Optimal. Leaf size=112 \[ \frac {\left (-8-23 x^3\right ) \left (1+x^3\right )^{2/3}}{80 x^5}+\frac {1}{16} \text {RootSum}\left [5-10 \text {$\#$1}^3+8 \text {$\#$1}^6\& ,\frac {-5 \log (x)+5 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+6 \log (x) \text {$\#$1}^3-6 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}+8 \text {$\#$1}^4}\& \right ] \]
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Rubi [C] Result contains complex when optimal does not.
time = 0.84, antiderivative size = 615, normalized size of antiderivative = 5.49, number of steps
used = 13, number of rules used = 6, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6860, 270,
283, 245, 399, 384} \begin {gather*} -\frac {\left (\sqrt {15}+i\right ) \text {ArcTan}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{16 \sqrt {5}}+\frac {\left (-\sqrt {15}+i\right ) \text {ArcTan}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{16 \sqrt {5}}+\frac {1}{8} \sqrt {3} \text {ArcTan}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )-\frac {\sqrt [3]{3} \left (-\sqrt {15}+i\right ) \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [6]{15} x}{\sqrt [3]{\sqrt {15}-3 i} \sqrt [3]{x^3+1}}}{\sqrt {3}}\right )}{16 \sqrt [6]{5} \left (\sqrt {15}-3 i\right )^{2/3}}+\frac {\sqrt [3]{3} \left (\sqrt {15}+i\right ) \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [6]{15} x}{\sqrt [3]{\sqrt {15}+3 i} \sqrt [3]{x^3+1}}}{\sqrt {3}}\right )}{16 \sqrt [6]{5} \left (\sqrt {15}+3 i\right )^{2/3}}+\frac {\left (\sqrt {15}+i\right ) \log \left (6 x^3+2 \left (3-i \sqrt {15}\right )\right )}{32 \sqrt [6]{15} \left (\sqrt {15}+3 i\right )^{2/3}}-\frac {\left (-\sqrt {15}+i\right ) \log \left (6 x^3+2 \left (3+i \sqrt {15}\right )\right )}{32 \sqrt [6]{15} \left (\sqrt {15}-3 i\right )^{2/3}}+\frac {3^{5/6} \left (-\sqrt {15}+i\right ) \log \left (-\sqrt [3]{x^3+1}+\frac {\sqrt [6]{15} x}{\sqrt [3]{\sqrt {15}-3 i}}\right )}{32 \sqrt [6]{5} \left (\sqrt {15}-3 i\right )^{2/3}}-\frac {3^{5/6} \left (\sqrt {15}+i\right ) \log \left (-\sqrt [3]{x^3+1}+\frac {\sqrt [6]{15} x}{\sqrt [3]{\sqrt {15}+3 i}}\right )}{32 \sqrt [6]{5} \left (\sqrt {15}+3 i\right )^{2/3}}+\frac {1}{160} \left (15+i \sqrt {15}\right ) \log \left (\sqrt [3]{x^3+1}-x\right )+\frac {1}{160} \left (15-i \sqrt {15}\right ) \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {3}{16} \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {\left (x^3+1\right )^{5/3}}{10 x^5}-\frac {3 \left (x^3+1\right )^{2/3}}{16 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 245
Rule 270
Rule 283
Rule 384
Rule 399
Rule 6860
Rubi steps
\begin {align*} \int \frac {\left (1+x^3\right )^{2/3} \left (4+6 x^3+3 x^6\right )}{x^6 \left (8+6 x^3+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 {3 \left (1+x^3\right )^{2/3} \left (2+3 x^3\right )}{8 \left (8+6 x^3+3 x^6\right )}\right ) \, dx\\ &=\frac {3}{8} \int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx-\frac {3}{8} \int \frac {\left (1+x^3\right )^{2/3} \left (2+3 x^3\right )}{8+6 x^3+3 x^6} \, 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 {3}{8} \int \frac {1}{\sqrt [3]{1+x^3}} \, dx-\frac {3}{8} \int \left (\frac {\left (3+i \sqrt {\frac {3}{5}}\right ) \left (1+x^3\right )^{2/3}}{6-2 i \sqrt {15}+6 x^3}+\frac {\left (3-i \sqrt {\frac {3}{5}}\right ) \left (1+x^3\right )^{2/3}}{6+2 i \sqrt {15}+6 x^3}\right ) \, 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}{40} \left (3 \left (15-i \sqrt {15}\right )\right ) \int \frac {\left (1+x^3\right )^{2/3}}{6+2 i \sqrt {15}+6 x^3} \, dx-\frac {1}{40} \left (3 \left (15+i \sqrt {15}\right )\right ) \int \frac {\left (1+x^3\right )^{2/3}}{6-2 i \sqrt {15}+6 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 {3 \left (15 i-\sqrt {15}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {3 x^3}{3-i \sqrt {15}}\right )}{80 \left (3 i+\sqrt {15}\right )}-\frac {3 \left (15 i+\sqrt {15}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {3 x^3}{3+i \sqrt {15}}\right )}{80 \left (3 i-\sqrt {15}\right )}+\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 )\\ \end {align*}
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Mathematica [A]
time = 0.17, 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}{16} \text {RootSum}\left [5-10 \text {$\#$1}^3+8 \text {$\#$1}^6\&,\frac {-5 \log (x)+5 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+6 \log (x) \text {$\#$1}^3-6 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}+8 \text {$\#$1}^4}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
1.
time = 230.80, size = 8003, normalized size = 71.46
method | result | size |
risch | \(\text {Expression too large to display}\) | \(8003\) |
trager | \(\text {Expression too large to display}\) | \(10918\) |
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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+1\right )}^{2/3}\,\left (3\,x^6+6\,x^3+4\right )}{x^6\,\left (3\,x^6+6\,x^3+8\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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