Optimal. Leaf size=110 \[ \frac {1}{54} \log \left (\sqrt [3]{x^6+1}-x^2\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x^2}{2 \sqrt [3]{x^6+1}+x^2}\right )}{18 \sqrt {3}}+\frac {1}{36} \sqrt [3]{x^6+1} \left (3 x^{10}+x^4\right )-\frac {1}{108} \log \left (\left (x^6+1\right )^{2/3}+x^4+\sqrt [3]{x^6+1} x^2\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 119, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 10, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {275, 279, 321, 331, 292, 31, 634, 618, 204, 628} \begin {gather*} \frac {1}{12} \sqrt [3]{x^6+1} x^{10}+\frac {1}{36} \sqrt [3]{x^6+1} x^4+\frac {1}{54} \log \left (1-\frac {x^2}{\sqrt [3]{x^6+1}}\right )+\frac {\tan ^{-1}\left (\frac {\frac {2 x^2}{\sqrt [3]{x^6+1}}+1}{\sqrt {3}}\right )}{18 \sqrt {3}}-\frac {1}{108} \log \left (\frac {x^4}{\left (x^6+1\right )^{2/3}}+\frac {x^2}{\sqrt [3]{x^6+1}}+1\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 204
Rule 275
Rule 279
Rule 292
Rule 321
Rule 331
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int x^9 \sqrt [3]{1+x^6} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^4 \sqrt [3]{1+x^3} \, dx,x,x^2\right )\\ &=\frac {1}{12} x^{10} \sqrt [3]{1+x^6}+\frac {1}{12} \operatorname {Subst}\left (\int \frac {x^4}{\left (1+x^3\right )^{2/3}} \, dx,x,x^2\right )\\ &=\frac {1}{36} x^4 \sqrt [3]{1+x^6}+\frac {1}{12} x^{10} \sqrt [3]{1+x^6}-\frac {1}{18} \operatorname {Subst}\left (\int \frac {x}{\left (1+x^3\right )^{2/3}} \, dx,x,x^2\right )\\ &=\frac {1}{36} x^4 \sqrt [3]{1+x^6}+\frac {1}{12} x^{10} \sqrt [3]{1+x^6}-\frac {1}{18} \operatorname {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {x^2}{\sqrt [3]{1+x^6}}\right )\\ &=\frac {1}{36} x^4 \sqrt [3]{1+x^6}+\frac {1}{12} x^{10} \sqrt [3]{1+x^6}-\frac {1}{54} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x^2}{\sqrt [3]{1+x^6}}\right )+\frac {1}{54} \operatorname {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {x^2}{\sqrt [3]{1+x^6}}\right )\\ &=\frac {1}{36} x^4 \sqrt [3]{1+x^6}+\frac {1}{12} x^{10} \sqrt [3]{1+x^6}+\frac {1}{54} \log \left (1-\frac {x^2}{\sqrt [3]{1+x^6}}\right )-\frac {1}{108} \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {x^2}{\sqrt [3]{1+x^6}}\right )+\frac {1}{36} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x^2}{\sqrt [3]{1+x^6}}\right )\\ &=\frac {1}{36} x^4 \sqrt [3]{1+x^6}+\frac {1}{12} x^{10} \sqrt [3]{1+x^6}+\frac {1}{54} \log \left (1-\frac {x^2}{\sqrt [3]{1+x^6}}\right )-\frac {1}{108} \log \left (1+\frac {x^4}{\left (1+x^6\right )^{2/3}}+\frac {x^2}{\sqrt [3]{1+x^6}}\right )-\frac {1}{18} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x^2}{\sqrt [3]{1+x^6}}\right )\\ &=\frac {1}{36} x^4 \sqrt [3]{1+x^6}+\frac {1}{12} x^{10} \sqrt [3]{1+x^6}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x^2}{\sqrt [3]{1+x^6}}}{\sqrt {3}}\right )}{18 \sqrt {3}}+\frac {1}{54} \log \left (1-\frac {x^2}{\sqrt [3]{1+x^6}}\right )-\frac {1}{108} \log \left (1+\frac {x^4}{\left (1+x^6\right )^{2/3}}+\frac {x^2}{\sqrt [3]{1+x^6}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 34, normalized size = 0.31 \begin {gather*} \frac {1}{12} x^4 \left (\left (x^6+1\right )^{4/3}-\, _2F_1\left (-\frac {1}{3},\frac {2}{3};\frac {5}{3};-x^6\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.88, size = 110, normalized size = 1.00 \begin {gather*} \frac {1}{54} \log \left (\sqrt [3]{x^6+1}-x^2\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x^2}{2 \sqrt [3]{x^6+1}+x^2}\right )}{18 \sqrt {3}}+\frac {1}{36} \sqrt [3]{x^6+1} \left (3 x^{10}+x^4\right )-\frac {1}{108} \log \left (\left (x^6+1\right )^{2/3}+x^4+\sqrt [3]{x^6+1} x^2\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 100, normalized size = 0.91 \begin {gather*} -\frac {1}{54} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2} + 2 \, \sqrt {3} {\left (x^{6} + 1\right )}^{\frac {1}{3}}}{3 \, x^{2}}\right ) + \frac {1}{36} \, {\left (3 \, x^{10} + x^{4}\right )} {\left (x^{6} + 1\right )}^{\frac {1}{3}} + \frac {1}{54} \, \log \left (-\frac {x^{2} - {\left (x^{6} + 1\right )}^{\frac {1}{3}}}{x^{2}}\right ) - \frac {1}{108} \, \log \left (\frac {x^{4} + {\left (x^{6} + 1\right )}^{\frac {1}{3}} x^{2} + {\left (x^{6} + 1\right )}^{\frac {2}{3}}}{x^{4}}\right ) \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 {\left (x^{6} + 1\right )}^{\frac {1}{3}} x^{9}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.29, size = 37, normalized size = 0.34 \begin {gather*} \frac {x^{4} \left (3 x^{6}+1\right ) \left (x^{6}+1\right )^{\frac {1}{3}}}{36}-\frac {x^{4} \hypergeom \left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -x^{6}\right )}{36} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 121, normalized size = 1.10 \begin {gather*} -\frac {1}{54} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{6} + 1\right )}^{\frac {1}{3}}}{x^{2}} + 1\right )}\right ) - \frac {\frac {2 \, {\left (x^{6} + 1\right )}^{\frac {1}{3}}}{x^{2}} + \frac {{\left (x^{6} + 1\right )}^{\frac {4}{3}}}{x^{8}}}{36 \, {\left (\frac {2 \, {\left (x^{6} + 1\right )}}{x^{6}} - \frac {{\left (x^{6} + 1\right )}^{2}}{x^{12}} - 1\right )}} - \frac {1}{108} \, \log \left (\frac {{\left (x^{6} + 1\right )}^{\frac {1}{3}}}{x^{2}} + \frac {{\left (x^{6} + 1\right )}^{\frac {2}{3}}}{x^{4}} + 1\right ) + \frac {1}{54} \, \log \left (\frac {{\left (x^{6} + 1\right )}^{\frac {1}{3}}}{x^{2}} - 1\right ) \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 x^9\,{\left (x^6+1\right )}^{1/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 1.18, size = 31, normalized size = 0.28 \begin {gather*} \frac {x^{10} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{6 \Gamma \left (\frac {8}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________