Optimal. Leaf size=243 \[ \frac {1}{6} \sqrt [3]{x^3-x^2} (3 x-1)-\frac {17}{9} \log \left (\sqrt [3]{x^3-x^2}-x\right )+\sqrt [3]{2} \log \left (2^{2/3} \sqrt [3]{x^3-x^2}-2 x\right )+\frac {17}{18} \log \left (x^2+\sqrt [3]{x^3-x^2} x+\left (x^3-x^2\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^3-x^2} x+\sqrt [3]{2} \left (x^3-x^2\right )^{2/3}\right )}{2^{2/3}}-\frac {17 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-x^2}+x}\right )}{3 \sqrt {3}}+\sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3-x^2}+x}\right ) \]
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Rubi [A] time = 1.01, antiderivative size = 385, normalized size of antiderivative = 1.58, number of steps used = 36, number of rules used = 14, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {2056, 1586, 6733, 6725, 331, 292, 31, 634, 618, 204, 628, 321, 494, 617} \begin {gather*} \frac {1}{2} \sqrt [3]{x^3-x^2} x-\frac {1}{6} \sqrt [3]{x^3-x^2}-\frac {17 \sqrt [3]{x^3-x^2} \log \left (1-\frac {\sqrt [3]{x}}{\sqrt [3]{x-1}}\right )}{9 \sqrt [3]{x-1} x^{2/3}}+\frac {\sqrt [3]{2} \sqrt [3]{x^3-x^2} \log \left (1-\frac {\sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{x-1}}\right )}{\sqrt [3]{x-1} x^{2/3}}+\frac {17 \sqrt [3]{x^3-x^2} \log \left (\frac {x^{2/3}}{(x-1)^{2/3}}+\frac {\sqrt [3]{x}}{\sqrt [3]{x-1}}+1\right )}{18 \sqrt [3]{x-1} x^{2/3}}-\frac {\sqrt [3]{x^3-x^2} \log \left (\frac {2^{2/3} x^{2/3}}{(x-1)^{2/3}}+\frac {\sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1\right )}{2^{2/3} \sqrt [3]{x-1} x^{2/3}}-\frac {17 \sqrt [3]{x^3-x^2} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{x-1} x^{2/3}}+\frac {\sqrt [3]{2} \sqrt {3} \sqrt [3]{x^3-x^2} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{\sqrt [3]{x-1} x^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 292
Rule 321
Rule 331
Rule 494
Rule 617
Rule 618
Rule 628
Rule 634
Rule 1586
Rule 2056
Rule 6725
Rule 6733
Rubi steps
\begin {align*} \int \frac {\left (1+x^2\right ) \sqrt [3]{-x^2+x^3}}{-1+x^2} \, dx &=\frac {\sqrt [3]{-x^2+x^3} \int \frac {\sqrt [3]{-1+x} x^{2/3} \left (1+x^2\right )}{-1+x^2} \, dx}{\sqrt [3]{-1+x} x^{2/3}}\\ &=\frac {\sqrt [3]{-x^2+x^3} \int \frac {x^{2/3} \left (1+x^2\right )}{(-1+x)^{2/3} (1+x)} \, dx}{\sqrt [3]{-1+x} x^{2/3}}\\ &=\frac {\left (3 \sqrt [3]{-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {x^4 \left (1+x^6\right )}{\left (-1+x^3\right )^{2/3} \left (1+x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-1+x} x^{2/3}}\\ &=\frac {\left (3 \sqrt [3]{-x^2+x^3}\right ) \operatorname {Subst}\left (\int \left (\frac {2 x}{\left (-1+x^3\right )^{2/3}}-\frac {x^4}{\left (-1+x^3\right )^{2/3}}+\frac {x^7}{\left (-1+x^3\right )^{2/3}}-\frac {2 x}{\left (-1+x^3\right )^{2/3} \left (1+x^3\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-1+x} x^{2/3}}\\ &=-\frac {\left (3 \sqrt [3]{-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (-1+x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-1+x} x^{2/3}}+\frac {\left (3 \sqrt [3]{-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {x^7}{\left (-1+x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-1+x} x^{2/3}}+\frac {\left (6 \sqrt [3]{-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-1+x} x^{2/3}}-\frac {\left (6 \sqrt [3]{-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (-1+x^3\right )^{2/3} \left (1+x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-1+x} x^{2/3}}\\ &=-\sqrt [3]{-x^2+x^3}+\frac {1}{2} x \sqrt [3]{-x^2+x^3}-\frac {\left (2 \sqrt [3]{-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-1+x} x^{2/3}}+\frac {\left (5 \sqrt [3]{-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (-1+x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-1+x} x^{2/3}}-\frac {\left (6 \sqrt [3]{-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {x}{1-2 x^3} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{\sqrt [3]{-1+x} x^{2/3}}+\frac {\left (6 \sqrt [3]{-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{\sqrt [3]{-1+x} x^{2/3}}\\ &=-\frac {1}{6} \sqrt [3]{-x^2+x^3}+\frac {1}{2} x \sqrt [3]{-x^2+x^3}+\frac {\left (5 \sqrt [3]{-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-1+x} x^{2/3}}+\frac {\left (2 \sqrt [3]{-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{\sqrt [3]{-1+x} x^{2/3}}-\frac {\left (2 \sqrt [3]{-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{\sqrt [3]{-1+x} x^{2/3}}-\frac {\left (2 \sqrt [3]{-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{\sqrt [3]{-1+x} x^{2/3}}-\frac {\left (2^{2/3} \sqrt [3]{-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [3]{2} x} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{\sqrt [3]{-1+x} x^{2/3}}+\frac {\left (2^{2/3} \sqrt [3]{-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1-\sqrt [3]{2} x}{1+\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{\sqrt [3]{-1+x} x^{2/3}}\\ &=-\frac {1}{6} \sqrt [3]{-x^2+x^3}+\frac {1}{2} x \sqrt [3]{-x^2+x^3}-\frac {2 \sqrt [3]{-x^2+x^3} \log \left (1-\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{\sqrt [3]{-1+x} x^{2/3}}+\frac {\sqrt [3]{2} \sqrt [3]{-x^2+x^3} \log \left (1-\frac {\sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{\sqrt [3]{-1+x} x^{2/3}}-\frac {\left (2 \sqrt [3]{-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{3 \sqrt [3]{-1+x} x^{2/3}}+\frac {\left (2 \sqrt [3]{-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{3 \sqrt [3]{-1+x} x^{2/3}}+\frac {\sqrt [3]{-x^2+x^3} \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{\sqrt [3]{-1+x} x^{2/3}}+\frac {\left (5 \sqrt [3]{-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{3 \sqrt [3]{-1+x} x^{2/3}}-\frac {\left (3 \sqrt [3]{-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{\sqrt [3]{-1+x} x^{2/3}}-\frac {\sqrt [3]{-x^2+x^3} \operatorname {Subst}\left (\int \frac {\sqrt [3]{2}+2\ 2^{2/3} x}{1+\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{2^{2/3} \sqrt [3]{-1+x} x^{2/3}}+\frac {\left (3 \sqrt [3]{-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{\sqrt [3]{2} \sqrt [3]{-1+x} x^{2/3}}\\ &=-\frac {1}{6} \sqrt [3]{-x^2+x^3}+\frac {1}{2} x \sqrt [3]{-x^2+x^3}-\frac {4 \sqrt [3]{-x^2+x^3} \log \left (1-\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{3 \sqrt [3]{-1+x} x^{2/3}}+\frac {\sqrt [3]{2} \sqrt [3]{-x^2+x^3} \log \left (1-\frac {\sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{\sqrt [3]{-1+x} x^{2/3}}+\frac {\sqrt [3]{-x^2+x^3} \log \left (1+\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}+\frac {x^{2/3}}{(-1+x)^{2/3}}\right )}{\sqrt [3]{-1+x} x^{2/3}}-\frac {\sqrt [3]{-x^2+x^3} \log \left (1+\frac {\sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{-1+x}}+\frac {2^{2/3} x^{2/3}}{(-1+x)^{2/3}}\right )}{2^{2/3} \sqrt [3]{-1+x} x^{2/3}}-\frac {\sqrt [3]{-x^2+x^3} \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{3 \sqrt [3]{-1+x} x^{2/3}}+\frac {\left (5 \sqrt [3]{-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{9 \sqrt [3]{-1+x} x^{2/3}}-\frac {\left (5 \sqrt [3]{-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{9 \sqrt [3]{-1+x} x^{2/3}}+\frac {\sqrt [3]{-x^2+x^3} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{\sqrt [3]{-1+x} x^{2/3}}+\frac {\left (6 \sqrt [3]{-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{\sqrt [3]{-1+x} x^{2/3}}-\frac {\left (3 \sqrt [3]{2} \sqrt [3]{-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{\sqrt [3]{-1+x} x^{2/3}}\\ &=-\frac {1}{6} \sqrt [3]{-x^2+x^3}+\frac {1}{2} x \sqrt [3]{-x^2+x^3}-\frac {2 \sqrt {3} \sqrt [3]{-x^2+x^3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{x}}{\sqrt [3]{-1+x}}}{\sqrt {3}}\right )}{\sqrt [3]{-1+x} x^{2/3}}+\frac {\sqrt [3]{2} \sqrt {3} \sqrt [3]{-x^2+x^3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{-1+x}}}{\sqrt {3}}\right )}{\sqrt [3]{-1+x} x^{2/3}}-\frac {17 \sqrt [3]{-x^2+x^3} \log \left (1-\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{9 \sqrt [3]{-1+x} x^{2/3}}+\frac {\sqrt [3]{2} \sqrt [3]{-x^2+x^3} \log \left (1-\frac {\sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{\sqrt [3]{-1+x} x^{2/3}}+\frac {2 \sqrt [3]{-x^2+x^3} \log \left (1+\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}+\frac {x^{2/3}}{(-1+x)^{2/3}}\right )}{3 \sqrt [3]{-1+x} x^{2/3}}-\frac {\sqrt [3]{-x^2+x^3} \log \left (1+\frac {\sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{-1+x}}+\frac {2^{2/3} x^{2/3}}{(-1+x)^{2/3}}\right )}{2^{2/3} \sqrt [3]{-1+x} x^{2/3}}+\frac {\left (5 \sqrt [3]{-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{18 \sqrt [3]{-1+x} x^{2/3}}-\frac {\left (5 \sqrt [3]{-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{6 \sqrt [3]{-1+x} x^{2/3}}-\frac {\left (2 \sqrt [3]{-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{\sqrt [3]{-1+x} x^{2/3}}\\ &=-\frac {1}{6} \sqrt [3]{-x^2+x^3}+\frac {1}{2} x \sqrt [3]{-x^2+x^3}+\frac {2 \sqrt [3]{-x^2+x^3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{x}}{\sqrt [3]{-1+x}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{-1+x} x^{2/3}}-\frac {2 \sqrt {3} \sqrt [3]{-x^2+x^3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{x}}{\sqrt [3]{-1+x}}}{\sqrt {3}}\right )}{\sqrt [3]{-1+x} x^{2/3}}+\frac {\sqrt [3]{2} \sqrt {3} \sqrt [3]{-x^2+x^3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{-1+x}}}{\sqrt {3}}\right )}{\sqrt [3]{-1+x} x^{2/3}}-\frac {17 \sqrt [3]{-x^2+x^3} \log \left (1-\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{9 \sqrt [3]{-1+x} x^{2/3}}+\frac {\sqrt [3]{2} \sqrt [3]{-x^2+x^3} \log \left (1-\frac {\sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{\sqrt [3]{-1+x} x^{2/3}}+\frac {17 \sqrt [3]{-x^2+x^3} \log \left (1+\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}+\frac {x^{2/3}}{(-1+x)^{2/3}}\right )}{18 \sqrt [3]{-1+x} x^{2/3}}-\frac {\sqrt [3]{-x^2+x^3} \log \left (1+\frac {\sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{-1+x}}+\frac {2^{2/3} x^{2/3}}{(-1+x)^{2/3}}\right )}{2^{2/3} \sqrt [3]{-1+x} x^{2/3}}+\frac {\left (5 \sqrt [3]{-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{3 \sqrt [3]{-1+x} x^{2/3}}\\ &=-\frac {1}{6} \sqrt [3]{-x^2+x^3}+\frac {1}{2} x \sqrt [3]{-x^2+x^3}+\frac {\sqrt [3]{-x^2+x^3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{x}}{\sqrt [3]{-1+x}}}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{-1+x} x^{2/3}}-\frac {2 \sqrt {3} \sqrt [3]{-x^2+x^3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{x}}{\sqrt [3]{-1+x}}}{\sqrt {3}}\right )}{\sqrt [3]{-1+x} x^{2/3}}+\frac {\sqrt [3]{2} \sqrt {3} \sqrt [3]{-x^2+x^3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{-1+x}}}{\sqrt {3}}\right )}{\sqrt [3]{-1+x} x^{2/3}}-\frac {17 \sqrt [3]{-x^2+x^3} \log \left (1-\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{9 \sqrt [3]{-1+x} x^{2/3}}+\frac {\sqrt [3]{2} \sqrt [3]{-x^2+x^3} \log \left (1-\frac {\sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{\sqrt [3]{-1+x} x^{2/3}}+\frac {17 \sqrt [3]{-x^2+x^3} \log \left (1+\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}+\frac {x^{2/3}}{(-1+x)^{2/3}}\right )}{18 \sqrt [3]{-1+x} x^{2/3}}-\frac {\sqrt [3]{-x^2+x^3} \log \left (1+\frac {\sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{-1+x}}+\frac {2^{2/3} x^{2/3}}{(-1+x)^{2/3}}\right )}{2^{2/3} \sqrt [3]{-1+x} x^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 85, normalized size = 0.35 \begin {gather*} \frac {3 \sqrt [3]{(x-1) x^2} \left ((x-1) \sqrt [3]{x} \, _2F_1\left (-\frac {2}{3},\frac {4}{3};\frac {7}{3};1-x\right )+8 \sqrt [3]{x} \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};1-x\right )-4 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {x-1}{2 x}\right )\right )}{4 x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.69, size = 243, normalized size = 1.00 \begin {gather*} \frac {1}{6} (-1+3 x) \sqrt [3]{-x^2+x^3}-\frac {17 \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x^2+x^3}}\right )}{3 \sqrt {3}}+\sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-x^2+x^3}}\right )-\frac {17}{9} \log \left (-x+\sqrt [3]{-x^2+x^3}\right )+\sqrt [3]{2} \log \left (-2 x+2^{2/3} \sqrt [3]{-x^2+x^3}\right )+\frac {17}{18} \log \left (x^2+x \sqrt [3]{-x^2+x^3}+\left (-x^2+x^3\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-x^2+x^3}+\sqrt [3]{2} \left (-x^2+x^3\right )^{2/3}\right )}{2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 222, normalized size = 0.91 \begin {gather*} -\sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} + \sqrt {3} x}{3 \, x}\right ) + \frac {17}{9} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{6} \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} {\left (3 \, x - 1\right )} + 2^{\frac {1}{3}} \log \left (-\frac {2^{\frac {1}{3}} x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{2} \cdot 2^{\frac {1}{3}} \log \left (\frac {2^{\frac {2}{3}} x^{2} + 2^{\frac {1}{3}} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - \frac {17}{9} \, \log \left (-\frac {x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {17}{18} \, \log \left (\frac {x^{2} + {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.86, size = 174, normalized size = 0.72 \begin {gather*} \frac {1}{6} \, {\left ({\left (-\frac {1}{x} + 1\right )}^{\frac {4}{3}} + 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}}\right )} x^{2} - \sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {17}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {1}{2} \cdot 2^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}}\right ) + 2^{\frac {1}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} \right |}\right ) + \frac {17}{18} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {17}{9} \, \log \left ({\left | {\left (-\frac {1}{x} + 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 = 6.89, size = 1278, normalized size = 5.26
method | result | size |
trager | \(\text {Expression too large to display}\) | \(1278\) |
risch | \(\text {Expression too large to display}\) | \(2369\) |
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^{2}\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )}}{x^{2} - 1}\,{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 = -0.00 \begin {gather*} \int \frac {\left (x^2+1\right )\,{\left (x^3-x^2\right )}^{1/3}}{x^2-1} \,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^{2} \left (x - 1\right )} \left (x^{2} + 1\right )}{\left (x - 1\right ) \left (x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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