Optimal. Leaf size=100 \[ -\frac {1}{3} \text {RootSum}\left [\text {$\#$1}^6-\text {$\#$1}^3-1\& ,\frac {-\text {$\#$1}^3 \log \left (\sqrt [3]{x^3+1}-\text {$\#$1} x\right )+\text {$\#$1}^3 \log (x)+\log \left (\sqrt [3]{x^3+1}-\text {$\#$1} x\right )-\log (x)}{2 \text {$\#$1}^4-\text {$\#$1}}\& \right ]-\frac {\left (x^3+1\right )^{2/3}}{2 x^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.59, antiderivative size = 172, normalized size of antiderivative = 1.72, number of steps used = 8, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6728, 277, 239, 429} \begin {gather*} \frac {\left (5+3 \sqrt {5}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,\frac {2 x^3}{1-\sqrt {5}}\right )}{5 \left (1-\sqrt {5}\right )}+\frac {\left (5-3 \sqrt {5}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,\frac {2 x^3}{1+\sqrt {5}}\right )}{5 \left (1+\sqrt {5}\right )}-\frac {1}{2} \log \left (\sqrt [3]{x^3+1}-x\right )+\frac {\tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\left (x^3+1\right )^{2/3}}{2 x^2} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 239
Rule 277
Rule 429
Rule 6728
Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^{2/3}}{x^3 \left (-1-x^3+x^6\right )} \, dx &=\int \left (\frac {\left (1+x^3\right )^{2/3}}{x^3}+\frac {\left (2-x^3\right ) \left (1+x^3\right )^{2/3}}{-1-x^3+x^6}\right ) \, dx\\ &=\int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx+\int \frac {\left (2-x^3\right ) \left (1+x^3\right )^{2/3}}{-1-x^3+x^6} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\int \frac {1}{\sqrt [3]{1+x^3}} \, dx+\int \left (\frac {\left (-1+\frac {3}{\sqrt {5}}\right ) \left (1+x^3\right )^{2/3}}{-1-\sqrt {5}+2 x^3}+\frac {\left (-1-\frac {3}{\sqrt {5}}\right ) \left (1+x^3\right )^{2/3}}{-1+\sqrt {5}+2 x^3}\right ) \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{5} \left (-5+3 \sqrt {5}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{-1-\sqrt {5}+2 x^3} \, dx-\frac {1}{5} \left (5+3 \sqrt {5}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{-1+\sqrt {5}+2 x^3} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (5+3 \sqrt {5}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,\frac {2 x^3}{1-\sqrt {5}}\right )}{5 \left (1-\sqrt {5}\right )}+\frac {\left (5-3 \sqrt {5}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,\frac {2 x^3}{1+\sqrt {5}}\right )}{5 \left (1+\sqrt {5}\right )}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (-x+\sqrt [3]{1+x^3}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.68, size = 342, normalized size = 3.42 \begin {gather*} -\frac {\left (x^3+1\right )^{2/3}}{2 x^2}-\frac {-2 \left (\sqrt {5}-1\right )^{4/3} \log \left (\sqrt [3]{\sqrt {5}-1}-\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3+1}}\right )+2 \left (1+\sqrt {5}\right )^{4/3} \log \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3+1}}+\sqrt [3]{1+\sqrt {5}}\right )+\left (\sqrt {5}-1\right )^{4/3} \left (2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {2}{\sqrt {5}-1}} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )+\log \left (\frac {\sqrt [3]{2 \left (\sqrt {5}-1\right )} x}{\sqrt [3]{x^3+1}}+\frac {2^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+\left (\sqrt {5}-1\right )^{2/3}\right )\right )+\left (1+\sqrt {5}\right )^{4/3} \left (2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {2}{1+\sqrt {5}}} x}{\sqrt [3]{x^3+1}}-1}{\sqrt {3}}\right )-\log \left (-\frac {\sqrt [3]{2 \left (1+\sqrt {5}\right )} x}{\sqrt [3]{x^3+1}}+\frac {2^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+\left (1+\sqrt {5}\right )^{2/3}\right )\right )}{12 \sqrt [3]{2} \sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.20, size = 100, normalized size = 1.00 \begin {gather*} -\frac {\left (1+x^3\right )^{2/3}}{2 x^2}-\frac {1}{3} \text {RootSum}\left [-1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}+2 \text {$\#$1}^4}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{3} - 1\right )}}{{\left (x^{6} - x^{3} - 1\right )} x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 141.11, size = 6585, normalized size = 65.85
method | result | size |
risch | \(\text {Expression too large to display}\) | \(6585\) |
trager | \(\text {Expression too large to display}\) | \(11049\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{3} - 1\right )}}{{\left (x^{6} - x^{3} - 1\right )} x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {\left (x^3-1\right )\,{\left (x^3+1\right )}^{2/3}}{x^3\,\left (-x^6+x^3+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] 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.
[In]
[Out]
________________________________________________________________________________________