Optimal. Leaf size=208 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^6-2 \text {$\#$1}^3+2\& ,\frac {\log \left (\sqrt [3]{x^3-x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]-\frac {3 \left (x^3-x^2\right )^{2/3}}{4 (x-1) x}+\frac {\log \left (2^{2/3} \sqrt [3]{x^3-x^2}-2 x\right )}{4 \sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^3-x^2} x+\sqrt [3]{2} \left (x^3-x^2\right )^{2/3}\right )}{8 \sqrt [3]{2}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3-x^2}+x}\right )}{4 \sqrt [3]{2}} \]
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Rubi [C] time = 0.48, antiderivative size = 538, normalized size of antiderivative = 2.59, number of steps used = 10, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2056, 6725, 848, 96, 91, 912} \begin {gather*} -\frac {3 x}{4 \sqrt [3]{x^3-x^2}}+\frac {3 \sqrt [3]{x-1} x^{2/3} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{x-1}}{\sqrt [3]{1-i}}\right )}{8 \sqrt [3]{1-i} \sqrt [3]{x^3-x^2}}+\frac {3 \sqrt [3]{x-1} x^{2/3} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{x-1}}{\sqrt [3]{1+i}}\right )}{8 \sqrt [3]{1+i} \sqrt [3]{x^3-x^2}}+\frac {3 \sqrt [3]{x-1} x^{2/3} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{2}}-\sqrt [3]{x}\right )}{8 \sqrt [3]{2} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log (-x-1)}{8 \sqrt [3]{2} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log (-x+i)}{8 \sqrt [3]{1+i} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log (x+i)}{8 \sqrt [3]{1-i} \sqrt [3]{x^3-x^2}}+\frac {\sqrt {3} \sqrt [3]{x-1} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{x-1}}{\sqrt [3]{1-i} \sqrt {3} \sqrt [3]{x}}\right )}{4 \sqrt [3]{1-i} \sqrt [3]{x^3-x^2}}+\frac {\sqrt {3} \sqrt [3]{x-1} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{x-1}}{\sqrt [3]{1+i} \sqrt {3} \sqrt [3]{x}}\right )}{4 \sqrt [3]{1+i} \sqrt [3]{x^3-x^2}}+\frac {\sqrt {3} \sqrt [3]{x-1} x^{2/3} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt [3]{x^3-x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 91
Rule 96
Rule 848
Rule 912
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{-x^2+x^3} \left (-1+x^4\right )} \, dx &=\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3} \left (-1+x^4\right )} \, dx}{\sqrt [3]{-x^2+x^3}}\\ &=\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \left (-\frac {1}{2 \sqrt [3]{-1+x} x^{2/3} \left (1-x^2\right )}-\frac {1}{2 \sqrt [3]{-1+x} x^{2/3} \left (1+x^2\right )}\right ) \, dx}{\sqrt [3]{-x^2+x^3}}\\ &=-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3} \left (1-x^2\right )} \, dx}{2 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3} \left (1+x^2\right )} \, dx}{2 \sqrt [3]{-x^2+x^3}}\\ &=-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{(-1-x) (-1+x)^{4/3} x^{2/3}} \, dx}{2 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \left (\frac {i}{2 (i-x) \sqrt [3]{-1+x} x^{2/3}}+\frac {i}{2 \sqrt [3]{-1+x} x^{2/3} (i+x)}\right ) \, dx}{2 \sqrt [3]{-x^2+x^3}}\\ &=-\frac {3 x}{4 \sqrt [3]{-x^2+x^3}}-\frac {\left (i \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{(i-x) \sqrt [3]{-1+x} x^{2/3}} \, dx}{4 \sqrt [3]{-x^2+x^3}}-\frac {\left (i \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3} (i+x)} \, dx}{4 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{(-1-x) \sqrt [3]{-1+x} x^{2/3}} \, dx}{4 \sqrt [3]{-x^2+x^3}}\\ &=-\frac {3 x}{4 \sqrt [3]{-x^2+x^3}}+\frac {\sqrt {3} \sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt [3]{1-i} \sqrt {3} \sqrt [3]{x}}\right )}{4 \sqrt [3]{1-i} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt {3} \sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt [3]{1+i} \sqrt {3} \sqrt [3]{x}}\right )}{4 \sqrt [3]{1+i} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt {3} \sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{4 \sqrt [3]{2} \sqrt [3]{-x^2+x^3}}+\frac {3 \sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{1-i}}-\sqrt [3]{x}\right )}{8 \sqrt [3]{1-i} \sqrt [3]{-x^2+x^3}}+\frac {3 \sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{1+i}}-\sqrt [3]{x}\right )}{8 \sqrt [3]{1+i} \sqrt [3]{-x^2+x^3}}+\frac {3 \sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{2}}-\sqrt [3]{x}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log (-1-x)}{8 \sqrt [3]{2} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log (i-x)}{8 \sqrt [3]{1+i} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log (i+x)}{8 \sqrt [3]{1-i} \sqrt [3]{-x^2+x^3}}\\ \end {align*}
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Mathematica [C] time = 1.13, size = 392, normalized size = 1.88 \begin {gather*} \frac {x \left (-12-\frac {-\frac {4 \log \left (1-\sqrt [3]{1-i} \sqrt [3]{\frac {x}{x-1}}\right )}{\sqrt [3]{1-i}}-\frac {4 \log \left (1-\sqrt [3]{1+i} \sqrt [3]{\frac {x}{x-1}}\right )}{\sqrt [3]{1+i}}-2\ 2^{2/3} \log \left (1-\sqrt [3]{2} \sqrt [3]{\frac {x}{x-1}}\right )+\frac {2 \log \left ((1-i)^{2/3} \left (\frac {x}{x-1}\right )^{2/3}+\sqrt [3]{1-i} \sqrt [3]{\frac {x}{x-1}}+1\right )}{\sqrt [3]{1-i}}+\frac {2 \log \left ((1+i)^{2/3} \left (\frac {x}{x-1}\right )^{2/3}+\sqrt [3]{1+i} \sqrt [3]{\frac {x}{x-1}}+1\right )}{\sqrt [3]{1+i}}+2^{2/3} \log \left (2^{2/3} \left (\frac {x}{x-1}\right )^{2/3}+\sqrt [3]{2} \sqrt [3]{\frac {x}{x-1}}+1\right )+\frac {4 \sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{1-i} \sqrt [3]{\frac {x}{x-1}}}{\sqrt {3}}\right )}{\sqrt [3]{1-i}}+\frac {4 \sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{1+i} \sqrt [3]{\frac {x}{x-1}}}{\sqrt {3}}\right )}{\sqrt [3]{1+i}}+2\ 2^{2/3} \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{2} \sqrt [3]{\frac {x}{x-1}}+1}{\sqrt {3}}\right )}{\sqrt [3]{\frac {x}{x-1}}}\right )}{16 \sqrt [3]{(x-1) x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.50, size = 208, normalized size = 1.00 \begin {gather*} -\frac {3 \left (-x^2+x^3\right )^{2/3}}{4 (-1+x) x}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-x^2+x^3}}\right )}{4 \sqrt [3]{2}}+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-x^2+x^3}\right )}{4 \sqrt [3]{2}}-\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 )}{8 \sqrt [3]{2}}+\frac {1}{4} \text {RootSum}\left [2-2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.93, size = 1790, normalized size = 8.61
result too large to display
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 {1}{{\left (x^{4} - 1\right )} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 61.45, size = 14413, normalized size = 69.29
method | result | size |
trager | \(\text {Expression too large to display}\) | \(14413\) |
risch | \(\text {Expression too large to display}\) | \(16404\) |
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 {1}{{\left (x^{4} - 1\right )} {\left (x^{3} - x^{2}\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 = -0.00 \begin {gather*} \int \frac {1}{\left (x^4-1\right )\,{\left (x^3-x^2\right )}^{1/3}} \,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 {1}{\sqrt [3]{x^{2} \left (x - 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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