Optimal. Leaf size=163 \[ \frac {1}{6} \text {RootSum}\left [\text {$\#$1}^6-\text {$\#$1}^3+1\& ,\frac {\log \left (\sqrt [3]{x^3+x}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]+\frac {\log \left (2^{2/3} \sqrt [3]{x^3+x}-2 x\right )}{6 \sqrt [3]{2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3+x}+x}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {\log \left (2^{2/3} \sqrt [3]{x^3+x} x+\sqrt [3]{2} \left (x^3+x\right )^{2/3}+2 x^2\right )}{12 \sqrt [3]{2}} \]
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Rubi [C] time = 1.77, antiderivative size = 1281, normalized size of antiderivative = 7.86, number of steps used = 25, number of rules used = 12, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {2056, 6715, 2074, 2148, 6728, 377, 200, 31, 634, 617, 204, 628} \begin {gather*} -\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \tan ^{-1}\left (\frac {2-\frac {\sqrt [3]{2} \left (-2 x^{2/3}-i \sqrt {3}+1\right )}{\sqrt [3]{x^2+1}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x^3+x}}-\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \tan ^{-1}\left (\frac {2-\frac {\sqrt [3]{2} \left (-2 x^{2/3}+i \sqrt {3}+1\right )}{\sqrt [3]{x^2+1}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x^3+x}}-\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{2} \left (x^{2/3}+1\right )}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x^3+x}}-\frac {\left (3 i-\sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{x^2+1} \tan ^{-1}\left (\frac {1-\frac {2 x^{2/3}}{\sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{x^2+1}}}{\sqrt {3}}\right )}{6\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}} \sqrt [3]{x^3+x}}+\frac {\left (1+i \sqrt {3}\right )^{2/3} \sqrt [3]{x} \sqrt [3]{x^2+1} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} x^{2/3}}{\sqrt [3]{x^2+1}}}{\sqrt {3}}\right )}{2\ 2^{2/3} \sqrt {3} \sqrt [3]{x^3+x}}-\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (\left (1-x^{2/3}\right )^2 \left (x^{2/3}+1\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{x^3+x}}-\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (\left (-2 x^{2/3}-i \sqrt {3}+1\right ) \left (2 x^{2/3}-i \sqrt {3}+1\right )^2\right )}{24 \sqrt [3]{2} \sqrt [3]{x^3+x}}-\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (\left (-2 x^{2/3}+i \sqrt {3}+1\right ) \left (2 x^{2/3}+i \sqrt {3}+1\right )^2\right )}{24 \sqrt [3]{2} \sqrt [3]{x^3+x}}+\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (\frac {\left (1-i \sqrt {3}\right )^{2/3} x^{4/3}}{\left (x^2+1\right )^{2/3}}-\frac {2^{2/3} x^{2/3}}{\sqrt [3]{x^2+1}}+\left (1+i \sqrt {3}\right )^{2/3}\right )}{12 \sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{x^3+x}}+\frac {\sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (\frac {\left (1+i \sqrt {3}\right )^{2/3} x^{4/3}}{\left (x^2+1\right )^{2/3}}-\frac {2^{2/3} x^{2/3}}{\sqrt [3]{x^2+1}}+\left (1-i \sqrt {3}\right )^{2/3}\right )}{12 \sqrt [3]{x^3+x}}-\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (\frac {\sqrt [3]{1-i \sqrt {3}} x^{2/3}}{\sqrt [3]{x^2+1}}+\sqrt [3]{1+i \sqrt {3}}\right )}{6 \sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{x^3+x}}-\frac {\sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (\frac {\sqrt [3]{1+i \sqrt {3}} x^{2/3}}{\sqrt [3]{x^2+1}}+\sqrt [3]{1-i \sqrt {3}}\right )}{6 \sqrt [3]{x^3+x}}+\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (x^{2/3}-2^{2/3} \sqrt [3]{x^2+1}+1\right )}{8 \sqrt [3]{2} \sqrt [3]{x^3+x}}+\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (-2 x^{2/3}+2\ 2^{2/3} \sqrt [3]{x^2+1}-i \sqrt {3}+1\right )}{8 \sqrt [3]{2} \sqrt [3]{x^3+x}}+\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (-2 x^{2/3}+2\ 2^{2/3} \sqrt [3]{x^2+1}+i \sqrt {3}+1\right )}{8 \sqrt [3]{2} \sqrt [3]{x^3+x}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 31
Rule 200
Rule 204
Rule 377
Rule 617
Rule 628
Rule 634
Rule 2056
Rule 2074
Rule 2148
Rule 6715
Rule 6728
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{x+x^3} \left (-1+x^6\right )} \, dx &=\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{1+x^2} \left (-1+x^6\right )} \, dx}{\sqrt [3]{x+x^3}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3} \left (-1+x^9\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^3}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{9 (-1+x) \sqrt [3]{1+x^3}}+\frac {-2-x}{9 \left (1+x+x^2\right ) \sqrt [3]{1+x^3}}+\frac {-2-x^3}{3 \sqrt [3]{1+x^3} \left (1+x^3+x^6\right )}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^3}}\\ &=\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 \sqrt [3]{x+x^3}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {-2-x}{\left (1+x+x^2\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 \sqrt [3]{x+x^3}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {-2-x^3}{\sqrt [3]{1+x^3} \left (1+x^3+x^6\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^3}}\\ &=-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {1+\frac {\sqrt [3]{2} \left (1+x^{2/3}\right )}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\left (1-x^{2/3}\right )^2 \left (1+x^{2/3}\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1+x^{2/3}-2^{2/3} \sqrt [3]{1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {-1+i \sqrt {3}}{\left (1-i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^3}}+\frac {-1-i \sqrt {3}}{\left (1+i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^3}}\right ) \, dx,x,x^{2/3}\right )}{6 \sqrt [3]{x+x^3}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {-1+i \sqrt {3}}{\sqrt [3]{1+x^3} \left (1-i \sqrt {3}+2 x^3\right )}+\frac {-1-i \sqrt {3}}{\sqrt [3]{1+x^3} \left (1+i \sqrt {3}+2 x^3\right )}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^3}}\\ &=-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {1+\frac {\sqrt [3]{2} \left (1+x^{2/3}\right )}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\left (1-x^{2/3}\right )^2 \left (1+x^{2/3}\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1+x^{2/3}-2^{2/3} \sqrt [3]{1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 \sqrt [3]{x+x^3}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3} \left (1+i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^3}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 \sqrt [3]{x+x^3}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3} \left (1-i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^3}}\\ &=-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {2-\frac {\sqrt [3]{2} \left (1-i \sqrt {3}-2 x^{2/3}\right )}{\sqrt [3]{1+x^2}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {2-\frac {\sqrt [3]{2} \left (1+i \sqrt {3}-2 x^{2/3}\right )}{\sqrt [3]{1+x^2}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {1+\frac {\sqrt [3]{2} \left (1+x^{2/3}\right )}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\left (1-x^{2/3}\right )^2 \left (1+x^{2/3}\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\left (1-i \sqrt {3}-2 x^{2/3}\right ) \left (1-i \sqrt {3}+2 x^{2/3}\right )^2\right )}{24 \sqrt [3]{2} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\left (1+i \sqrt {3}-2 x^{2/3}\right ) \left (1+i \sqrt {3}+2 x^{2/3}\right )^2\right )}{24 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1+x^{2/3}-2^{2/3} \sqrt [3]{1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1-i \sqrt {3}-2 x^{2/3}+2\ 2^{2/3} \sqrt [3]{1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1+i \sqrt {3}-2 x^{2/3}+2\ 2^{2/3} \sqrt [3]{1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+i \sqrt {3}-\left (-1+i \sqrt {3}\right ) x^3} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 \sqrt [3]{x+x^3}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-i \sqrt {3}-\left (-1-i \sqrt {3}\right ) x^3} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 \sqrt [3]{x+x^3}}\\ &=-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {2-\frac {\sqrt [3]{2} \left (1-i \sqrt {3}-2 x^{2/3}\right )}{\sqrt [3]{1+x^2}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {2-\frac {\sqrt [3]{2} \left (1+i \sqrt {3}-2 x^{2/3}\right )}{\sqrt [3]{1+x^2}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {1+\frac {\sqrt [3]{2} \left (1+x^{2/3}\right )}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\left (1-x^{2/3}\right )^2 \left (1+x^{2/3}\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\left (1-i \sqrt {3}-2 x^{2/3}\right ) \left (1-i \sqrt {3}+2 x^{2/3}\right )^2\right )}{24 \sqrt [3]{2} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\left (1+i \sqrt {3}-2 x^{2/3}\right ) \left (1+i \sqrt {3}+2 x^{2/3}\right )^2\right )}{24 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1+x^{2/3}-2^{2/3} \sqrt [3]{1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1-i \sqrt {3}-2 x^{2/3}+2\ 2^{2/3} \sqrt [3]{1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1+i \sqrt {3}-2 x^{2/3}+2\ 2^{2/3} \sqrt [3]{1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{1+i \sqrt {3}} x} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{6 \left (1-i \sqrt {3}\right )^{2/3} \sqrt [3]{x+x^3}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {2 \sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{1+i \sqrt {3}} x}{\left (1-i \sqrt {3}\right )^{2/3}-\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+i \sqrt {3}} x+\left (1+i \sqrt {3}\right )^{2/3} x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{6 \left (1-i \sqrt {3}\right )^{2/3} \sqrt [3]{x+x^3}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+i \sqrt {3}}+\sqrt [3]{1-i \sqrt {3}} x} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{6 \left (1+i \sqrt {3}\right )^{2/3} \sqrt [3]{x+x^3}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {2 \sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{1-i \sqrt {3}} x}{\left (1+i \sqrt {3}\right )^{2/3}-\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+i \sqrt {3}} x+\left (1-i \sqrt {3}\right )^{2/3} x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{6 \left (1+i \sqrt {3}\right )^{2/3} \sqrt [3]{x+x^3}}\\ &=-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {2-\frac {\sqrt [3]{2} \left (1-i \sqrt {3}-2 x^{2/3}\right )}{\sqrt [3]{1+x^2}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {2-\frac {\sqrt [3]{2} \left (1+i \sqrt {3}-2 x^{2/3}\right )}{\sqrt [3]{1+x^2}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {1+\frac {\sqrt [3]{2} \left (1+x^{2/3}\right )}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\left (1-x^{2/3}\right )^2 \left (1+x^{2/3}\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\left (1-i \sqrt {3}-2 x^{2/3}\right ) \left (1-i \sqrt {3}+2 x^{2/3}\right )^2\right )}{24 \sqrt [3]{2} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\left (1+i \sqrt {3}-2 x^{2/3}\right ) \left (1+i \sqrt {3}+2 x^{2/3}\right )^2\right )}{24 \sqrt [3]{2} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\sqrt [3]{1+i \sqrt {3}}+\frac {\sqrt [3]{1-i \sqrt {3}} x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{6 \sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\sqrt [3]{1-i \sqrt {3}}+\frac {\sqrt [3]{1+i \sqrt {3}} x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{6 \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1+x^{2/3}-2^{2/3} \sqrt [3]{1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1-i \sqrt {3}-2 x^{2/3}+2\ 2^{2/3} \sqrt [3]{1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1+i \sqrt {3}-2 x^{2/3}+2\ 2^{2/3} \sqrt [3]{1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-i \sqrt {3}\right )^{2/3}-\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+i \sqrt {3}} x+\left (1+i \sqrt {3}\right )^{2/3} x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 \sqrt [3]{1-i \sqrt {3}} \sqrt [3]{x+x^3}}-\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {-\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+i \sqrt {3}}+2 \left (1-i \sqrt {3}\right )^{2/3} x}{\left (1+i \sqrt {3}\right )^{2/3}-\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+i \sqrt {3}} x+\left (1-i \sqrt {3}\right )^{2/3} x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{12 \sqrt [3]{1-i \sqrt {3}} \left (1+i \sqrt {3}\right )^{2/3} \sqrt [3]{x+x^3}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}\right )^{2/3}-\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+i \sqrt {3}} x+\left (1-i \sqrt {3}\right )^{2/3} x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{x+x^3}}-\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {-\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+i \sqrt {3}}+2 \left (1+i \sqrt {3}\right )^{2/3} x}{\left (1-i \sqrt {3}\right )^{2/3}-\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+i \sqrt {3}} x+\left (1+i \sqrt {3}\right )^{2/3} x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{12 \left (1-i \sqrt {3}\right )^{2/3} \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{x+x^3}}\\ &=-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {2-\frac {\sqrt [3]{2} \left (1-i \sqrt {3}-2 x^{2/3}\right )}{\sqrt [3]{1+x^2}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {2-\frac {\sqrt [3]{2} \left (1+i \sqrt {3}-2 x^{2/3}\right )}{\sqrt [3]{1+x^2}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {1+\frac {\sqrt [3]{2} \left (1+x^{2/3}\right )}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\left (1-x^{2/3}\right )^2 \left (1+x^{2/3}\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\left (1-i \sqrt {3}-2 x^{2/3}\right ) \left (1-i \sqrt {3}+2 x^{2/3}\right )^2\right )}{24 \sqrt [3]{2} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\left (1+i \sqrt {3}-2 x^{2/3}\right ) \left (1+i \sqrt {3}+2 x^{2/3}\right )^2\right )}{24 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\left (1+i \sqrt {3}\right )^{2/3}+\frac {\left (1-i \sqrt {3}\right )^{2/3} x^{4/3}}{\left (1+x^2\right )^{2/3}}-\frac {2^{2/3} x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{12 \sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\left (1-i \sqrt {3}\right )^{2/3}+\frac {\left (1+i \sqrt {3}\right )^{2/3} x^{4/3}}{\left (1+x^2\right )^{2/3}}-\frac {2^{2/3} x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{12 \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\sqrt [3]{1+i \sqrt {3}}+\frac {\sqrt [3]{1-i \sqrt {3}} x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{6 \sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\sqrt [3]{1-i \sqrt {3}}+\frac {\sqrt [3]{1+i \sqrt {3}} x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{6 \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1+x^{2/3}-2^{2/3} \sqrt [3]{1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1-i \sqrt {3}-2 x^{2/3}+2\ 2^{2/3} \sqrt [3]{1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1+i \sqrt {3}-2 x^{2/3}+2\ 2^{2/3} \sqrt [3]{1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{1+i \sqrt {3}} x^{2/3}}{\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+x^2}}\right )}{2\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}} \sqrt [3]{x+x^3}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{1-i \sqrt {3}} x^{2/3}}{\sqrt [3]{1+i \sqrt {3}} \sqrt [3]{1+x^2}}\right )}{2\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{x+x^3}}\\ &=-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {2-\frac {\sqrt [3]{2} \left (1-i \sqrt {3}-2 x^{2/3}\right )}{\sqrt [3]{1+x^2}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {2-\frac {\sqrt [3]{2} \left (1+i \sqrt {3}-2 x^{2/3}\right )}{\sqrt [3]{1+x^2}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {1+\frac {\sqrt [3]{2} \left (1+x^{2/3}\right )}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\left (3 i-\sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {1-\frac {2 x^{2/3}}{\sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{6\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}} \sqrt [3]{x+x^3}}+\frac {\left (3 i+\sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{6\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\left (1-x^{2/3}\right )^2 \left (1+x^{2/3}\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\left (1-i \sqrt {3}-2 x^{2/3}\right ) \left (1-i \sqrt {3}+2 x^{2/3}\right )^2\right )}{24 \sqrt [3]{2} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\left (1+i \sqrt {3}-2 x^{2/3}\right ) \left (1+i \sqrt {3}+2 x^{2/3}\right )^2\right )}{24 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\left (1+i \sqrt {3}\right )^{2/3}+\frac {\left (1-i \sqrt {3}\right )^{2/3} x^{4/3}}{\left (1+x^2\right )^{2/3}}-\frac {2^{2/3} x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{12 \sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\left (1-i \sqrt {3}\right )^{2/3}+\frac {\left (1+i \sqrt {3}\right )^{2/3} x^{4/3}}{\left (1+x^2\right )^{2/3}}-\frac {2^{2/3} x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{12 \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\sqrt [3]{1+i \sqrt {3}}+\frac {\sqrt [3]{1-i \sqrt {3}} x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{6 \sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\sqrt [3]{1-i \sqrt {3}}+\frac {\sqrt [3]{1+i \sqrt {3}} x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{6 \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1+x^{2/3}-2^{2/3} \sqrt [3]{1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1-i \sqrt {3}-2 x^{2/3}+2\ 2^{2/3} \sqrt [3]{1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1+i \sqrt {3}-2 x^{2/3}+2\ 2^{2/3} \sqrt [3]{1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{x+x^3}}\\ \end {align*}
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Mathematica [A] time = 11.35, size = 155, normalized size = 0.95 \begin {gather*} \frac {\sqrt [3]{\frac {1}{x^2}+1} x \left (4 \text {RootSum}\left [\text {$\#$1}^6-\text {$\#$1}^3+1\&,\frac {\log \left (\sqrt [3]{\frac {1}{x^2}+1}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]+2^{2/3} \left (2 \log \left (2-2^{2/3} \sqrt [3]{\frac {1}{x^2}+1}\right )-\log \left (\sqrt [3]{2} \left (\frac {1}{x^2}+1\right )^{2/3}+2^{2/3} \sqrt [3]{\frac {1}{x^2}+1}+2\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{\frac {1}{x^2}+1}+1}{\sqrt {3}}\right )\right )\right )}{24 \sqrt [3]{x^3+x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.50, size = 163, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{x+x^3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{x+x^3}\right )}{6 \sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{x+x^3}+\sqrt [3]{2} \left (x+x^3\right )^{2/3}\right )}{12 \sqrt [3]{2}}+\frac {1}{6} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{x+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.32, size = 941, normalized size = 5.77
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 79.49, size = 5190, normalized size = 31.84
method | result | size |
trager | \(\text {Expression too large to display}\) | \(5190\) |
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*} -\frac {3 \, {\left (9 \, x^{7} + 3 \, x^{5} - x^{3} + 5 \, x\right )}}{80 \, {\left (x^{\frac {19}{3}} - x^{\frac {1}{3}}\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}} - \int \frac {9 \, {\left (9 \, x^{6} + 3 \, x^{4} - x^{2} + 5\right )}}{40 \, {\left (x^{\frac {37}{3}} - 2 \, x^{\frac {19}{3}} + x^{\frac {1}{3}}\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (x^6-1\right )\,{\left (x^3+x\right )}^{1/3}} \,d x \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}{\sqrt [3]{x \left (x^{2} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - 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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