Optimal. Leaf size=121 \[ \frac {\log \left (2^{2/3} \sqrt [3]{x^4+x^2}-2 x\right )}{\sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^4+x^2} x+\sqrt [3]{2} \left (x^4+x^2\right )^{2/3}\right )}{2 \sqrt [3]{2}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^4+x^2}+x}\right )}{\sqrt [3]{2}} \]
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Rubi [C] time = 0.65, antiderivative size = 127, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2056, 6733, 6725, 245, 1438, 429, 465, 510} \begin {gather*} -\frac {3 \sqrt [3]{x^2+1} x^2 F_1\left (\frac {2}{3};1,\frac {1}{3};\frac {5}{3};x^2,-x^2\right )}{2 \sqrt [3]{x^4+x^2}}-\frac {6 \sqrt [3]{x^2+1} x F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};x^2,-x^2\right )}{\sqrt [3]{x^4+x^2}}+\frac {3 \sqrt [3]{x^2+1} 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 245
Rule 429
Rule 465
Rule 510
Rule 1438
Rule 2056
Rule 6725
Rule 6733
Rubi steps
\begin {align*} \int \frac {1+x}{(-1+x) \sqrt [3]{x^2+x^4}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {1+x}{(-1+x) x^{2/3} \sqrt [3]{1+x^2}} \, dx}{\sqrt [3]{x^2+x^4}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1+x^3}{\left (-1+x^3\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt [3]{1+x^6}}+\frac {2}{\left (-1+x^3\right ) \sqrt [3]{1+x^6}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}+\frac {\left (6 x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^3\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 (6 x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\left (-1+x^6\right ) \sqrt [3]{1+x^6}}+\frac {x^3}{\left (-1+x^6\right ) \sqrt [3]{1+x^6}}\right ) \, 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 (6 x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^6\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}+\frac {\left (6 x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (-1+x^6\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}\\ &=-\frac {6 x \sqrt [3]{1+x^2} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};x^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 x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (-1+x^3\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x^2+x^4}}\\ &=-\frac {6 x \sqrt [3]{1+x^2} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};x^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};x^2,-x^2\right )}{2 \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 = 0.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x}{(-1+x) \sqrt [3]{x^2+x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.40, size = 121, normalized size = 1.00 \begin {gather*} \frac {\log \left (2^{2/3} \sqrt [3]{x^4+x^2}-2 x\right )}{\sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^4+x^2} x+\sqrt [3]{2} \left (x^4+x^2\right )^{2/3}\right )}{2 \sqrt [3]{2}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^4+x^2}+x}\right )}{\sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 5.34, size = 307, normalized size = 2.54 \begin {gather*} -\frac {1}{6} \cdot 4^{\frac {1}{3}} \sqrt {3} \arctan \left (\frac {3 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (x^{4} + 2 \, x^{3} - 6 \, x^{2} + 2 \, x + 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} + 6 \cdot 4^{\frac {1}{3}} \sqrt {3} {\left (x^{5} + 14 \, x^{4} + 6 \, x^{3} + 14 \, x^{2} + x\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} + \sqrt {3} {\left (x^{7} + 30 \, x^{6} + 51 \, x^{5} + 52 \, x^{4} + 51 \, x^{3} + 30 \, x^{2} + x\right )}}{3 \, {\left (x^{7} - 6 \, x^{6} - 93 \, x^{5} - 20 \, x^{4} - 93 \, x^{3} - 6 \, x^{2} + x\right )}}\right ) - \frac {1}{12} \cdot 4^{\frac {1}{3}} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} {\left (x^{2} + 4 \, x + 1\right )} + 4^{\frac {2}{3}} {\left (x^{5} + 14 \, x^{4} + 6 \, x^{3} + 14 \, x^{2} + x\right )} + 24 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + x^{2} + x\right )}}{x^{5} - 4 \, x^{4} + 6 \, x^{3} - 4 \, x^{2} + x}\right ) + \frac {1}{6} \cdot 4^{\frac {1}{3}} \log \left (-\frac {3 \cdot 4^{\frac {2}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x + 4^{\frac {1}{3}} {\left (x^{3} - 2 \, x^{2} + x\right )} - 6 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{x^{3} - 2 \, 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 + 1}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 9.50, size = 971, normalized size = 8.02
result too large to display
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 + 1}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (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+1}{{\left (x^4+x^2\right )}^{1/3}\,\left (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 + 1}{\sqrt [3]{x^{2} \left (x^{2} + 1\right )} \left (x - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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