Optimal. Leaf size=116 \[ \frac {3 \left (x^2-1\right )^{2/3}}{x+1}-\frac {7}{4} \log \left (2 \sqrt [3]{x^2-1}+x-1\right )+\frac {7}{8} \log \left (x^2+4 \left (x^2-1\right )^{2/3}+(2-2 x) \sqrt [3]{x^2-1}-2 x+1\right )-\frac {7}{4} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^2-1}}{\sqrt [3]{x^2-1}-x+1}\right ) \]
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Rubi [F] time = 0.37, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {2-x+x^2}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {2-x+x^2}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx &=\int \left (\frac {1}{\sqrt [3]{-1+x^2}}-\frac {1+5 x}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )}\right ) \, dx\\ &=\int \frac {1}{\sqrt [3]{-1+x^2}} \, dx-\int \frac {1+5 x}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx\\ &=\frac {\left (3 \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1+x^3}} \, dx,x,\sqrt [3]{-1+x^2}\right )}{2 x}-\int \frac {1+5 x}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx\\ &=\frac {\left (3 \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \frac {1-\sqrt {3}+x}{\sqrt {1+x^3}} \, dx,x,\sqrt [3]{-1+x^2}\right )}{2 x}+\frac {\left (3 \sqrt {\frac {1}{2} \left (2-\sqrt {3}\right )} \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^3}} \, dx,x,\sqrt [3]{-1+x^2}\right )}{x}-\int \frac {1+5 x}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx\\ &=\frac {3 x}{1+\sqrt {3}+\sqrt [3]{-1+x^2}}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+\sqrt [3]{-1+x^2}\right ) \sqrt {\frac {1-\sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+\sqrt [3]{-1+x^2}}{1+\sqrt {3}+\sqrt [3]{-1+x^2}}\right )|-7-4 \sqrt {3}\right )}{2 x \sqrt {\frac {1+\sqrt [3]{-1+x^2}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^2}\right )^2}}}+\frac {\sqrt {2} 3^{3/4} \left (1+\sqrt [3]{-1+x^2}\right ) \sqrt {\frac {1-\sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+\sqrt [3]{-1+x^2}}{1+\sqrt {3}+\sqrt [3]{-1+x^2}}\right )|-7-4 \sqrt {3}\right )}{x \sqrt {\frac {1+\sqrt [3]{-1+x^2}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^2}\right )^2}}}-\int \frac {1+5 x}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx\\ \end {align*}
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Mathematica [C] time = 0.17, size = 71, normalized size = 0.61 \begin {gather*} \frac {3 \left (x^2-1\right )^{2/3}}{x+1}-\frac {21 \left (x^2-1\right )^{2/3} F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};\frac {1-x}{2},\frac {1-x}{4}\right )}{8 \sqrt [3]{2} (x+1)^{2/3}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.37, size = 116, normalized size = 1.00 \begin {gather*} \frac {3 \left (x^2-1\right )^{2/3}}{x+1}-\frac {7}{4} \log \left (2 \sqrt [3]{x^2-1}+x-1\right )+\frac {7}{8} \log \left (x^2+4 \left (x^2-1\right )^{2/3}+(2-2 x) \sqrt [3]{x^2-1}-2 x+1\right )-\frac {7}{4} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^2-1}}{\sqrt [3]{x^2-1}-x+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 127, normalized size = 1.09 \begin {gather*} \frac {14 \, \sqrt {3} {\left (x + 1\right )} \arctan \left (\frac {286273 \, \sqrt {3} {\left (x^{2} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + \sqrt {3} {\left (66978 \, x^{2} + 434719 \, x + 635653\right )} + 539695 \, \sqrt {3} {\left (x^{2} - 1\right )}^{\frac {2}{3}}}{226981 \, x^{2} - 1974837 \, x - 1293894}\right ) - 7 \, {\left (x + 1\right )} \log \left (\frac {x^{2} + 6 \, {\left (x^{2} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + 6 \, x + 12 \, {\left (x^{2} - 1\right )}^{\frac {2}{3}} + 9}{x^{2} + 6 \, x + 9}\right ) + 24 \, {\left (x^{2} - 1\right )}^{\frac {2}{3}}}{8 \, {\left (x + 1\right )}} \end {gather*}
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 \begin {gather*} \int \frac {x^{2} - x + 2}{{\left (x^{2} + 4 \, x + 3\right )} {\left (x^{2} - 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.03, size = 594, normalized size = 5.12 \begin {gather*} \frac {-3+3 x}{\left (x^{2}-1\right )^{\frac {1}{3}}}+\frac {7 \ln \left (-\frac {96 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x^{2}-288 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x -864 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {2}{3}}+432 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}} x -278 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x^{2}-432 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}+492 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x -516 \left (x^{2}-1\right )^{\frac {2}{3}}+258 x \left (x^{2}-1\right )^{\frac {1}{3}}+17 x^{2}-342 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )-258 \left (x^{2}-1\right )^{\frac {1}{3}}+918 x +969}{\left (3+x \right )^{2}}\right )}{4}-\frac {7 \ln \left (-\frac {96 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x^{2}-288 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x -864 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {2}{3}}+432 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}} x -278 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x^{2}-432 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}+492 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x -516 \left (x^{2}-1\right )^{\frac {2}{3}}+258 x \left (x^{2}-1\right )^{\frac {1}{3}}+17 x^{2}-342 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )-258 \left (x^{2}-1\right )^{\frac {1}{3}}+918 x +969}{\left (3+x \right )^{2}}\right ) \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )}{2}+\frac {7 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \ln \left (-\frac {48 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x^{2}-144 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x +432 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {2}{3}}-216 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}} x +91 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x^{2}+216 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}-102 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x -474 \left (x^{2}-1\right )^{\frac {2}{3}}+237 x \left (x^{2}-1\right )^{\frac {1}{3}}-49 x^{2}+171 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )-237 \left (x^{2}-1\right )^{\frac {1}{3}}+546 x +399}{\left (3+x \right )^{2}}\right )}{2} \end {gather*}
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 \begin {gather*} \int \frac {x^{2} - x + 2}{{\left (x^{2} + 4 \, x + 3\right )} {\left (x^{2} - 1\right )}^{\frac {1}{3}}}\,{d x} \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 = -0.01 \begin {gather*} \int \frac {x^2-x+2}{{\left (x^2-1\right )}^{1/3}\,\left (x^2+4\,x+3\right )} \,d x \end {gather*}
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 \begin {gather*} \int \frac {x^{2} - x + 2}{\sqrt [3]{\left (x - 1\right ) \left (x + 1\right )} \left (x + 1\right ) \left (x + 3\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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