Optimal. Leaf size=93 \[ -\log \left (\sqrt [3]{x^4-x^2}+x\right )+\frac {1}{2} \log \left (x^2-\sqrt [3]{x^4-x^2} x+\left (x^4-x^2\right )^{2/3}\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^4-x^2}-x}\right ) \]
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Rubi [C] time = 1.25, antiderivative size = 293, normalized size of antiderivative = 3.15, number of steps used = 23, number of rules used = 11, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {2056, 6728, 365, 364, 959, 466, 430, 429, 465, 511, 510} \begin {gather*} \frac {3 \sqrt [3]{1-x^2} x^2 F_1\left (\frac {2}{3};1,\frac {1}{3};\frac {5}{3};\frac {4 x^2}{\left (1-\sqrt {5}\right )^2},x^2\right )}{2 \left (1-\sqrt {5}\right ) \sqrt [3]{x^4-x^2}}+\frac {3 \sqrt [3]{1-x^2} x^2 F_1\left (\frac {2}{3};1,\frac {1}{3};\frac {5}{3};\frac {4 x^2}{\left (1+\sqrt {5}\right )^2},x^2\right )}{2 \left (1+\sqrt {5}\right ) \sqrt [3]{x^4-x^2}}-\frac {3 \sqrt [3]{1-x^2} x F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};\frac {4 x^2}{\left (1-\sqrt {5}\right )^2},x^2\right )}{\sqrt [3]{x^4-x^2}}-\frac {3 \sqrt [3]{1-x^2} x F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};\frac {4 x^2}{\left (1+\sqrt {5}\right )^2},x^2\right )}{\sqrt [3]{x^4-x^2}}+\frac {3 \sqrt [3]{1-x^2} x \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^2\right )}{\sqrt [3]{x^4-x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 364
Rule 365
Rule 429
Rule 430
Rule 465
Rule 466
Rule 510
Rule 511
Rule 959
Rule 2056
Rule 6728
Rubi steps
\begin {align*} \int \frac {1+x^2}{\left (-1+x+x^2\right ) \sqrt [3]{-x^2+x^4}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {1+x^2}{x^{2/3} \sqrt [3]{-1+x^2} \left (-1+x+x^2\right )} \, dx}{\sqrt [3]{-x^2+x^4}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \left (\frac {1}{x^{2/3} \sqrt [3]{-1+x^2}}+\frac {2-x}{x^{2/3} \sqrt [3]{-1+x^2} \left (-1+x+x^2\right )}\right ) \, dx}{\sqrt [3]{-x^2+x^4}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-1+x^2}} \, dx}{\sqrt [3]{-x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {2-x}{x^{2/3} \sqrt [3]{-1+x^2} \left (-1+x+x^2\right )} \, dx}{\sqrt [3]{-x^2+x^4}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{1-x^2}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{1-x^2}} \, dx}{\sqrt [3]{-x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \left (\frac {-1+\sqrt {5}}{x^{2/3} \left (1-\sqrt {5}+2 x\right ) \sqrt [3]{-1+x^2}}+\frac {-1-\sqrt {5}}{x^{2/3} \left (1+\sqrt {5}+2 x\right ) \sqrt [3]{-1+x^2}}\right ) \, dx}{\sqrt [3]{-x^2+x^4}}\\ &=\frac {3 x \sqrt [3]{1-x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^2\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1-\sqrt {5}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {1}{x^{2/3} \left (1+\sqrt {5}+2 x\right ) \sqrt [3]{-1+x^2}} \, dx}{\sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1+\sqrt {5}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {1}{x^{2/3} \left (1-\sqrt {5}+2 x\right ) \sqrt [3]{-1+x^2}} \, dx}{\sqrt [3]{-x^2+x^4}}\\ &=\frac {3 x \sqrt [3]{1-x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^2\right )}{\sqrt [3]{-x^2+x^4}}-\frac {\left (2 \left (-1-\sqrt {5}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {\sqrt [3]{x}}{\left (\left (1+\sqrt {5}\right )^2-4 x^2\right ) \sqrt [3]{-1+x^2}} \, dx}{\sqrt [3]{-x^2+x^4}}-\frac {\left (2 \left (-1+\sqrt {5}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {\sqrt [3]{x}}{\left (\left (1-\sqrt {5}\right )^2-4 x^2\right ) \sqrt [3]{-1+x^2}} \, dx}{\sqrt [3]{-x^2+x^4}}+\frac {\left (\left (1-\sqrt {5}\right ) \left (-1+\sqrt {5}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {1}{x^{2/3} \left (\left (1-\sqrt {5}\right )^2-4 x^2\right ) \sqrt [3]{-1+x^2}} \, dx}{\sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1-\sqrt {5}\right ) \left (1+\sqrt {5}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {1}{x^{2/3} \left (\left (1+\sqrt {5}\right )^2-4 x^2\right ) \sqrt [3]{-1+x^2}} \, dx}{\sqrt [3]{-x^2+x^4}}\\ &=\frac {3 x \sqrt [3]{1-x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^2\right )}{\sqrt [3]{-x^2+x^4}}-\frac {\left (6 \left (-1-\sqrt {5}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (\left (1+\sqrt {5}\right )^2-4 x^6\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}}-\frac {\left (6 \left (-1+\sqrt {5}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (\left (1-\sqrt {5}\right )^2-4 x^6\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (3 \left (1-\sqrt {5}\right ) \left (-1+\sqrt {5}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (1-\sqrt {5}\right )^2-4 x^6\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (3 \left (-1-\sqrt {5}\right ) \left (1+\sqrt {5}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (1+\sqrt {5}\right )^2-4 x^6\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}}\\ &=\frac {3 x \sqrt [3]{1-x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^2\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (3 \left (1-\sqrt {5}\right ) \left (-1+\sqrt {5}\right ) x^{2/3} \sqrt [3]{1-x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (1-\sqrt {5}\right )^2-4 x^6\right ) \sqrt [3]{1-x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (3 \left (-1-\sqrt {5}\right ) \left (1+\sqrt {5}\right ) x^{2/3} \sqrt [3]{1-x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (1+\sqrt {5}\right )^2-4 x^6\right ) \sqrt [3]{1-x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}}-\frac {\left (3 \left (-1-\sqrt {5}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (\left (1+\sqrt {5}\right )^2-4 x^3\right ) \sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{-x^2+x^4}}-\frac {\left (3 \left (-1+\sqrt {5}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (\left (1-\sqrt {5}\right )^2-4 x^3\right ) \sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{-x^2+x^4}}\\ &=-\frac {3 x \sqrt [3]{1-x^2} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};\frac {4 x^2}{\left (1-\sqrt {5}\right )^2},x^2\right )}{\sqrt [3]{-x^2+x^4}}-\frac {3 x \sqrt [3]{1-x^2} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};\frac {4 x^2}{\left (1+\sqrt {5}\right )^2},x^2\right )}{\sqrt [3]{-x^2+x^4}}+\frac {3 x \sqrt [3]{1-x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^2\right )}{\sqrt [3]{-x^2+x^4}}-\frac {\left (3 \left (-1-\sqrt {5}\right ) x^{2/3} \sqrt [3]{1-x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (\left (1+\sqrt {5}\right )^2-4 x^3\right ) \sqrt [3]{1-x^3}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{-x^2+x^4}}-\frac {\left (3 \left (-1+\sqrt {5}\right ) x^{2/3} \sqrt [3]{1-x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (\left (1-\sqrt {5}\right )^2-4 x^3\right ) \sqrt [3]{1-x^3}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{-x^2+x^4}}\\ &=-\frac {3 x \sqrt [3]{1-x^2} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};\frac {4 x^2}{\left (1-\sqrt {5}\right )^2},x^2\right )}{\sqrt [3]{-x^2+x^4}}-\frac {3 x \sqrt [3]{1-x^2} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};\frac {4 x^2}{\left (1+\sqrt {5}\right )^2},x^2\right )}{\sqrt [3]{-x^2+x^4}}+\frac {3 x^2 \sqrt [3]{1-x^2} F_1\left (\frac {2}{3};1,\frac {1}{3};\frac {5}{3};\frac {4 x^2}{\left (1-\sqrt {5}\right )^2},x^2\right )}{2 \left (1-\sqrt {5}\right ) \sqrt [3]{-x^2+x^4}}+\frac {3 x^2 \sqrt [3]{1-x^2} F_1\left (\frac {2}{3};1,\frac {1}{3};\frac {5}{3};\frac {4 x^2}{\left (1+\sqrt {5}\right )^2},x^2\right )}{2 \left (1+\sqrt {5}\right ) \sqrt [3]{-x^2+x^4}}+\frac {3 x \sqrt [3]{1-x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^2\right )}{\sqrt [3]{-x^2+x^4}}\\ \end {align*}
