 \, {\left (2 \, \sqrt {3} {\left (23 \, x^{3} + 9 \, x\right )} {\left (3 \, x^{2} + 1\right )}^{\frac {2}{3}} + \sqrt {3} {\left (x^{5} - 80 \, x^{3} - 9 \, x\right )} {\left (3 \, x^{2} + 1\right )}^{\frac {1}{3}} + \sqrt {3} {\left (11 \, x^{5} + 10 \, x^{3} - 9 \, x\right )}\right )}}{x^{6} - 657 \, x^{4} - 189 \, x^{2} - 27}\right ) + \frac {1}{24} \, \log \left (\frac {x^{6} + 108 \, x^{5} + 549 \, x^{4} + 360 \, x^{3} + 99 \, x^{2} + 6 \, {\left (3 \, x^{4} + 32 \, x^{3} + 42 \, x^{2} + 3\right )} {\left (3 \, x^{2} + 1\right )}^{\frac {2}{3}} + 6 \, {\left (x^{5} + 27 \, x^{4} + 70 \, x^{3} + 18 \, x^{2} + 9 \, x + 3\right )} {\left (3 \, x^{2} + 1\right )}^{\frac {1}{3}} + 108 \, x - 9}{x^{6} + 9 \, x^{4} + 27 \, x^{2} + 27}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (3 \, x^{2} + 1\right )}^{\frac {1}{3}} {\left (x^{2} + 3\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.46, size = 444, normalized size = 5.48 \[ \RootOf \left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \ln \left (\frac {-12 x^{2} \RootOf \left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-x^{2}+24 \left (3 x^{2}+1\right )^{\frac {1}{3}} x \RootOf \left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-48 x \RootOf \left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+4 \left (3 x^{2}+1\right )^{\frac {1}{3}} x -4 x +24 \left (3 x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+12 \RootOf \left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-2 \left (3 x^{2}+1\right )^{\frac {2}{3}}+4 \left (3 x^{2}+1\right )^{\frac {1}{3}}+1}{x^{2}+3}\right )-\RootOf \left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \ln \left (-\frac {-6 x^{2} \RootOf \left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-x^{2}+12 \left (3 x^{2}+1\right )^{\frac {1}{3}} x \RootOf \left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-24 x \RootOf \left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+\left (3 x^{2}+1\right )^{\frac {1}{3}} x -4 x +12 \left (3 x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+6 \RootOf \left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+\left (3 x^{2}+1\right )^{\frac {2}{3}}+\left (3 x^{2}+1\right )^{\frac {1}{3}}+1}{x^{2}+3}\right )-\frac {\ln \left (-\frac {-6 x^{2} \RootOf \left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-x^{2}+12 \left (3 x^{2}+1\right )^{\frac {1}{3}} x \RootOf \left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-24 x \RootOf \left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+\left (3 x^{2}+1\right )^{\frac {1}{3}} x -4 x +12 \left (3 x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+6 \RootOf \left (48 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+\left (3 x^{2}+1\right )^{\frac {2}{3}}+\left (3 x^{2}+1\right )^{\frac {1}{3}}+1}{x^{2}+3}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (3 \, x^{2} + 1\right )}^{\frac {1}{3}} {\left (x^{2} + 3\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\left (x^2+3\right )\,{\left (3\,x^2+1\right )}^{1/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (x^{2} + 3\right ) \sqrt [3]{3 x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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