Optimal. Leaf size=459 \[ \frac {1}{3} \text {RootSum}\left [\text {$\#$1}^6+\text {$\#$1}^3+7\& ,\frac {\text {$\#$1}^3 \log \left (x^3+1\right )-\text {$\#$1}^3 \log \left (-\text {$\#$1} x^3-\text {$\#$1}+\sqrt [3]{-3 x^{10}+x^9-9 x^7+3 x^6-9 x^4+3 x^3-3 x+1}\right )-5 \log \left (-\text {$\#$1} x^3-\text {$\#$1}+\sqrt [3]{-3 x^{10}+x^9-9 x^7+3 x^6-9 x^4+3 x^3-3 x+1}\right )+5 \log \left (x^3+1\right )}{2 \text {$\#$1}^4+\text {$\#$1}}\& \right ]-\frac {\log \left (x^3+1\right )}{3\ 2^{2/3}}+\frac {\log \left (x^6+2 x^3+1\right )}{6\ 2^{2/3}}+\frac {\log \left (-2 x^3+\sqrt [3]{2} \sqrt [3]{-3 x^{10}+x^9-9 x^7+3 x^6-9 x^4+3 x^3-3 x+1}-2\right )}{3\ 2^{2/3}}-\frac {\log \left (4 x^6+8 x^3+2^{2/3} \left (-3 x^{10}+x^9-9 x^7+3 x^6-9 x^4+3 x^3-3 x+1\right )^{2/3}+\left (2 \sqrt [3]{2} x^3+2 \sqrt [3]{2}\right ) \sqrt [3]{-3 x^{10}+x^9-9 x^7+3 x^6-9 x^4+3 x^3-3 x+1}+4\right )}{6\ 2^{2/3}}-\frac {\tan ^{-1}\left (\frac {\left (\sqrt {3} x+\sqrt {3}\right ) \left (x^2-x+1\right )}{x^3+\sqrt [3]{2} \sqrt [3]{-3 x^{10}+x^9-9 x^7+3 x^6-9 x^4+3 x^3-3 x+1}+1}\right )}{2^{2/3} \sqrt {3}} \]
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Rubi [C] time = 1.05, antiderivative size = 1245, normalized size of antiderivative = 2.71, number of steps used = 31, number of rules used = 11, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.297, Rules used = {6688, 6719, 2074, 56, 617, 204, 31, 843, 711, 50, 55} \begin {gather*} -\frac {\sqrt [3]{3 x-1} \left (x^3+1\right ) \tan ^{-1}\left (\frac {1-\sqrt [3]{2} \sqrt [3]{3 x-1}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}-\frac {i \left (1-3 i \sqrt {3}\right )^{2/3} \sqrt [3]{3 x-1} \left (x^3+1\right ) \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{3 x-1}}{\sqrt [3]{1-3 i \sqrt {3}}}+1}{\sqrt {3}}\right )}{9\ 2^{2/3} \sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}+\frac {5 i \sqrt [3]{2} \sqrt [3]{3 x-1} \left (x^3+1\right ) \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{3 x-1}}{\sqrt [3]{1-3 i \sqrt {3}}}+1}{\sqrt {3}}\right )}{9 \sqrt [3]{1-3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}+\frac {i \left (1+3 i \sqrt {3}\right )^{2/3} \sqrt [3]{3 x-1} \left (x^3+1\right ) \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{3 x-1}}{\sqrt [3]{1+3 i \sqrt {3}}}+1}{\sqrt {3}}\right )}{9\ 2^{2/3} \sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}-\frac {5 i \sqrt [3]{2} \sqrt [3]{3 x-1} \left (x^3+1\right ) \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{3 x-1}}{\sqrt [3]{1+3 i \sqrt {3}}}+1}{\sqrt {3}}\right )}{9 \sqrt [3]{1+3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}+\frac {i \left (1-3 i \sqrt {3}\right )^{2/3} \sqrt [3]{3 x-1} \left (x^3+1\right ) \log \left (-2 x-i \sqrt {3}+1\right )}{18\ 2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}-\frac {5 i \sqrt [3]{3 x-1} \left (x^3+1\right ) \log \left (-2 x-i \sqrt {3}+1\right )}{9\ 2^{2/3} \sqrt {3} \sqrt [3]{1-3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}-\frac {i \left (1+3 i \sqrt {3}\right )^{2/3} \sqrt [3]{3 x-1} \left (x^3+1\right ) \log \left (-2 x+i \sqrt {3}+1\right )}{18\ 2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}+\frac {5 i \sqrt [3]{3 x-1} \left (x^3+1\right ) \log \left (-2 x+i \sqrt {3}+1\right )}{9\ 2^{2/3} \sqrt {3} \sqrt [3]{1+3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}+\frac {\sqrt [3]{3 x-1} \left (x^3+1\right ) \log (x+1)}{6\ 2^{2/3} \sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}-\frac {\sqrt [3]{3 x-1} \left (x^3+1\right ) \log \left (\sqrt [3]{3 x-1}+2^{2/3}\right )}{2\ 2^{2/3} \sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}-\frac {i \left (1-3 i \sqrt {3}\right )^{2/3} \sqrt [3]{3 x-1} \left (x^3+1\right ) \log \left (\sqrt [3]{1-3 i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{3 x-1}\right )}{6\ 2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}+\frac {5 i \sqrt [3]{3 x-1} \left (x^3+1\right ) \log \left (\sqrt [3]{1-3 i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{3 x-1}\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1-3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}+\frac {i \left (1+3 i \sqrt {3}\right )^{2/3} \sqrt [3]{3 x-1} \left (x^3+1\right ) \log \left (\sqrt [3]{1+3 i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{3 x-1}\right )}{6\ 2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}-\frac {5 i \sqrt [3]{3 x-1} \left (x^3+1\right ) \log \left (\sqrt [3]{1+3 i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{3 x-1}\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1+3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (x^3+1\right )^3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 50
