Integrand size = 22, antiderivative size = 172 \[ \int \frac {1}{x^5 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=-\frac {\left (1-x^2\right )^{2/3}}{12 x^4}-\frac {\left (1-x^2\right )^{2/3}}{18 x^2}-\frac {\arctan \left (\frac {1+\sqrt [3]{2-2 x^2}}{\sqrt {3}}\right )}{18\ 2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {1+2 \sqrt [3]{1-x^2}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {\log (x)}{27}+\frac {\log \left (3+x^2\right )}{108\ 2^{2/3}}+\frac {1}{18} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac {\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{36\ 2^{2/3}} \]
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Time = 0.10 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {457, 105, 156, 162, 57, 632, 210, 31, 631} \[ \int \frac {1}{x^5 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{2-2 x^2}+1}{\sqrt {3}}\right )}{18\ 2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {2 \sqrt [3]{1-x^2}+1}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {\left (1-x^2\right )^{2/3}}{18 x^2}+\frac {\log \left (x^2+3\right )}{108\ 2^{2/3}}+\frac {1}{18} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac {\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{36\ 2^{2/3}}-\frac {\left (1-x^2\right )^{2/3}}{12 x^4}-\frac {\log (x)}{27} \]
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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*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} x^3 (3+x)} \, dx,x,x^2\right ) \\ & = -\frac {\left (1-x^2\right )^{2/3}}{12 x^4}-\frac {1}{12} \text {Subst}\left (\int \frac {-2-\frac {4 x}{3}}{\sqrt [3]{1-x} x^2 (3+x)} \, dx,x,x^2\right ) \\ & = -\frac {\left (1-x^2\right )^{2/3}}{12 x^4}-\frac {\left (1-x^2\right )^{2/3}}{18 x^2}+\frac {1}{36} \text {Subst}\left (\int \frac {4+\frac {2 x}{3}}{\sqrt [3]{1-x} x (3+x)} \, dx,x,x^2\right ) \\ & = -\frac {\left (1-x^2\right )^{2/3}}{12 x^4}-\frac {\left (1-x^2\right )^{2/3}}{18 x^2}-\frac {1}{54} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} (3+x)} \, dx,x,x^2\right )+\frac {1}{27} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} x} \, dx,x,x^2\right ) \\ & = -\frac {\left (1-x^2\right )^{2/3}}{12 x^4}-\frac {\left (1-x^2\right )^{2/3}}{18 x^2}-\frac {\log (x)}{27}+\frac {\log \left (3+x^2\right )}{108\ 2^{2/3}}-\frac {1}{36} \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 {1}{18} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1-x^2}\right )+\frac {1}{18} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1-x^2}\right )+\frac {\text {Subst}\left (\int \frac {1}{2^{2/3}-x} \, dx,x,\sqrt [3]{1-x^2}\right )}{36\ 2^{2/3}} \\ & = -\frac {\left (1-x^2\right )^{2/3}}{12 x^4}-\frac {\left (1-x^2\right )^{2/3}}{18 x^2}-\frac {\log (x)}{27}+\frac {\log \left (3+x^2\right )}{108\ 2^{2/3}}+\frac {1}{18} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac {\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{36\ 2^{2/3}}-\frac {1}{9} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1-x^2}\right )+\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\sqrt [3]{2-2 x^2}\right )}{18\ 2^{2/3}} \\ & = -\frac {\left (1-x^2\right )^{2/3}}{12 x^4}-\frac {\left (1-x^2\right )^{2/3}}{18 x^2}-\frac {\tan ^{-1}\left (\frac {1+\sqrt [3]{2-2 x^2}}{\sqrt {3}}\right )}{18\ 2^{2/3} \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {1+2 \sqrt [3]{1-x^2}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {\log (x)}{27}+\frac {\log \left (3+x^2\right )}{108\ 2^{2/3}}+\frac {1}{18} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac {\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{36\ 2^{2/3}} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.24 \[ \int \frac {1}{x^5 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=-\frac {18 \left (1-x^2\right )^{2/3}+12 x^2 \left (1-x^2\right )^{2/3}+2 \sqrt [3]{2} \sqrt {3} x^4 \arctan \left (\frac {1+\sqrt [3]{2-2 x^2}}{\sqrt {3}}\right )-8 \sqrt {3} x^4 \arctan \left (\frac {1+2 \sqrt [3]{1-x^2}}{\sqrt {3}}\right )+2 \sqrt [3]{2} x^4 \log \left (-2+\sqrt [3]{2-2 x^2}\right )-\sqrt [3]{2} x^4 \log \left (4+2 \sqrt [3]{2-2 x^2}+\left (2-2 x^2\right )^{2/3}\right )-8 x^4 \log \left (-1+\sqrt [3]{1-x^2}\right )+4 x^4 \log \left (1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}\right )}{216 x^4} \]
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Time = 4.51 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.31
method | result | size |
pseudoelliptic | \(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (1+2^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {1}{3}}\right )}{3}\right ) 2^{\frac {1}{3}} x^{4}+8 \sqrt {3}\, \arctan \left (\frac {\left (1+2 \left (-x^{2}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) x^{4}-2 \ln \left (\left (-x^{2}+1\right )^{\frac {1}{3}}-2^{\frac {2}{3}}\right ) 2^{\frac {1}{3}} x^{4}+\ln \left (\left (-x^{2}+1\right )^{\frac {2}{3}}+2^{\frac {2}{3}} \left (-x^{2}+1\right )^{\frac {1}{3}}+2 \,2^{\frac {1}{3}}\right ) 2^{\frac {1}{3}} x^{4}+8 \ln \left (-1+\left (-x^{2}+1\right )^{\frac {1}{3}}\right ) x^{4}-4 \ln \left (1+\left (-x^{2}+1\right )^{\frac {1}{3}}+\left (-x^{2}+1\right )^{\frac {2}{3}}\right ) x^{4}-12 \left (-x^{2}+1\right )^{\frac {2}{3}} x^{2}-18 \left (-x^{2}+1\right )^{\frac {2}{3}}}{216 {\left (-1+\left (-x^{2}+1\right )^{\frac {1}{3}}\right )}^{2} {\left (1+\left (-x^{2}+1\right )^{\frac {1}{3}}+\left (-x^{2}+1\right )^{\frac {2}{3}}\right )}^{2}}\) | \(226\) |
