Optimal. Leaf size=108 \[ -\frac {4}{9} \log \left (\sqrt [3]{x^3+1}-x\right )+\frac {4 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+1}+x}\right )}{3 \sqrt {3}}+\frac {2}{9} \log \left (\sqrt [3]{x^3+1} x+\left (x^3+1\right )^{2/3}+x^2\right )+\frac {\left (x^3+1\right )^{2/3} \left (10 x^6-3 x^3-3\right )}{15 x^5} \]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 97, normalized size of antiderivative = 0.90, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1487, 451, 277, 239} \begin {gather*} -\frac {2}{3} \log \left (\sqrt [3]{x^3+1}-x\right )+\frac {4 \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\left (x^3+1\right )^{5/3}}{5 x^5}+\frac {2 \left (x^3+1\right )^{5/3}}{3 x^2}-\frac {2 \left (x^3+1\right )^{2/3}}{3 x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 239
Rule 277
Rule 451
Rule 1487
Rubi steps
\begin {align*} \int \frac {\left (1+x^3\right )^{2/3} \left (1+2 x^6\right )}{x^6} \, dx &=\frac {2 \left (1+x^3\right )^{5/3}}{3 x^2}+\frac {1}{3} \int \frac {\left (1+x^3\right )^{2/3} \left (3+4 x^3\right )}{x^6} \, dx\\ &=-\frac {\left (1+x^3\right )^{5/3}}{5 x^5}+\frac {2 \left (1+x^3\right )^{5/3}}{3 x^2}+\frac {4}{3} \int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{3 x^2}-\frac {\left (1+x^3\right )^{5/3}}{5 x^5}+\frac {2 \left (1+x^3\right )^{5/3}}{3 x^2}+\frac {4}{3} \int \frac {1}{\sqrt [3]{1+x^3}} \, dx\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{3 x^2}-\frac {\left (1+x^3\right )^{5/3}}{5 x^5}+\frac {2 \left (1+x^3\right )^{5/3}}{3 x^2}+\frac {4 \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {2}{3} \log \left (-x+\sqrt [3]{1+x^3}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.03, size = 35, normalized size = 0.32 \begin {gather*} 2 x \, _2F_1\left (-\frac {2}{3},\frac {1}{3};\frac {4}{3};-x^3\right )-\frac {\left (x^3+1\right )^{5/3}}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.21, size = 108, normalized size = 1.00 \begin {gather*} -\frac {4}{9} \log \left (\sqrt [3]{x^3+1}-x\right )+\frac {4 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+1}+x}\right )}{3 \sqrt {3}}+\frac {2}{9} \log \left (\sqrt [3]{x^3+1} x+\left (x^3+1\right )^{2/3}+x^2\right )+\frac {\left (x^3+1\right )^{2/3} \left (10 x^6-3 x^3-3\right )}{15 x^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.86, size = 117, normalized size = 1.08 \begin {gather*} \frac {20 \, \sqrt {3} x^{5} \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 13720 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (5831 \, x^{3} + 7200\right )}}{58653 \, x^{3} + 8000}\right ) - 10 \, x^{5} \log \left (3 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + 1\right ) + 3 \, {\left (10 \, x^{6} - 3 \, x^{3} - 3\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{45 \, x^{5}} \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 (2 \, x^{6} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.28, size = 45, normalized size = 0.42 \begin {gather*} \frac {10 x^{9}+7 x^{6}-6 x^{3}-3}{15 x^{5} \left (x^{3}+1\right )^{\frac {1}{3}}}+\frac {4 x \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -x^{3}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.41, size = 106, normalized size = 0.98 \begin {gather*} -\frac {4}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) + \frac {2 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{3 \, x^{2} {\left (\frac {x^{3} + 1}{x^{3}} - 1\right )}} - \frac {{\left (x^{3} + 1\right )}^{\frac {5}{3}}}{5 \, x^{5}} + \frac {2}{9} \, \log \left (\frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) - \frac {4}{9} \, \log \left (\frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.12, size = 38, normalized size = 0.35 \begin {gather*} 2\,x\,{{}}_2{\mathrm {F}}_1\left (-\frac {2}{3},\frac {1}{3};\ \frac {4}{3};\ -x^3\right )-\frac {{\left (x^3+1\right )}^{2/3}+x^3\,{\left (x^3+1\right )}^{2/3}}{5\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 2.39, size = 85, normalized size = 0.79 \begin {gather*} \frac {2 x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {\left (1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} + \frac {\left (1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 x^{3} \Gamma \left (- \frac {2}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________