Rule 55
Rule 56
Rule 204
Rule 617
Rule 711
Rule 843
Rule 2074
Rule 6688
Rule 6719
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}} \, dx &=\int \frac {1}{\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}} \, dx\\ &=\frac {\left (\sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \int \frac {1}{\sqrt [3]{-1+3 x} \left (1+x^3\right )} \, dx}{\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}\\ &=\frac {\left (\sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \int \left (\frac {1}{3 (1+x) \sqrt [3]{-1+3 x}}+\frac {2-x}{3 \sqrt [3]{-1+3 x} \left (1-x+x^2\right )}\right ) \, dx}{\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}\\ &=\frac {\left (\sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \int \frac {1}{(1+x) \sqrt [3]{-1+3 x}} \, dx}{3 \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}+\frac {\left (\sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \int \frac {2-x}{\sqrt [3]{-1+3 x} \left (1-x+x^2\right )} \, dx}{3 \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}\\ &=\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \log (1+x)}{6\ 2^{2/3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {\left (\sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \int \frac {(-1+3 x)^{2/3}}{1-x+x^2} \, dx}{9 \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}+\frac {\left (\sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2 \sqrt [3]{2}-2^{2/3} x+x^2} \, dx,x,\sqrt [3]{-1+3 x}\right )}{2 \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}+\frac {\left (5 \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \int \frac {1}{\sqrt [3]{-1+3 x} \left (1-x+x^2\right )} \, dx}{9 \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}-\frac {\left (\sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2^{2/3}+x} \, dx,x,\sqrt [3]{-1+3 x}\right )}{2\ 2^{2/3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}\\ &=\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \log (1+x)}{6\ 2^{2/3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (2^{2/3}+\sqrt [3]{-1+3 x}\right )}{2\ 2^{2/3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {\left (\sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \int \left (\frac {2 i (-1+3 x)^{2/3}}{\sqrt {3} \left (1+i \sqrt {3}-2 x\right )}+\frac {2 i (-1+3 x)^{2/3}}{\sqrt {3} \left (-1+i \sqrt {3}+2 x\right )}\right ) \, dx}{9 \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}+\frac {\left (5 \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \int \left (\frac {2 i}{\sqrt {3} \left (1+i \sqrt {3}-2 x\right ) \sqrt [3]{-1+3 x}}+\frac {2 i}{\sqrt {3} \left (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{-1+3 x}}\right ) \, dx}{9 \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}+\frac {\left (\sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\sqrt [3]{-2+6 x}\right )}{2^{2/3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}\\ &=-\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \tan ^{-1}\left (\frac {1-\sqrt [3]{2} \sqrt [3]{-1+3 x}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \log (1+x)}{6\ 2^{2/3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (2^{2/3}+\sqrt [3]{-1+3 x}\right )}{2\ 2^{2/3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {\left (2 i \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \int \frac {(-1+3 x)^{2/3}}{1+i \sqrt {3}-2 x} \, dx}{9 \sqrt {3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}-\frac {\left (2 i \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \int \frac {(-1+3 x)^{2/3}}{-1+i \sqrt {3}+2 x} \, dx}{9 \sqrt {3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}+\frac {\left (10 i \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \int \frac {1}{\left (1+i \sqrt {3}-2 x\right ) \sqrt [3]{-1+3 x}} \, dx}{9 \sqrt {3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}+\frac {\left (10 i \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{-1+3 x}} \, dx}{9 \sqrt {3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}\\ &=-\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \tan ^{-1}\left (\frac {1-\sqrt [3]{2} \sqrt [3]{-1+3 