Optimal. Leaf size=112 \[ \frac {1}{96} \text {RootSum}\left [8 \text {$\#$1}^6-20 \text {$\#$1}^3+13\& ,\frac {-32 \text {$\#$1}^3 \log \left (\sqrt [3]{x^3+1}-\text {$\#$1} x\right )+32 \text {$\#$1}^3 \log (x)+39 \log \left (\sqrt [3]{x^3+1}-\text {$\#$1} x\right )-39 \log (x)}{4 \text {$\#$1}^4-5 \text {$\#$1}}\& \right ]+\frac {\left (x^3+1\right )^{2/3} \left (-23 x^3-8\right )}{80 x^5} \]
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Rubi [F] time = 1.07, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (1+x^3\right )^{2/3} \left (4+x^3\right )}{x^6 \left (8-4 x^3+x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {\left (1+x^3\right )^{2/3} \left (4+x^3\right )}{x^6 \left (8-4 x^3+x^6\right )} \, dx &=\int \left (\frac {\left (1+x^3\right )^{2/3}}{2 x^6}+\frac {3 \left (1+x^3\right )^{2/3}}{8 x^3}+\frac {(4+x) \left (1+x^3\right )^{2/3}}{48 \left (2+2 x+x^2\right )}+\frac {\left (16-10 x-x^3\right ) \left (1+x^3\right )^{2/3}}{48 \left (4-4 x+2 x^2-2 x^3+x^4\right )}\right ) \, dx\\ &=\frac {1}{48} \int \frac {(4+x) \left (1+x^3\right )^{2/3}}{2+2 x+x^2} \, dx+\frac {1}{48} \int \frac {\left (16-10 x-x^3\right ) \left (1+x^3\right )^{2/3}}{4-4 x+2 x^2-2 x^3+x^4} \, dx+\frac {3}{8} \int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx+\frac {1}{2} \int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx\\ &=-\frac {3 \left (1+x^3\right )^{2/3}}{16 x^2}-\frac {\left (1+x^3\right )^{5/3}}{10 x^5}+\frac {1}{48} \int \left (\frac {(1-3 i) \left (1+x^3\right )^{2/3}}{(2-2 i)+2 x}+\frac {(1+3 i) \left (1+x^3\right )^{2/3}}{(2+2 i)+2 x}\right ) \, dx+\frac {1}{48} \int \left (\frac {16 \left (1+x^3\right )^{2/3}}{4-4 x+2 x^2-2 x^3+x^4}-\frac {10 x \left (1+x^3\right )^{2/3}}{4-4 x+2 x^2-2 x^3+x^4}-\frac {x^3 \left (1+x^3\right )^{2/3}}{4-4 x+2 x^2-2 x^3+x^4}\right ) \, dx+\frac {3}{8} \int \frac {1}{\sqrt [3]{1+x^3}} \, dx\\ &=-\frac {3 \left (1+x^3\right )^{2/3}}{16 x^2}-\frac {\left (1+x^3\right )^{5/3}}{10 x^5}+\frac {1}{8} \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )-\frac {3}{16} \log \left (-x+\sqrt [3]{1+x^3}\right )-\frac {1}{48} \int \frac {x^3 \left (1+x^3\right )^{2/3}}{4-4 x+2 x^2-2 x^3+x^4} \, dx+\left (\frac {1}{48}-\frac {i}{16}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{(2-2 i)+2 x} \, dx+\left (\frac {1}{48}+\frac {i}{16}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{(2+2 i)+2 x} \, dx-\frac {5}{24} \int \frac {x \left (1+x^3\right )^{2/3}}{4-4 x+2 x^2-2 x^3+x^4} \, dx+\frac {1}{3} \int \frac {\left (1+x^3\right )^{2/3}}{4-4 x+2 x^2-2 x^3+x^4} \, dx\\ \end {align*}
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Mathematica [C] time = 0.28, size = 337, normalized size = 3.01 \begin {gather*} \left (x^3+1\right )^{2/3} \left (-\frac {1}{10 x^5}-\frac {23}{80 x^2}\right )+\frac {-(16-2 i) \sqrt [3]{5-i} \log \left (-\frac {\sqrt [3]{13} x}{\sqrt [3]{x^3+1}}+\sqrt [3]{10-2 i}\right )-(16+2 i) \sqrt [3]{5+i} \log \left (-\frac {\sqrt [3]{13} x}{\sqrt [3]{x^3+1}}+\sqrt [3]{10+2 i}\right )+(16-2 i) \sqrt {3} \sqrt [3]{5-i} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} x}{\sqrt [3]{\frac {5}{13}-\frac {i}{13}} \sqrt {3} \sqrt [3]{x^3+1}}\right )+(16+2 i) \sqrt {3} \sqrt [3]{5+i} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} x}{\sqrt [3]{\frac {5}{13}+\frac {i}{13}} \sqrt {3} \sqrt [3]{x^3+1}}\right )+(8-i) \sqrt [3]{5-i} \log \left (\frac {\sqrt [3]{130-26 i} x}{\sqrt [3]{x^3+1}}+\frac {13^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+(10-2 i)^{2/3}\right )+(8+i) \sqrt [3]{5+i} \log \left (\frac {\sqrt [3]{130+26 i} x}{\sqrt [3]{x^3+1}}+\frac {13^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+(10+2 i)^{2/3}\right )}{96\ 2^{2/3} \sqrt [3]{13}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 112, normalized size = 1.00 \begin {gather*} \frac {\left (-8-23 x^3\right ) \left (1+x^3\right )^{2/3}}{80 x^5}+\frac {1}{96} \text {RootSum}\left [13-20 \text {$\#$1}^3+8 \text {$\#$1}^6\&,\frac {-39 \log (x)+39 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+32 \log (x) \text {$\#$1}^3-32 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}+4 \text {$\#$1}^4}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} + 4\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - 4 \, x^{3} + 8\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 228.83, size = 8085, normalized size = 72.19
method | result | size |
risch | \(\text {Expression too large to display}\) | \(8085\) |
trager | \(\text {Expression too large to display}\) | \(12523\) |
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} + 4\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - 4 \, x^{3} + 8\right )} x^{6}}\,{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+1\right )}^{2/3}\,\left (x^3+4\right )}{x^6\,\left (x^6-4\,x^3+8\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 {\left (\left (x + 1\right ) \left (x^{2} - x + 1\right )\right )^{\frac {2}{3}} \left (x^{3} + 4\right )}{x^{6} \left (x^{2} + 2 x + 2\right ) \left (x^{4} - 2 x^{3} + 2 x^{2} - 4 x + 4\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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