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Mathematica [F] time = 0.64, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^2}{\left (-1+x+x^2\right ) \sqrt [3]{-x^2+x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.35, size = 93, normalized size = 1.00 \begin {gather*} -\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-x^2+x^4}}\right )-\log \left (x+\sqrt [3]{-x^2+x^4}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{-x^2+x^4}+\left (-x^2+x^4\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.42, size = 130, normalized size = 1.40 \begin {gather*} -\sqrt {3} \arctan \left (-\frac {128537192 \, \sqrt {3} {\left (x^{4} - x^{2}\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (1454911 \, x^{3} - 69864736 \, x^{2} - 1454911 \, x\right )} - 14102102 \, \sqrt {3} {\left (x^{4} - x^{2}\right )}^{\frac {2}{3}}}{226981 \, x^{3} + 171879616 \, x^{2} - 226981 \, x}\right ) - \frac {1}{2} \, \log \left (\frac {x^{3} + x^{2} + 3 \, {\left (x^{4} - x^{2}\right )}^{\frac {1}{3}} x - x + 3 \, {\left (x^{4} - x^{2}\right )}^{\frac {2}{3}}}{x^{3} + x^{2} - x}\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} + 1}{{\left (x^{4} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} + x - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.52, size = 581, normalized size = 6.25
method | result | size |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \ln \left (\frac {494 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{3}-741 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}-2222 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{3}+1356 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{4}-x^{2}\right )^{\frac {2}{3}}-1356 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{4}-x^{2}\right )^{\frac {1}{3}} x -494 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x +1604 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}+2220 x^{3}+1980 \left (x^{4}-x^{2}\right )^{\frac {2}{3}}-1980 x \left (x^{4}-x^{2}\right )^{\frac {1}{3}}+2222 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x -740 x^{2}-2220 x}{x \left (x^{2}+x -1\right )}\right )}{2}-\frac {\ln \left (-\frac {-494 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{3}+741 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}-246 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{3}+1356 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{4}-x^{2}\right )^{\frac {2}{3}}-1356 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{4}-x^{2}\right )^{\frac {1}{3}} x +494 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x -1360 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}+248 x^{3}-4692 \left (x^{4}-x^{2}\right )^{\frac {2}{3}}+4692 x \left (x^{4}-x^{2}\right )^{\frac {1}{3}}+246 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x +496 x^{2}-248 x}{x \left (x^{2}+x -1\right )}\right ) \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )}{2}+\ln \left (-\frac {-494 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{3}+741 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}-246 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{3}+1356 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{4}-x^{2}\right )^{\frac {2}{3}}-1356 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{4}-x^{2}\right )^{\frac {1}{3}} x +494 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x -1360 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}+248 x^{3}-4692 \left (x^{4}-x^{2}\right )^{\frac {2}{3}}+4692 x \left (x^{4}-x^{2}\right )^{\frac {1}{3}}+246 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x +496 x^{2}-248 x}{x \left (x^{2}+x -1\right )}\right )\) | \(581\) |
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} + 1}{{\left (x^{4} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} + x - 1\right )}}\,{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+1}{{\left (x^4-x^2\right )}^{1/3}\,\left (x^2+x-1\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} + 1}{\sqrt [3]{x^{2} \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + x - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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