Optimal. Leaf size=157 \[ -\frac {\left (1-x^3\right )^{2/3}}{3 x^3}-\frac {2 \tan ^{-1}\left (\frac {1+2 \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log (x)}{3}-\frac {\log \left (1+x^3\right )}{6 \sqrt [3]{2}}-\frac {1}{3} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {457, 105, 162,
57, 632, 210, 31, 631} \begin {gather*} -\frac {2 \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 )}{\sqrt [3]{2} \sqrt {3}}-\frac {\left (1-x^3\right )^{2/3}}{3 x^3}-\frac {\log \left (x^3+1\right )}{6 \sqrt [3]{2}}-\frac {1}{3} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}+\frac {\log (x)}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 57
Rule 105
Rule 162
Rule 210
Rule 457
Rule 631
Rule 632
Rubi steps
\begin {align*} \int \frac {1}{x^4 \sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} x^2 (1+x)} \, dx,x,x^3\right )\\ &=-\frac {\left (1-x^3\right )^{2/3}}{3 x^3}-\frac {1}{3} \text {Subst}\left (\int \frac {\frac {2}{3}-\frac {x}{3}}{\sqrt [3]{1-x} x (1+x)} \, dx,x,x^3\right )\\ &=-\frac {\left (1-x^3\right )^{2/3}}{3 x^3}-\frac {2}{9} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} x} \, dx,x,x^3\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} (1+x)} \, dx,x,x^3\right )\\ &=-\frac {\left (1-x^3\right )^{2/3}}{3 x^3}+\frac {\log (x)}{3}-\frac {\log \left (1+x^3\right )}{6 \sqrt [3]{2}}+\frac {1}{3} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1-x^3}\right )-\frac {1}{3} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1-x^3}\right )+\frac {1}{2} \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 )}{2 \sqrt [3]{2}}\\ &=-\frac {\left (1-x^3\right )^{2/3}}{3 x^3}+\frac {\log (x)}{3}-\frac {\log \left (1+x^3\right )}{6 \sqrt [3]{2}}-\frac {1}{3} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}+\frac {2}{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 )}{\sqrt [3]{2}}\\ &=-\frac {\left (1-x^3\right )^{2/3}}{3 x^3}-\frac {2 \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 )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log (x)}{3}-\frac {\log \left (1+x^3\right )}{6 \sqrt [3]{2}}-\frac {1}{3} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 202, normalized size = 1.29 \begin {gather*} \frac {1}{36} \left (-\frac {12 \left (1-x^3\right )^{2/3}}{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.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{4} \left (-x^{3}+1\right )^{\frac {1}{3}} \left (x^{3}+1\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A]
time = 2.43, size = 187, normalized size = 1.19 \begin {gather*} \frac {6 \, \sqrt {6} 2^{\frac {1}{6}} x^{3} \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}} x^{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 ) + 6 \cdot 2^{\frac {2}{3}} x^{3} \log \left (-2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) - 8 \, \sqrt {3} x^{3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + 4 \, x^{3} \log \left ({\left (-x^{3} + 1\right )}^{\frac {2}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) - 8 \, x^{3} \log \left ({\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) - 12 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{36 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{4} \sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.72, size = 163, normalized size = 1.04 \begin {gather*} \frac {1}{6} \, \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}{12} \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}{6} \cdot 2^{\frac {2}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} \right |}\right ) - \frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{3 \, x^{3}} + \frac {1}{9} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {2}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {2}{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.
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Mupad [B]
time = 4.86, size = 382, normalized size = 2.43 \begin {gather*} \frac {2^{2/3}\,\ln \left (\frac {2^{1/3}\,\left (\frac {2^{2/3}\,\left (81\,2^{1/3}-75\,{\left (1-x^3\right )}^{1/3}\right )}{6}-\frac {38}{3}\right )}{18}+\frac {16\,{\left (1-x^3\right )}^{1/3}}{27}\right )}{6}-\frac {{\left (1-x^3\right )}^{2/3}}{3\,x^3}-\frac {2\,\ln \left (\frac {344\,{\left (1-x^3\right )}^{1/3}}{243}-\frac {344}{243}\right )}{9}+\ln \left ({\left (\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}^2\,\left (\left (\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )\,\left (1458\,{\left (\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}^2-75\,{\left (1-x^3\right )}^{1/3}\right )-\frac {38}{3}\right )+\frac {16\,{\left (1-x^3\right )}^{1/3}}{27}\right )\,\left (\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )-\ln \left (\frac {16\,{\left (1-x^3\right )}^{1/3}}{27}-{\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}^2\,\left (\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )\,\left (1458\,{\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}^2-75\,{\left (1-x^3\right )}^{1/3}\right )+\frac {38}{3}\right )\right )\,\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )+\frac {2^{2/3}\,\ln \left (\frac {16\,{\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}{4}-75\,{\left (1-x^3\right )}^{1/3}\right )}{12}-\frac {38}{3}\right )}{72}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{12}-\frac {2^{2/3}\,\ln \left (\frac {16\,{\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}{4}-75\,{\left (1-x^3\right )}^{1/3}\right )}{12}+\frac {38}{3}\right )}{72}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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