x}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {5 i \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (1-i \sqrt {3}-2 x\right )}{9\ 2^{2/3} \sqrt {3} \sqrt [3]{1-3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {5 i \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (1+i \sqrt {3}-2 x\right )}{9\ 2^{2/3} \sqrt {3} \sqrt [3]{1+3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \log (1+x)}{6\ 2^{2/3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (2^{2/3}+\sqrt [3]{-1+3 x}\right )}{2\ 2^{2/3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {\left (5 i \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{2} \left (1-3 i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1-3 i \sqrt {3}\right )} x+x^2} \, dx,x,\sqrt [3]{-1+3 x}\right )}{6 \sqrt {3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}-\frac {\left (5 i \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{2} \left (1+3 i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1+3 i \sqrt {3}\right )} x+x^2} \, dx,x,\sqrt [3]{-1+3 x}\right )}{6 \sqrt {3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}-\frac {\left (5 i \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{\frac {1}{2} \left (1-3 i \sqrt {3}\right )}-x} \, dx,x,\sqrt [3]{-1+3 x}\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1-3 i \sqrt {3}} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}-\frac {\left (i \left (1-3 i \sqrt {3}\right ) \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{-1+3 x}} \, dx}{9 \sqrt {3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}+\frac {\left (5 i \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{\frac {1}{2} \left (1+3 i \sqrt {3}\right )}-x} \, dx,x,\sqrt [3]{-1+3 x}\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1+3 i \sqrt {3}} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}-\frac {\left (i \left (1+3 i \sqrt {3}\right ) \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \int \frac {1}{\left (1+i \sqrt {3}-2 x\right ) \sqrt [3]{-1+3 x}} \, dx}{9 \sqrt {3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}\\ &=-\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \tan ^{-1}\left (\frac {1-\sqrt [3]{2} \sqrt [3]{-1+3 x}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {5 i \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (1-i \sqrt {3}-2 x\right )}{9\ 2^{2/3} \sqrt {3} \sqrt [3]{1-3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {i \left (1-3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (1-i \sqrt {3}-2 x\right )}{18\ 2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {5 i \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (1+i \sqrt {3}-2 x\right )}{9\ 2^{2/3} \sqrt {3} \sqrt [3]{1+3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {i \left (1+3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (1+i \sqrt {3}-2 x\right )}{18\ 2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \log (1+x)}{6\ 2^{2/3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (2^{2/3}+\sqrt [3]{-1+3 x}\right )}{2\ 2^{2/3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {5 i \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (\sqrt [3]{1-3 i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{-1+3 x}\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1-3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {5 i \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (\sqrt [3]{1+3 i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{-1+3 x}\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1+3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {\left (5 i \sqrt [3]{2} \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{-2+6 x}}{\sqrt [3]{1-3 i \sqrt {3}}}\right )}{3 \sqrt {3} \sqrt [3]{1-3 i \sqrt {3}} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}+\frac {\left (i \left (1-3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{\frac {1}{2} \left (1-3 i \sqrt {3}\right )}-x} \, dx,x,\sqrt [3]{-1+3 x}\right )}{6\ 2^{2/3} \sqrt {3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}-\frac {\left (i \left (1-3 i \sqrt {3}\right ) \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{2} \left (1-3 i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1-3 i \sqrt {3}\right )} x+x^2} \, dx,x,\sqrt [3]{-1+3 x}\right )}{12 \sqrt {3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}+\frac {\left (5 i \sqrt [3]{2} \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{-2+6 