Optimal. Leaf size=141 \[ \frac {1}{8} \left (1-x^3\right )^{8/3}-\frac {2}{5} \left (1-x^3\right )^{5/3}+\left (1-x^3\right )^{2/3}+\frac {1}{2 \sqrt [3]{1-x^3}}-\frac {\log \left (x^3+1\right )}{12 \sqrt [3]{2}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}}+\frac {\tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}} \]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {446, 87, 43, 627, 51, 55, 617, 204, 31} \begin {gather*} \frac {1}{8} \left (1-x^3\right )^{8/3}-\frac {2}{5} \left (1-x^3\right )^{5/3}+\left (1-x^3\right )^{2/3}+\frac {1}{2 \sqrt [3]{1-x^3}}-\frac {\log \left (x^3+1\right )}{12 \sqrt [3]{2}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}}+\frac {\tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 43
Rule 51
Rule 55
Rule 87
Rule 204
Rule 446
Rule 617
Rule 627
Rubi steps
\begin {align*} \int \frac {x^{14}}{\left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^4}{(1-x)^{4/3} (1+x)} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (-\frac {1}{\sqrt [3]{1-x}}-\frac {x^2}{\sqrt [3]{1-x}}+\frac {1}{\sqrt [3]{1-x} \left (1-x^2\right )}\right ) \, dx,x,x^3\right )\\ &=\frac {1}{2} \left (1-x^3\right )^{2/3}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [3]{1-x}} \, dx,x,x^3\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} \left (1-x^2\right )} \, dx,x,x^3\right )\\ &=\frac {1}{2} \left (1-x^3\right )^{2/3}-\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt [3]{1-x}}-2 (1-x)^{2/3}+(1-x)^{5/3}\right ) \, dx,x,x^3\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{(1-x)^{4/3} (1+x)} \, dx,x,x^3\right )\\ &=\frac {1}{2 \sqrt [3]{1-x^3}}+\left (1-x^3\right )^{2/3}-\frac {2}{5} \left (1-x^3\right )^{5/3}+\frac {1}{8} \left (1-x^3\right )^{8/3}+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} (1+x)} \, dx,x,x^3\right )\\ &=\frac {1}{2 \sqrt [3]{1-x^3}}+\left (1-x^3\right )^{2/3}-\frac {2}{5} \left (1-x^3\right )^{5/3}+\frac {1}{8} \left (1-x^3\right )^{8/3}-\frac {\log \left (1+x^3\right )}{12 \sqrt [3]{2}}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{1-x^3}\right )-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{2}-x} \, dx,x,\sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}}\\ &=\frac {1}{2 \sqrt [3]{1-x^3}}+\left (1-x^3\right )^{2/3}-\frac {2}{5} \left (1-x^3\right )^{5/3}+\frac {1}{8} \left (1-x^3\right )^{8/3}-\frac {\log \left (1+x^3\right )}{12 \sqrt [3]{2}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2^{2/3} \sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\\ &=\frac {1}{2 \sqrt [3]{1-x^3}}+\left (1-x^3\right )^{2/3}-\frac {2}{5} \left (1-x^3\right )^{5/3}+\frac {1}{8} \left (1-x^3\right )^{8/3}+\frac {\tan ^{-1}\left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {\log \left (1+x^3\right )}{12 \sqrt [3]{2}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.04, size = 53, normalized size = 0.38 \begin {gather*} \frac {20 \, _2F_1\left (-\frac {1}{3},1;\frac {2}{3};\frac {1}{2} \left (1-x^3\right )\right )-5 x^9-x^6-23 x^3+29}{40 \sqrt [3]{1-x^3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.18, size = 156, normalized size = 1.11 \begin {gather*} \frac {\log \left (2^{2/3} \sqrt [3]{1-x^3}-2\right )}{6 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2} \left (1-x^3\right )^{2/3}+2^{2/3} \sqrt [3]{1-x^3}+2\right )}{12 \sqrt [3]{2}}+\frac {\tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\left (1-x^3\right )^{2/3} \left (5 x^9+x^6+23 x^3-49\right )}{40 \left (x^3-1\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 140, normalized size = 0.99 \begin {gather*} \frac {10 \, \sqrt {6} 2^{\frac {1}{6}} {\left (x^{3} - 1\right )} \arctan \left (\frac {1}{6} \cdot 2^{\frac {1}{6}} {\left (\sqrt {6} 2^{\frac {1}{3}} + 2 \, \sqrt {6} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) - 5 \cdot 2^{\frac {2}{3}} {\left (x^{3} - 1\right )} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) + 10 \cdot 2^{\frac {2}{3}} {\left (x^{3} - 1\right )} \log \left (-2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) + 3 \, {\left (5 \, x^{9} + x^{6} + 23 \, x^{3} - 49\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{120 \, {\left (x^{3} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.19, size = 136, normalized size = 0.96 \begin {gather*} \frac {1}{8} \, {\left (x^{3} - 1\right )}^{2} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + \frac {1}{12} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) - \frac {2}{5} \, {\left (-x^{3} + 1\right )}^{\frac {5}{3}} - \frac {1}{24} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} \right |}\right ) + {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + \frac {1}{2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 3.78, size = 682, normalized size = 4.84
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.07, size = 128, normalized size = 0.91 \begin {gather*} \frac {1}{8} \, {\left (-x^{3} + 1\right )}^{\frac {8}{3}} + \frac {1}{12} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) - \frac {2}{5} \, {\left (-x^{3} + 1\right )}^{\frac {5}{3}} - \frac {1}{24} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (-2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) + {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + \frac {1}{2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.86, size = 148, normalized size = 1.05 \begin {gather*} \frac {2^{2/3}\,\ln \left (\frac {{\left (1-x^3\right )}^{1/3}}{4}-\frac {2^{1/3}}{4}\right )}{12}+\frac {1}{2\,{\left (1-x^3\right )}^{1/3}}+{\left (1-x^3\right )}^{2/3}-\frac {2\,{\left (1-x^3\right )}^{5/3}}{5}+\frac {{\left (1-x^3\right )}^{8/3}}{8}+\frac {2^{2/3}\,\ln \left (\frac {{\left (1-x^3\right )}^{1/3}}{4}-\frac {2^{1/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{16}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{24}-\frac {2^{2/3}\,\ln \left (\frac {{\left (1-x^3\right )}^{1/3}}{4}-\frac {2^{1/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{16}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{24} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{14}}{\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {4}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________