Optimal. Leaf size=120 \[ \frac {3 \left (-1+4 x^2\right ) \sqrt [3]{-x+x^3}}{4 x^3}-\frac {1}{8} \text {RootSum}\left [3-6 \text {$\#$1}^3+4 \text {$\#$1}^6\& ,\frac {-9 \log (x)+9 \log \left (\sqrt [3]{-x+x^3}-x \text {$\#$1}\right )+4 \log (x) \text {$\#$1}^3-4 \log \left (\sqrt [3]{-x+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-3 \text {$\#$1}^2+4 \text {$\#$1}^5}\& \right ] \]
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Rubi [C] Result contains complex when optimal does not.
time = 1.21, antiderivative size = 725, normalized size of antiderivative = 6.04, number of steps
used = 18, number of rules used = 9, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.243, Rules used = {2081, 6860,
270, 477, 476, 486, 597, 12, 503} \begin {gather*} -\frac {\sqrt [6]{3} \left (-2 \sqrt {3}+i\right ) \sqrt [3]{x^3-x} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [6]{3} x^{2/3}}{\sqrt [3]{\sqrt {3}-i} \sqrt [3]{x^2-1}}}{\sqrt {3}}\right )}{\left (\sqrt {3}-i\right )^{7/3} \sqrt [3]{x} \sqrt [3]{x^2-1}}+\frac {\sqrt [6]{3} \left (2 \sqrt {3}+i\right ) \sqrt [3]{x^3-x} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [6]{3} x^{2/3}}{\sqrt [3]{\sqrt {3}+i} \sqrt [3]{x^2-1}}}{\sqrt {3}}\right )}{\left (\sqrt {3}+i\right )^{7/3} \sqrt [3]{x} \sqrt [3]{x^2-1}}+\frac {3 \left (2 \sqrt {3}+i\right ) \sqrt [3]{x^3-x}}{4 \left (\sqrt {3}+i\right ) x}+\frac {3 \left (-2 \sqrt {3}+i\right ) \sqrt [3]{x^3-x}}{4 \left (-\sqrt {3}+i\right ) x}-\frac {\left (\sqrt {3}+6 i\right ) \sqrt [3]{x^3-x}}{4 \left (-\sqrt {3}+i\right ) x^3}-\frac {\left (-\sqrt {3}+6 i\right ) \sqrt [3]{x^3-x}}{4 \left (\sqrt {3}+i\right ) x^3}-\frac {3 \sqrt [3]{x^3-x} \left (1-x^2\right )}{8 x^3}-\frac {\left (2 \sqrt {3}+i\right ) \sqrt [3]{x^3-x} \log \left (-x^2-i \sqrt {3}+1\right )}{2 \sqrt [3]{3} \left (\sqrt {3}+i\right )^{7/3} \sqrt [3]{x} \sqrt [3]{x^2-1}}+\frac {\left (-2 \sqrt {3}+i\right ) \sqrt [3]{x^3-x} \log \left (-x^2+i \sqrt {3}+1\right )}{2 \sqrt [3]{3} \left (\sqrt {3}-i\right )^{7/3} \sqrt [3]{x} \sqrt [3]{x^2-1}}-\frac {3^{2/3} \left (-2 \sqrt {3}+i\right ) \sqrt [3]{x^3-x} \log \left (\sqrt [6]{3} x^{2/3}-\sqrt [3]{\sqrt {3}-i} \sqrt [3]{x^2-1}\right )}{2 \left (\sqrt {3}-i\right )^{7/3} \sqrt [3]{x} \sqrt [3]{x^2-1}}+\frac {3^{2/3} \left (2 \sqrt {3}+i\right ) \sqrt [3]{x^3-x} \log \left (\sqrt [6]{3} x^{2/3}-\sqrt [3]{\sqrt {3}+i} \sqrt [3]{x^2-1}\right )}{2 \left (\sqrt {3}+i\right )^{7/3} \sqrt [3]{x} \sqrt [3]{x^2-1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 270
Rule 476
Rule 477
Rule 486
Rule 503
Rule 597
Rule 2081
Rule 6860
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx &=\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{-1+x^2} \left (8-10 x^2+x^4\right )}{x^{11/3} \left (4-2 x^2+x^4\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=\frac {\sqrt [3]{-x+x^3} \int \left (\frac {\sqrt [3]{-1+x^2}}{x^{11/3}}+\frac {4 \left (1-2 x^2\right ) \sqrt [3]{-1+x^2}}{x^{11/3} \left (4-2 x^2+x^4\right )}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{-1+x^2}}{x^{11/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\left (4 \sqrt [3]{-x+x^3}\right ) \int \frac {\left (1-2 x^2\right ) \sqrt [3]{-1+x^2}}{x^{11/3} \left (4-2 x^2+x^4\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=-\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}+\frac {\left (4 \sqrt [3]{-x+x^3}\right ) \int \left (\frac {\left (-2+\frac {i}{\sqrt {3}}\right ) \sqrt [3]{-1+x^2}}{x^{11/3} \left (-2-2 i \sqrt {3}+2 x^2\right )}+\frac {\left (-2-\frac {i}{\sqrt {3}}\right ) \sqrt [3]{-1+x^2}}{x^{11/3} \left (-2+2 i \sqrt {3}+2 x^2\right )}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=-\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}+\frac {\left (4 \left (-6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \int \frac {\sqrt [3]{-1+x^2}}{x^{11/3} \left (-2-2 i \sqrt {3}+2 x^2\right )} \, dx}{3 \sqrt [3]{x} \sqrt [3]{-1+x^2}}-\frac {\left (4 \left (6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \int \frac {\sqrt [3]{-1+x^2}}{x^{11/3} \left (-2+2 i \sqrt {3}+2 x^2\right )} \, dx}{3 \sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=-\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}+\frac {\left (4 \left (-6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1+x^6}}{x^9 \left (-2-2 i \sqrt {3}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}-\frac {\left (4 \left (6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1+x^6}}{x^9 \left (-2+2 i \sqrt {3}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=-\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}+\frac {\left (2 \left (-6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1+x^3}}{x^5 \left (-2-2 i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}-\frac {\left (2 \left (6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1+x^3}}{x^5 \left (-2+2 i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=-\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}+\frac {\left (2 \left (-6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1-x^3}}{x^5 \left (-2-2 i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^2}}-\frac {\left (2 \left (6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1-x^3}}{x^5 \left (-2+2 i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^2}}\\ &=-\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}+\frac {\left (6 i+\sqrt {3}\right ) \sqrt [3]{-x+x^3} \left (2 \left (1-x^2\right ) \left (2 \left (i+\sqrt {3}\right )-3 \left (i-\sqrt {3}\right ) x^2\right )+x^2 \left (3 i+\sqrt {3}+3 \sqrt {3} x^2\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {\sqrt {3} x^2}{\left (i-\sqrt {3}\right ) \left (1-x^2\right )}\right )-3 x^2 \left (3 i+\sqrt {3}-\sqrt {3} x^2\right ) \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {\sqrt {3} x^2}{\left (i-\sqrt {3}\right ) \left (1-x^2\right )}\right )\right )}{64 x^3 \left (1-x^2\right )}+\frac {\left (6 i-\sqrt {3}\right ) \sqrt [3]{-x+x^3} \left (2 \left (1-x^2\right ) \left (2 \left (i-\sqrt {3}\right )-3 \left (i+\sqrt {3}\right ) x^2\right )+x^2 \left (3 i-\sqrt {3}-3 \sqrt {3} x^2\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {\sqrt {3} x^2}{\left (i+\sqrt {3}\right ) \left (1-x^2\right )}\right )-3 x^2 \left (3 i-\sqrt {3}+\sqrt {3} x^2\right ) \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};-\frac {\sqrt {3} x^2}{\left (i+\sqrt {3}\right ) \left (1-x^2\right )}\right )\right )}{64 x^3 \left (1-x^2\right )}\\ \end {align*}
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Mathematica [A]
time = 2.86, size = 146, normalized size = 1.22 \begin {gather*} -\frac {\sqrt [3]{x \left (-1+x^2\right )} \left (18 \left (1-4 x^2\right ) \sqrt [3]{-1+x^2}+x^{8/3} \text {RootSum}\left [3-6 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-18 \log (x)+27 \log \left (\sqrt [3]{-1+x^2}-x^{2/3} \text {$\#$1}\right )+8 \log (x) \text {$\#$1}^3-12 \log \left (\sqrt [3]{-1+x^2}-x^{2/3} \text {$\#$1}\right ) \text {$\#$1}^3}{-3 \text {$\#$1}^2+4 \text {$\#$1}^5}\&\right ]\right )}{24 x^3 \sqrt [3]{-1+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{3}-x \right )^{\frac {1}{3}} \left (x^{4}-10 x^{2}+8\right )}{x^{4} \left (x^{4}-2 x^{2}+4\right )}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 )} \left (x^{4} - 10 x^{2} + 8\right )}{x^{4} \left (x^{4} - 2 x^{2} + 4\right )}\, dx \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*} \text {could not integrate} \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}\,\left (x^4-10\,x^2+8\right )}{x^4\,\left (x^4-2\,x^2+4\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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