Optimal. Leaf size=194 \[ -\frac {1}{8} \tan ^{-1}\left (\frac {-\frac {\sqrt [3]{1-3 x^2}}{\sqrt {3}}+x+\frac {1}{\sqrt {3}}}{\sqrt [3]{1-3 x^2}}\right )-\frac {1}{8} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{1-3 x^2}}{\sqrt {3}}+x-\frac {1}{\sqrt {3}}}{\sqrt [3]{1-3 x^2}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{1-3 x^2}+1}\right )}{4 \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {2 \sqrt {3} x \sqrt [3]{1-3 x^2}-2 \sqrt {3} x}{3 x^2+4 \left (1-3 x^2\right )^{2/3}-2 \sqrt [3]{1-3 x^2}+1}\right )}{8 \sqrt {3}} \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 81, normalized size of antiderivative = 0.42, 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 = {395} \begin {gather*} -\frac {1}{4} \tan ^{-1}\left (\frac {1-\sqrt [3]{1-3 x^2}}{x}\right )+\frac {\tanh ^{-1}\left (\frac {\left (1-\sqrt [3]{1-3 x^2}\right )^2}{3 \sqrt {3} x}\right )}{4 \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{4 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 395
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{1-3 x^2} \left (-3+x^2\right )} \, dx &=-\frac {1}{4} \tan ^{-1}\left (\frac {1-\sqrt [3]{1-3 x^2}}{x}\right )-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {\tanh ^{-1}\left (\frac {\left (1-\sqrt [3]{1-3 x^2}\right )^2}{3 \sqrt {3} x}\right )}{4 \sqrt {3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.10, size = 126, normalized size = 0.65 \begin {gather*} \frac {9 x F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};3 x^2,\frac {x^2}{3}\right )}{\sqrt [3]{1-3 x^2} \left (x^2-3\right ) \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 )} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 4.09, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{1-3 x^2} \left (-3+x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 3.29, size = 1790, normalized size = 9.23
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{2} - 3\right )} {\left (-3 \, x^{2} + 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 6.24, size = 643, normalized size = 3.31 \begin {gather*} 48 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{3} \ln \left (\frac {-18432 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{5} x +36864 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{5} x -768 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{3} x +2304 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{3} x +48 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{2} x^{2}+96 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{2} \left (-3 x^{2}+1\right )^{\frac {1}{3}}+2 \left (-3 x^{2}+1\right )^{\frac {2}{3}}+48 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{2}+32 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right ) x +x^{2}+1}{x^{2}-3}\right )+\RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right ) \ln \left (\frac {-18432 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{5} x +36864 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{5} x -768 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{3} x +2304 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{3} x +48 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{2} x^{2}+96 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{2} \left (-3 x^{2}+1\right )^{\frac {1}{3}}+2 \left (-3 x^{2}+1\right )^{\frac {2}{3}}+48 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{2}+32 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right ) x +x^{2}+1}{x^{2}-3}\right )+\RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right ) \ln \left (\frac {9216 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{5} x -18432 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{5} x +576 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{3} x -768 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{3} x -24 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{2} x^{2}-48 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{2} \left (-3 x^{2}+1\right )^{\frac {1}{3}}+8 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right ) x +\left (-3 x^{2}+1\right )^{\frac {2}{3}}-24 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{2}-\left (-3 x^{2}+1\right )^{\frac {1}{3}}}{x^{2}-3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{2} - 3\right )} {\left (-3 \, x^{2} + 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (x^2-3\right )\,{\left (1-3\,x^2\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{1 - 3 x^{2}} \left (x^{2} - 3\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________