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Time = 0.27 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.26 \[ \int \frac {1}{x^5 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=-\frac {4 \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} x^{4} \arctan \left (\frac {1}{6} \cdot 4^{\frac {1}{6}} {\left (2 \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} - 4^{\frac {1}{3}} \sqrt {3}\right )}\right ) + 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{4} \log \left (4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} - 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {2}{3}}\right ) - 2 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{4} \log \left (-4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right ) - 16 \, \sqrt {3} x^{4} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + 8 \, x^{4} \log \left ({\left (-x^{2} + 1\right )}^{\frac {2}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) - 16 \, x^{4} \log \left ({\left (-x^{2} + 1\right )}^{\frac {1}{3}} - 1\right ) + 12 \, {\left (2 \, x^{2} + 3\right )} {\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{432 \, x^{4}} \]
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\[ \int \frac {1}{x^5 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\int \frac {1}{x^{5} \sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )}\, dx \]
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\[ \int \frac {1}{x^5 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\int { \frac {1}{{\left (x^{2} + 3\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} x^{5}} \,d x } \]
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Time = 0.31 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.03 \[ \int \frac {1}{x^5 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=-\frac {1}{216} \cdot 4^{\frac {2}{3}} \sqrt {3} \arctan \left (\frac {1}{12} \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (4^{\frac {1}{3}} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {1}{432} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {2}{3}}\right ) - \frac {1}{216} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {1}{3}} - {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right ) + \frac {1}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {2 \, {\left (-x^{2} + 1\right )}^{\frac {5}{3}} - 5 \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{36 \, x^{4}} - \frac {1}{54} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac {2}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{27} \, \log \left (-{\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) \]
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Time = 5.58 (sec) , antiderivative size = 397, normalized size of antiderivative = 2.31 \[ \int \frac {1}{x^5 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\frac {\ln \left (\frac {11}{486}-\frac {11\,{\left (1-x^2\right )}^{1/3}}{486}\right )}{27}-\frac {2^{1/3}\,\ln \left (-\frac {2^{2/3}\,\left (\frac {2^{1/3}\,\left (\frac {135\,2^{2/3}}{4}-\frac {1755\,{\left (1-x^2\right )}^{1/3}}{4}\right )}{108}+\frac {7}{2}\right )}{11664}-\frac {{\left (1-x^2\right )}^{1/3}}{2916}\right )}{108}+\ln \left ({\left (-\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right )}^2\,\left (\left (-\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right )\,\left (393660\,{\left (-\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right )}^2-\frac {1755\,{\left (1-x^2\right )}^{1/3}}{4}\right )-\frac {7}{2}\right )-\frac {{\left (1-x^2\right )}^{1/3}}{2916}\right )\,\left (-\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right )-\ln \left (-{\left (\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right )}^2\,\left (\left (\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right )\,\left (393660\,{\left (\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right )}^2-\frac {1755\,{\left (1-x^2\right )}^{1/3}}{4}\right )+\frac {7}{2}\right )-\frac {{\left (1-x^2\right )}^{1/3}}{2916}\right )\,\left (\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right )-\frac {\frac {5\,{\left (1-x^2\right )}^{2/3}}{36}-\frac {{\left (1-x^2\right )}^{5/3}}{18}}{{\left (x^2-1\right )}^2+2\,x^2-1}+\frac {{\left (-1\right )}^{1/3}\,2^{1/3}\,\ln \left (\frac {{\left (-1\right )}^{2/3}\,2^{2/3}\,\left (\frac {{\left (-1\right )}^{1/3}\,2^{1/3}\,\left (\frac {135\,{\left (-1\right )}^{2/3}\,2^{2/3}}{4}-\frac {1755\,{\left (1-x^2\right )}^{1/3}}{4}\right )}{108}-\frac {7}{2}\right )}{11664}-\frac {{\left (1-x^2\right )}^{1/3}}{2916}\right )}{108}-\frac {{\left (-1\right )}^{1/3}\,2^{1/3}\,\ln \left (-\frac {{\left (1-x^2\right )}^{1/3}}{2916}+\frac {{\left (-1\right )}^{2/3}\,2^{2/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (\frac {{\left (-1\right )}^{1/3}\,2^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (\frac {1755\,{\left (1-x^2\right )}^{1/3}}{4}-\frac {135\,{\left (-1\right )}^{2/3}\,2^{2/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{16}\right )}{216}-\frac {7}{2}\right )}{46656}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{216} \]
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