Optimal. Leaf size=208 \[ \frac {\left (1-x^2\right )^{2/3}}{216 \left (3+x^2\right )}-\frac {\left (1-x^2\right )^{2/3}}{12 x^4 \left (3+x^2\right )}-\frac {\left (1-x^2\right )^{2/3}}{36 x^2 \left (3+x^2\right )}-\frac {13 \tan ^{-1}\left (\frac {1+\sqrt [3]{2-2 x^2}}{\sqrt {3}}\right )}{216\ 2^{2/3} \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {1+2 \sqrt [3]{1-x^2}}{\sqrt {3}}\right )}{18 \sqrt {3}}-\frac {\log (x)}{54}+\frac {13 \log \left (3+x^2\right )}{1296\ 2^{2/3}}+\frac {1}{36} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac {13 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{432\ 2^{2/3}} \]
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Rubi [A]
time = 0.11, antiderivative size = 208, 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, 156,
162, 57, 632, 210, 31, 631} \begin {gather*} -\frac {13 \text {ArcTan}\left (\frac {\sqrt [3]{2-2 x^2}+1}{\sqrt {3}}\right )}{216\ 2^{2/3} \sqrt {3}}+\frac {\text {ArcTan}\left (\frac {2 \sqrt [3]{1-x^2}+1}{\sqrt {3}}\right )}{18 \sqrt {3}}-\frac {\left (1-x^2\right )^{2/3}}{36 x^2 \left (x^2+3\right )}+\frac {\left (1-x^2\right )^{2/3}}{216 \left (x^2+3\right )}+\frac {13 \log \left (x^2+3\right )}{1296\ 2^{2/3}}+\frac {1}{36} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac {13 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{432\ 2^{2/3}}-\frac {\left (1-x^2\right )^{2/3}}{12 x^4 \left (x^2+3\right )}-\frac {\log (x)}{54} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 57
Rule 105
Rule 156
Rule 162
Rule 210
Rule 457
Rule 631
Rule 632
Rubi steps
\begin {align*} \int \frac {1}{x^5 \sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} x^3 (3+x)^2} \, dx,x,x^2\right )\\ &=-\frac {\left (1-x^2\right )^{2/3}}{12 x^4 \left (3+x^2\right )}-\frac {1}{12} \text {Subst}\left (\int \frac {-1-\frac {7 x}{3}}{\sqrt [3]{1-x} x^2 (3+x)^2} \, dx,x,x^2\right )\\ &=-\frac {\left (1-x^2\right )^{2/3}}{12 x^4 \left (3+x^2\right )}-\frac {\left (1-x^2\right )^{2/3}}{36 x^2 \left (3+x^2\right )}+\frac {1}{36} \text {Subst}\left (\int \frac {6+\frac {4 x}{3}}{\sqrt [3]{1-x} x (3+x)^2} \, dx,x,x^2\right )\\ &=\frac {\left (1-x^2\right )^{2/3}}{216 \left (3+x^2\right )}-\frac {\left (1-x^2\right )^{2/3}}{12 x^4 \left (3+x^2\right )}-\frac {\left (1-x^2\right )^{2/3}}{36 x^2 \left (3+x^2\right )}+\frac {1}{432} \text {Subst}\left (\int \frac {24-\frac {2 x}{3}}{\sqrt [3]{1-x} x (3+x)} \, dx,x,x^2\right )\\ &=\frac {\left (1-x^2\right )^{2/3}}{216 \left (3+x^2\right )}-\frac {\left (1-x^2\right )^{2/3}}{12 x^4 \left (3+x^2\right )}-\frac {\left (1-x^2\right )^{2/3}}{36 x^2 \left (3+x^2\right )}+\frac {1}{54} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} x} \, dx,x,x^2\right )-\frac {13}{648} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} (3+x)} \, dx,x,x^2\right )\\ &=\frac {\left (1-x^2\right )^{2/3}}{216 \left (3+x^2\right )}-\frac {\left (1-x^2\right )^{2/3}}{12 x^4 \left (3+x^2\right )}-\frac {\left (1-x^2\right )^{2/3}}{36 x^2 \left (3+x^2\right )}-\frac {\log (x)}{54}+\frac {13 \log \left (3+x^2\right )}{1296\ 2^{2/3}}-\frac {1}{36} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1-x^2}\right )+\frac {1}{36} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1-x^2}\right )-\frac {13}{432} \text {Subst}\left (\int \frac {1}{2 \sqrt [3]{2}+2^{2/3} x+x^2} \, dx,x,\sqrt [3]{1-x^2}\right )+\frac {13 \text {Subst}\left (\int \frac {1}{2^{2/3}-x} \, dx,x,\sqrt [3]{1-x^2}\right )}{432\ 2^{2/3}}\\ &=\frac {\left (1-x^2\right )^{2/3}}{216 \left (3+x^2\right )}-\frac {\left (1-x^2\right )^{2/3}}{12 x^4 \left (3+x^2\right )}-\frac {\left (1-x^2\right )^{2/3}}{36 x^2 \left (3+x^2\right )}-\frac {\log (x)}{54}+\frac {13 \log \left (3+x^2\right )}{1296\ 2^{2/3}}+\frac {1}{36} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac {13 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{432\ 2^{2/3}}-\frac {1}{18} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1-x^2}\right )+\frac {13 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\sqrt [3]{2-2 x^2}\right )}{216\ 2^{2/3}}\\ &=\frac {\left (1-x^2\right )^{2/3}}{216 \left (3+x^2\right )}-\frac {\left (1-x^2\right )^{2/3}}{12 x^4 \left (3+x^2\right )}-\frac {\left (1-x^2\right )^{2/3}}{36 x^2 \left (3+x^2\right )}-\frac {13 \tan ^{-1}\left (\frac {1+\sqrt [3]{2-2 x^2}}{\sqrt {3}}\right )}{216\ 2^{2/3} \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {1+2 \sqrt [3]{1-x^2}}{\sqrt {3}}\right )}{18 \sqrt {3}}-\frac {\log (x)}{54}+\frac {13 \log \left (3+x^2\right )}{1296\ 2^{2/3}}+\frac {1}{36} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac {13 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{432\ 2^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 197, normalized size = 0.95 \begin {gather*} \frac {\frac {12 \left (1-x^2\right )^{2/3} \left (-18-6 x^2+x^4\right )}{x^4 \left (3+x^2\right )}-26 \sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {1+\sqrt [3]{2-2 x^2}}{\sqrt {3}}\right )+48 \sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{1-x^2}}{\sqrt {3}}\right )-26 \sqrt [3]{2} \log \left (-2+\sqrt [3]{2-2 x^2}\right )+13 \sqrt [3]{2} \log \left (4+2 \sqrt [3]{2-2 x^2}+\left (2-2 x^2\right )^{2/3}\right )+48 \log \left (-1+\sqrt [3]{1-x^2}\right )-24 \log \left (1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}\right )}{2592} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{5} \left (-x^{2}+1\right )^{\frac {1}{3}} \left (x^{2}+3\right )^{2}}\, dx\]
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 = 0.49, size = 265, normalized size = 1.27 \begin {gather*} -\frac {52 \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} {\left (x^{6} + 3 \, x^{4}\right )} \arctan \left (\frac {1}{6} \cdot 4^{\frac {1}{6}} {\left (2 \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} - 4^{\frac {1}{3}} \sqrt {3}\right )}\right ) + 13 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{6} + 3 \, x^{4}\right )} \log \left (4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} - 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {2}{3}}\right ) - 26 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{6} + 3 \, x^{4}\right )} \log \left (-4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right ) - 96 \, \sqrt {3} {\left (x^{6} + 3 \, x^{4}\right )} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + 48 \, {\left (x^{6} + 3 \, x^{4}\right )} \log \left ({\left (-x^{2} + 1\right )}^{\frac {2}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) - 96 \, {\left (x^{6} + 3 \, x^{4}\right )} \log \left ({\left (-x^{2} + 1\right )}^{\frac {1}{3}} - 1\right ) - 24 \, {\left (x^{4} - 6 \, x^{2} - 18\right )} {\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{5184 \, {\left (x^{6} + 3 \, x^{4}\right )}} \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^{5} \sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.81, size = 181, normalized size = 0.87 \begin {gather*} -\frac {13}{2592} \cdot 4^{\frac {2}{3}} \sqrt {3} \arctan \left (\frac {1}{12} \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (4^{\frac {1}{3}} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {13}{5184} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {2}{3}}\right ) - \frac {13}{2592} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {1}{3}} - {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right ) + \frac {1}{54} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {{\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{216 \, {\left (x^{2} + 3\right )}} - \frac {{\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{36 \, x^{4}} - \frac {1}{108} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac {2}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{54} \, \log \left (-{\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.56, size = 416, normalized size = 2.00 \begin {gather*} \frac {\ln \left (\frac {9109}{10077696}-\frac {9109\,{\left (1-x^2\right )}^{1/3}}{10077696}\right )}{54}-\frac {13\,2^{1/3}\,\ln \left (-\frac {169\,2^{2/3}\,\left (\frac {13\,2^{1/3}\,\left (\frac {2535\,2^{2/3}}{64}-\frac {7419\,{\left (1-x^2\right )}^{1/3}}{64}\right )}{1296}-\frac {469}{3456}\right )}{1679616}-\frac {845\,{\left (1-x^2\right )}^{1/3}}{5038848}\right )}{1296}+\frac {\frac {{\left (1-x^2\right )}^{5/3}}{54}-\frac {23\,{\left (1-x^2\right )}^{2/3}}{216}+\frac {{\left (1-x^2\right )}^{8/3}}{216}}{6\,{\left (x^2-1\right )}^2+{\left (x^2-1\right )}^3+9\,x^2-5}+\ln \left ({\left (-\frac {1}{108}+\frac {\sqrt {3}\,1{}\mathrm {i}}{108}\right )}^2\,\left (\left (-\frac {1}{108}+\frac {\sqrt {3}\,1{}\mathrm {i}}{108}\right )\,\left (393660\,{\left (-\frac {1}{108}+\frac {\sqrt {3}\,1{}\mathrm {i}}{108}\right )}^2-\frac {7419\,{\left (1-x^2\right )}^{1/3}}{64}\right )+\frac {469}{3456}\right )-\frac {845\,{\left (1-x^2\right )}^{1/3}}{5038848}\right )\,\left (-\frac {1}{108}+\frac {\sqrt {3}\,1{}\mathrm {i}}{108}\right )-\ln \left (-{\left (\frac {1}{108}+\frac {\sqrt {3}\,1{}\mathrm {i}}{108}\right )}^2\,\left (\left (\frac {1}{108}+\frac {\sqrt {3}\,1{}\mathrm {i}}{108}\right )\,\left (393660\,{\left (\frac {1}{108}+\frac {\sqrt {3}\,1{}\mathrm {i}}{108}\right )}^2-\frac {7419\,{\left (1-x^2\right )}^{1/3}}{64}\right )-\frac {469}{3456}\right )-\frac {845\,{\left (1-x^2\right )}^{1/3}}{5038848}\right )\,\left (\frac {1}{108}+\frac {\sqrt {3}\,1{}\mathrm {i}}{108}\right )+\frac {13\,{\left (-1\right )}^{1/3}\,2^{1/3}\,\ln \left (\frac {169\,{\left (-1\right )}^{2/3}\,2^{2/3}\,\left (\frac {13\,{\left (-1\right )}^{1/3}\,2^{1/3}\,\left (\frac {2535\,{\left (-1\right )}^{2/3}\,2^{2/3}}{64}-\frac {7419\,{\left (1-x^2\right )}^{1/3}}{64}\right )}{1296}+\frac {469}{3456}\right )}{1679616}-\frac {845\,{\left (1-x^2\right )}^{1/3}}{5038848}\right )}{1296}-\frac {13\,{\left (-1\right )}^{1/3}\,2^{1/3}\,\ln \left (-\frac {845\,{\left (1-x^2\right )}^{1/3}}{5038848}+\frac {169\,{\left (-1\right )}^{2/3}\,2^{2/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (\frac {13\,{\left (-1\right )}^{1/3}\,2^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (\frac {7419\,{\left (1-x^2\right )}^{1/3}}{64}-\frac {2535\,{\left (-1\right )}^{2/3}\,2^{2/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{256}\right )}{2592}+\frac {469}{3456}\right )}{6718464}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2592} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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