x}}{\sqrt [3]{1+3 i \sqrt {3}}}\right )}{3 \sqrt {3} \sqrt [3]{1+3 i \sqrt {3}} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}-\frac {\left (i \left (1+3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{\frac {1}{2} \left (1+3 i \sqrt {3}\right )}-x} \, dx,x,\sqrt [3]{-1+3 x}\right )}{6\ 2^{2/3} \sqrt {3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}+\frac {\left (i \left (1+3 i \sqrt {3}\right ) \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{2} \left (1+3 i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1+3 i \sqrt {3}\right )} x+x^2} \, dx,x,\sqrt [3]{-1+3 x}\right )}{12 \sqrt {3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}\\ &=-\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \tan ^{-1}\left (\frac {1-\sqrt [3]{2} \sqrt [3]{-1+3 x}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {5 i \sqrt [3]{2} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{-1+3 x}}{\sqrt [3]{1-3 i \sqrt {3}}}}{\sqrt {3}}\right )}{9 \sqrt [3]{1-3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {5 i \sqrt [3]{2} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{-1+3 x}}{\sqrt [3]{1+3 i \sqrt {3}}}}{\sqrt {3}}\right )}{9 \sqrt [3]{1+3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {5 i \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (1-i \sqrt {3}-2 x\right )}{9\ 2^{2/3} \sqrt {3} \sqrt [3]{1-3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {i \left (1-3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (1-i \sqrt {3}-2 x\right )}{18\ 2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {5 i \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (1+i \sqrt {3}-2 x\right )}{9\ 2^{2/3} \sqrt {3} \sqrt [3]{1+3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {i \left (1+3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (1+i \sqrt {3}-2 x\right )}{18\ 2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \log (1+x)}{6\ 2^{2/3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (2^{2/3}+\sqrt [3]{-1+3 x}\right )}{2\ 2^{2/3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {5 i \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (\sqrt [3]{1-3 i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{-1+3 x}\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1-3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {i \left (1-3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (\sqrt [3]{1-3 i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{-1+3 x}\right )}{6\ 2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {5 i \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (\sqrt [3]{1+3 i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{-1+3 x}\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1+3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {i \left (1+3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (\sqrt [3]{1+3 i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{-1+3 x}\right )}{6\ 2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {\left (i \left (1-3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{-2+6 x}}{\sqrt [3]{1-3 i \sqrt {3}}}\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}-\frac {\left (i \left (1+3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{-2+6 x}}{\sqrt [3]{1+3 i \sqrt {3}}}\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}\\ &=-\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \tan ^{-1}\left (\frac {1-\sqrt [3]{2} \sqrt [3]{-1+3 x}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {5 i \sqrt [3]{2} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{-1+3 x}}{\sqrt [3]{1-3 i \sqrt {3}}}}{\sqrt {3}}\right )}{9 \sqrt [3]{1-3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {i \left (1-3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{-1+3 x}}{\sqrt [3]{1-3 i \sqrt {3}}}}{\sqrt {3}}\right )}{9\ 2^{2/3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {5 i \sqrt [3]{2} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{-1+3 x}}{\sqrt [3]{1+3 i \sqrt {3}}}}{\sqrt {3}}\right )}{9 \sqrt [3]{1+3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {i \left (1+3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{-1+3 x}}{\sqrt [3]{1+3 i \sqrt {3}}}}{\sqrt {3}}\right )}{9\ 2^{2/3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {5 i \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (1-i \sqrt {3}-2 x\right )}{9\ 2^{2/3} \sqrt {3} \sqrt [3]{1-3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {i \left (1-3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (1-i \sqrt {3}-2 x\right )}{18\ 2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {5 i \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (1+i \sqrt {3}-2 x\right )}{9\ 2^{2/3} \sqrt {3} \sqrt [3]{1+3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {i \left (1+3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (1+i \sqrt {3}-2 x\right )}{18\ 2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \log (1+x)}{6\ 2^{2/3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (2^{2/3}+\sqrt [3]{-1+3 x}\right )}{2\ 2^{2/3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {5 i \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (\sqrt [3]{1-3 i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{-1+3 x}\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1-3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {i \left (1-3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (\sqrt [3]{1-3 i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{-1+3 x}\right )}{6\ 2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {5 i \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (\sqrt [3]{1+3 i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{-1+3 x}\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1+3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {i \left (1+3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (\sqrt [3]{1+3 i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{-1+3 x}\right )}{6\ 2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}\\ \end {align*}
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Mathematica [A] time = 10.13, size = 175, normalized size = 0.38 \begin {gather*} \frac {\sqrt [3]{3 x-1} \left (x^3+1\right ) \left (\sqrt [3]{2} \left (-2 \log \left (\sqrt [3]{6 x-2}+2\right )+\log \left ((6 x-2)^{2/3}-2 \sqrt [3]{6 x-2}+4\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt [3]{6 x-2}-1}{\sqrt {3}}\right )\right )-4 \text {RootSum}\left [\text {$\#$1}^6-\text {$\#$1}^3+7\&,\frac {\text {$\#$1}^3 \log \left (\sqrt [3]{3 x-1}-\text {$\#$1}\right )-5 \log \left (\sqrt [3]{3 x-1}-\text {$\#$1}\right )}{2 \text {$\#$1}^4-\text {$\#$1}}\&\right ]\right )}{12 \sqrt [3]{-\left ((3 x-1) \left (x^3+1\right )^3\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.65, size = 459, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\left (\sqrt {3}+\sqrt {3} x\right ) \left (1-x+x^2\right )}{1+x^3+\sqrt [3]{2} \sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log \left (1+x^3\right )}{3\ 2^{2/3}}+\frac {\log \left (1+2 x^3+x^6\right )}{6\ 2^{2/3}}+\frac {\log \left (-2-2 x^3+\sqrt [3]{2} \sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}\right )}{3\ 2^{2/3}}-\frac {\log \left (4+8 x^3+4 x^6+\left (2 \sqrt [3]{2}+2 \sqrt [3]{2} x^3\right ) \sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}+2^{2/3} \left (1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}\right )^{2/3}\right )}{6\ 2^{2/3}}+\frac {1}{3} \text {RootSum}\left [7+\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {5 \log \left (1+x^3\right )-5 \log \left (\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}-\text {$\#$1}-x^3 \text {$\#$1}\right )+\log \left (1+x^3\right ) \text {$\#$1}^3-\log \left (\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}-\text {$\#$1}-x^3 \text {$\#$1}\right ) \text {$\#$1}^3}{\text {$\#$1}+2 \text {$\#$1}^4}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.98, size = 3411, normalized size = 7.43
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (-3 \, x^{10} + x^{9} - 9 \, x^{7} + 3 \, x^{6} - 9 \, x^{4} + 3 \, x^{3} - 3 \, x + 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 12.28, size = 17220, normalized size = 37.52
method | result | size |
trager | \(\text {Expression too large to display}\) | \(17220\) |
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*} \int \frac {1}{{\left (-3 \, x^{10} + x^{9} - 9 \, x^{7} + 3 \, x^{6} - 9 \, x^{4} + 3 \, x^{3} - 3 \, x + 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 = -0.00 \begin {gather*} \int \frac {1}{{\left (-3\,x^{10}+x^9-9\,x^7+3\,x^6-9\,x^4+3\,x^3-3\,x+1\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]{- 3 x^{10} + x^{9} - 9 x^{7} + 3 x^{6} - 9 x^{4} + 3 x^{3} - 3 x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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