Optimal. Leaf size=175 \[ \frac {5}{6 \sqrt [3]{1-x^3}}-\frac {1}{3 x^3 \sqrt [3]{1-x^3}}+\frac {\tan ^{-1}\left (\frac {1+2 \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {\log (x)}{6}-\frac {\log \left (1+x^3\right )}{12 \sqrt [3]{2}}+\frac {1}{6} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {457, 105, 162,
53, 57, 632, 210, 31, 631} \begin {gather*} \frac {\text {ArcTan}\left (\frac {2 \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {\text {ArcTan}\left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {1}{3 x^3 \sqrt [3]{1-x^3}}+\frac {5}{6 \sqrt [3]{1-x^3}}-\frac {\log \left (x^3+1\right )}{12 \sqrt [3]{2}}+\frac {1}{6} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}}-\frac {\log (x)}{6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 53
Rule 57
Rule 105
Rule 162
Rule 210
Rule 457
Rule 631
Rule 632
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{(1-x)^{4/3} x^2 (1+x)} \, dx,x,x^3\right )\\ &=-\frac {1}{3 x^3 \sqrt [3]{1-x^3}}-\frac {1}{3} \text {Subst}\left (\int \frac {-\frac {1}{3}-\frac {4 x}{3}}{(1-x)^{4/3} x (1+x)} \, dx,x,x^3\right )\\ &=-\frac {1}{3 x^3 \sqrt [3]{1-x^3}}+\frac {1}{9} \text {Subst}\left (\int \frac {1}{(1-x)^{4/3} x} \, dx,x,x^3\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{(1-x)^{4/3} (1+x)} \, dx,x,x^3\right )\\ &=\frac {5}{6 \sqrt [3]{1-x^3}}-\frac {1}{3 x^3 \sqrt [3]{1-x^3}}+\frac {1}{9} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} x} \, dx,x,x^3\right )+\frac {1}{6} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} (1+x)} \, dx,x,x^3\right )\\ &=\frac {5}{6 \sqrt [3]{1-x^3}}-\frac {1}{3 x^3 \sqrt [3]{1-x^3}}-\frac {\log (x)}{6}-\frac {\log \left (1+x^3\right )}{12 \sqrt [3]{2}}-\frac {1}{6} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1-x^3}\right )+\frac {1}{6} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1-x^3}\right )+\frac {1}{4} \text {Subst}\left (\int \frac {1}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{1-x^3}\right )-\frac {\text {Subst}\left (\int \frac {1}{\sqrt [3]{2}-x} \, dx,x,\sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}}\\ &=\frac {5}{6 \sqrt [3]{1-x^3}}-\frac {1}{3 x^3 \sqrt [3]{1-x^3}}-\frac {\log (x)}{6}-\frac {\log \left (1+x^3\right )}{12 \sqrt [3]{2}}+\frac {1}{6} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}}-\frac {1}{3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1-x^3}\right )-\frac {\text {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 {5}{6 \sqrt [3]{1-x^3}}-\frac {1}{3 x^3 \sqrt [3]{1-x^3}}+\frac {\tan ^{-1}\left (\frac {1+2 \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{3 \sqrt {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 (x)}{6}-\frac {\log \left (1+x^3\right )}{12 \sqrt [3]{2}}+\frac {1}{6} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.33, size = 209, normalized size = 1.19 \begin {gather*} \frac {1}{72} \left (\frac {12 \left (-2+5 x^3\right )}{x^3 \sqrt [3]{1-x^3}}+8 \sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{1-x^3}}{\sqrt {3}}\right )+6\ 2^{2/3} \sqrt {3} \tan ^{-1}\left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )+8 \log \left (-1+\sqrt [3]{1-x^3}\right )+6\ 2^{2/3} \log \left (-2+2^{2/3} \sqrt [3]{1-x^3}\right )-4 \log \left (1+\sqrt [3]{1-x^3}+\left (1-x^3\right )^{2/3}\right )-3\ 2^{2/3} \log \left (2+2^{2/3} \sqrt [3]{1-x^3}+\sqrt [3]{2} \left (1-x^3\right )^{2/3}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{4} \left (-x^{3}+1\right )^{\frac {4}{3}} \left (x^{3}+1\right )}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 3.00, size = 238, normalized size = 1.36 \begin {gather*} \frac {6 \, \sqrt {6} 2^{\frac {1}{6}} {\left (x^{6} - x^{3}\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 ) - 3 \cdot 2^{\frac {2}{3}} {\left (x^{6} - x^{3}\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 ) + 6 \cdot 2^{\frac {2}{3}} {\left (x^{6} - x^{3}\right )} \log \left (-2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) + 8 \, \sqrt {3} {\left (x^{6} - x^{3}\right )} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - 4 \, {\left (x^{6} - x^{3}\right )} \log \left ({\left (-x^{3} + 1\right )}^{\frac {2}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + 8 \, {\left (x^{6} - x^{3}\right )} \log \left ({\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) - 12 \, {\left (5 \, x^{3} - 2\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{72 \, {\left (x^{6} - x^{3}\right )}} \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 {1}{x^{4} \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]
________________________________________________________________________________________
Giac [A]
time = 1.23, size = 181, normalized size = 1.03 \begin {gather*} \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 {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 ) + \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {5 \, x^{3} - 2}{6 \, {\left ({\left (-x^{3} + 1\right )}^{\frac {4}{3}} - {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}} - \frac {1}{18} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {2}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{9} \, \log \left ({\left | {\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 5.02, size = 399, normalized size = 2.28 \begin {gather*} \frac {\ln \left (\frac {11\,{\left (1-x^3\right )}^{1/3}}{972}-\frac {11}{972}\right )}{9}+\frac {2^{2/3}\,\ln \left (\frac {2^{1/3}\,\left (\frac {2^{2/3}\,\left (\frac {81\,2^{1/3}}{4}-\frac {75\,{\left (1-x^3\right )}^{1/3}}{4}\right )}{12}-\frac {35}{12}\right )}{72}+\frac {{\left (1-x^3\right )}^{1/3}}{27}\right )}{12}+\ln \left ({\left (-\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )}^2\,\left (\left (-\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )\,\left (1458\,{\left (-\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )}^2-\frac {75\,{\left (1-x^3\right )}^{1/3}}{4}\right )-\frac {35}{12}\right )+\frac {{\left (1-x^3\right )}^{1/3}}{27}\right )\,\left (-\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )-\ln \left (\frac {{\left (1-x^3\right )}^{1/3}}{27}-{\left (\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )}^2\,\left (\left (\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )\,\left (1458\,{\left (\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )}^2-\frac {75\,{\left (1-x^3\right )}^{1/3}}{4}\right )+\frac {35}{12}\right )\right )\,\left (\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )+\frac {\frac {5\,x^3}{6}-\frac {1}{3}}{{\left (1-x^3\right )}^{1/3}-{\left (1-x^3\right )}^{4/3}}+\frac {2^{2/3}\,\ln \left (\frac {{\left (1-x^3\right )}^{1/3}}{27}+\frac {2^{1/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (\frac {2^{2/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (\frac {81\,2^{1/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{16}-\frac {75\,{\left (1-x^3\right )}^{1/3}}{4}\right )}{24}-\frac {35}{12}\right )}{288}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{24}-\frac {2^{2/3}\,\ln \left (\frac {{\left (1-x^3\right )}^{1/3}}{27}-\frac {2^{1/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (\frac {2^{2/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (\frac {81\,2^{1/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{16}-\frac {75\,{\left (1-x^3\right )}^{1/3}}{4}\right )}{24}+\frac {35}{12}\right )}{288}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{24} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________