Optimal. Leaf size=106 \[ -\frac {3 \sqrt [3]{x^3-x}}{2 x}-\frac {1}{2} \log \left (\sqrt [3]{x^3-x}-x\right )-\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-x}+x}\right )+\frac {1}{4} \log \left (\sqrt [3]{x^3-x} x+\left (x^3-x\right )^{2/3}+x^2\right ) \]
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Rubi [A] time = 0.15, antiderivative size = 188, normalized size of antiderivative = 1.77, number of steps used = 11, number of rules used = 11, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.733, Rules used = {2020, 2032, 329, 275, 331, 292, 31, 634, 618, 204, 628} \begin {gather*} -\frac {3 \sqrt [3]{x^3-x}}{2 x}-\frac {x^{2/3} \left (x^2-1\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{x^2-1}}\right )}{2 \left (x^3-x\right )^{2/3}}+\frac {x^{2/3} \left (x^2-1\right )^{2/3} \log \left (\frac {x^{4/3}}{\left (x^2-1\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{x^2-1}}+1\right )}{4 \left (x^3-x\right )^{2/3}}-\frac {\sqrt {3} x^{2/3} \left (x^2-1\right )^{2/3} \tan ^{-1}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{2 \left (x^3-x\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 275
Rule 292
Rule 329
Rule 331
Rule 618
Rule 628
Rule 634
Rule 2020
Rule 2032
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{-x+x^3}}{x^2} \, dx &=-\frac {3 \sqrt [3]{-x+x^3}}{2 x}+\int \frac {x}{\left (-x+x^3\right )^{2/3}} \, dx\\ &=-\frac {3 \sqrt [3]{-x+x^3}}{2 x}+\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\left (-1+x^2\right )^{2/3}} \, dx}{\left (-x+x^3\right )^{2/3}}\\ &=-\frac {3 \sqrt [3]{-x+x^3}}{2 x}+\frac {\left (3 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (-1+x^6\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\left (-x+x^3\right )^{2/3}}\\ &=-\frac {3 \sqrt [3]{-x+x^3}}{2 x}+\frac {\left (3 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{2 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {3 \sqrt [3]{-x+x^3}}{2 x}+\frac {\left (3 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{2 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {3 \sqrt [3]{-x+x^3}}{2 x}+\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{2 \left (-x+x^3\right )^{2/3}}-\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{2 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {3 \sqrt [3]{-x+x^3}}{2 x}-\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{2 \left (-x+x^3\right )^{2/3}}+\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{4 \left (-x+x^3\right )^{2/3}}-\frac {\left (3 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{4 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {3 \sqrt [3]{-x+x^3}}{2 x}-\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{2 \left (-x+x^3\right )^{2/3}}+\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1+\frac {x^{4/3}}{\left (-1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{4 \left (-x+x^3\right )^{2/3}}+\frac {\left (3 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{2 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {3 \sqrt [3]{-x+x^3}}{2 x}-\frac {\sqrt {3} x^{2/3} \left (-1+x^2\right )^{2/3} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{2 \left (-x+x^3\right )^{2/3}}-\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{2 \left (-x+x^3\right )^{2/3}}+\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1+\frac {x^{4/3}}{\left (-1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{4 \left (-x+x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 42, normalized size = 0.40 \begin {gather*} -\frac {3 \sqrt [3]{x \left (x^2-1\right )} \, _2F_1\left (-\frac {1}{3},-\frac {1}{3};\frac {2}{3};x^2\right )}{2 x \sqrt [3]{1-x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.20, size = 106, normalized size = 1.00 \begin {gather*} -\frac {3 \sqrt [3]{x^3-x}}{2 x}-\frac {1}{2} \log \left (\sqrt [3]{x^3-x}-x\right )-\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-x}+x}\right )+\frac {1}{4} \log \left (\sqrt [3]{x^3-x} x+\left (x^3-x\right )^{2/3}+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 104, normalized size = 0.98 \begin {gather*} -\frac {2 \, \sqrt {3} x \arctan \left (-\frac {44032959556 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (16754327161 \, x^{2} - 2707204793\right )} - 10524305234 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {2}{3}}}{81835897185 \, x^{2} - 1102302937}\right ) + x \log \left (-3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} x + 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} + 1\right ) + 6 \, {\left (x^{3} - x\right )}^{\frac {1}{3}}}{4 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 74, normalized size = 0.70 \begin {gather*} \frac {1}{2} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {3}{2} \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + \frac {1}{4} \, \log \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{2} \, \log \left ({\left | {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.93, size = 787, normalized size = 7.42
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 {{\left (x^{3} - x\right )}^{\frac {1}{3}}}{x^{2}}\,{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 {{\left (x^3-x\right )}^{1/3}}{x^2} \,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 {\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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