Integrand size = 22, antiderivative size = 499 \[ \int \frac {1+2 x}{\sqrt [3]{-1+x^2} \left (3+x^2\right )} \, dx=-\frac {\sqrt [6]{-253+1260 i \sqrt {3}} \arctan \left (\frac {3 \sqrt [3]{-1+x^2}}{-2^{2/3} \sqrt {3}-i 2^{2/3} x+\sqrt {3} \sqrt [3]{-1+x^2}}\right )}{6\ 2^{2/3}}-\frac {\sqrt [6]{-253-1260 i \sqrt {3}} \arctan \left (\frac {3 \sqrt [3]{-1+x^2}}{-2^{2/3} \sqrt {3}+i 2^{2/3} x+\sqrt {3} \sqrt [3]{-1+x^2}}\right )}{6\ 2^{2/3}}+\frac {(-1)^{2/3} \sqrt [3]{-54+35 i \sqrt {3}} \log \left (3\ 2^{2/3}-i 2^{2/3} \sqrt {3} x+6 \sqrt [3]{-1+x^2}\right )}{6\ 6^{2/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{-54-35 i \sqrt {3}} \log \left (3\ 2^{2/3}+i 2^{2/3} \sqrt {3} x+6 \sqrt [3]{-1+x^2}\right )}{6\ 6^{2/3}}+\frac {\sqrt [3]{54+35 i \sqrt {3}} \log \left (3 \sqrt [3]{2}+2 i \sqrt [3]{2} \sqrt {3} x-\sqrt [3]{2} x^2-3\ 2^{2/3} \sqrt [3]{-1+x^2}-i 2^{2/3} \sqrt {3} x \sqrt [3]{-1+x^2}+6 \left (-1+x^2\right )^{2/3}\right )}{12\ 6^{2/3}}+\frac {\sqrt [3]{54-35 i \sqrt {3}} \log \left (3 \sqrt [3]{2}-2 i \sqrt [3]{2} \sqrt {3} x-\sqrt [3]{2} x^2-3\ 2^{2/3} \sqrt [3]{-1+x^2}+i 2^{2/3} \sqrt {3} x \sqrt [3]{-1+x^2}+6 \left (-1+x^2\right )^{2/3}\right )}{12\ 6^{2/3}} \]
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Time = 0.06 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.43, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1024, 402, 455, 58, 631, 210, 31} \[ \int \frac {1+2 x}{\sqrt [3]{-1+x^2} \left (3+x^2\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {1-\sqrt [3]{2} \sqrt [3]{x^2-1}}{\sqrt {3}}\right )}{2^{2/3}}-\frac {(-1)^{2/3} \arctan \left (\frac {\sqrt {3} \left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{x^2-1}+1\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {(-1)^{2/3} \arctan \left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {1}{2} \left (-\frac {1}{2}\right )^{2/3} \text {arctanh}\left (\frac {\sqrt [3]{-1} x}{\sqrt [3]{2} \sqrt [3]{x^2-1}+\sqrt [3]{-1}}\right )+\frac {1}{6} \left (-\frac {1}{2}\right )^{2/3} \text {arctanh}(x)+\frac {\log \left (x^2+3\right )}{2\ 2^{2/3}}-\frac {3 \log \left (\sqrt [3]{x^2-1}+2^{2/3}\right )}{2\ 2^{2/3}} \]
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Rule 31
Rule 58
Rule 210
Rule 402
Rule 455
Rule 631
Rule 1024
Rubi steps \begin{align*} \text {integral}& = 2 \int \frac {x}{\sqrt [3]{-1+x^2} \left (3+x^2\right )} \, dx+\int \frac {1}{\sqrt [3]{-1+x^2} \left (3+x^2\right )} \, dx \\ & = -\frac {(-1)^{2/3} \arctan \left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {(-1)^{2/3} \arctan \left (\frac {\sqrt {3} \left (1+(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{-1+x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {1}{6} \left (-\frac {1}{2}\right )^{2/3} \text {arctanh}(x)-\frac {1}{2} \left (-\frac {1}{2}\right )^{2/3} \text {arctanh}\left (\frac {\sqrt [3]{-1} x}{\sqrt [3]{-1}+\sqrt [3]{2} \sqrt [3]{-1+x^2}}\right )+\text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} (3+x)} \, dx,x,x^2\right ) \\ & = -\frac {(-1)^{2/3} \arctan \left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {(-1)^{2/3} \arctan \left (\frac {\sqrt {3} \left (1+(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{-1+x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {1}{6} \left (-\frac {1}{2}\right )^{2/3} \text {arctanh}(x)-\frac {1}{2} \left (-\frac {1}{2}\right )^{2/3} \text {arctanh}\left (\frac {\sqrt [3]{-1} x}{\sqrt [3]{-1}+\sqrt [3]{2} \sqrt [3]{-1+x^2}}\right )+\frac {\log \left (3+x^2\right )}{2\ 2^{2/3}}+\frac {3}{2} \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 {3 \text {Subst}\left (\int \frac {1}{2^{2/3}+x} \, dx,x,\sqrt [3]{-1+x^2}\right )}{2\ 2^{2/3}} \\ & = -\frac {(-1)^{2/3} \arctan \left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {(-1)^{2/3} \arctan \left (\frac {\sqrt {3} \left (1+(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{-1+x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {1}{6} \left (-\frac {1}{2}\right )^{2/3} \text {arctanh}(x)-\frac {1}{2} \left (-\frac {1}{2}\right )^{2/3} \text {arctanh}\left (\frac {\sqrt [3]{-1} x}{\sqrt [3]{-1}+\sqrt [3]{2} \sqrt [3]{-1+x^2}}\right )+\frac {\log \left (3+x^2\right )}{2\ 2^{2/3}}-\frac {3 \log \left (2^{2/3}+\sqrt [3]{-1+x^2}\right )}{2\ 2^{2/3}}+\frac {3 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\sqrt [3]{2} \sqrt [3]{-1+x^2}\right )}{2^{2/3}} \\ & = -\frac {(-1)^{2/3} \arctan \left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {\sqrt {3} \arctan \left (\frac {1-\sqrt [3]{2} \sqrt [3]{-1+x^2}}{\sqrt {3}}\right )}{2^{2/3}}-\frac {(-1)^{2/3} \arctan \left (\frac {\sqrt {3} \left (1+(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{-1+x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {1}{6} \left (-\frac {1}{2}\right )^{2/3} \text {arctanh}(x)-\frac {1}{2} \left (-\frac {1}{2}\right )^{2/3} \text {arctanh}\left (\frac {\sqrt [3]{-1} x}{\sqrt [3]{-1}+\sqrt [3]{2} \sqrt [3]{-1+x^2}}\right )+\frac {\log \left (3+x^2\right )}{2\ 2^{2/3}}-\frac {3 \log \left (2^{2/3}+\sqrt [3]{-1+x^2}\right )}{2\ 2^{2/3}} \\ \end{align*}
Time = 6.87 (sec) , antiderivative size = 444, normalized size of antiderivative = 0.89 \[ \int \frac {1+2 x}{\sqrt [3]{-1+x^2} \left (3+x^2\right )} \, dx=\frac {-6 \sqrt [6]{-253+1260 i \sqrt {3}} \arctan \left (\frac {3 \sqrt [3]{-1+x^2}}{-i 2^{2/3} x+\sqrt {3} \left (-2^{2/3}+\sqrt [3]{-1+x^2}\right )}\right )-6 \sqrt [6]{-253-1260 i \sqrt {3}} \arctan \left (\frac {3 \sqrt [3]{-1+x^2}}{i 2^{2/3} x+\sqrt {3} \left (-2^{2/3}+\sqrt [3]{-1+x^2}\right )}\right )+\sqrt [3]{3} \left (2 (-1)^{2/3} \sqrt [3]{-54+35 i \sqrt {3}} \log \left (3\ 2^{2/3}-i 2^{2/3} \sqrt {3} x+6 \sqrt [3]{-1+x^2}\right )-2 \sqrt [3]{54+35 i \sqrt {3}} \log \left (3\ 2^{2/3}+i 2^{2/3} \sqrt {3} x+6 \sqrt [3]{-1+x^2}\right )+\sqrt [3]{54+35 i \sqrt {3}} \log \left (3 \sqrt [3]{2}-\sqrt [3]{2} x^2-3\ 2^{2/3} \sqrt [3]{-1+x^2}+6 \left (-1+x^2\right )^{2/3}-i \sqrt [3]{2} \sqrt {3} x \left (-2+\sqrt [3]{2} \sqrt [3]{-1+x^2}\right )\right )+\sqrt [3]{54-35 i \sqrt {3}} \log \left (3 \sqrt [3]{2}-\sqrt [3]{2} x^2-3\ 2^{2/3} \sqrt [3]{-1+x^2}+6 \left (-1+x^2\right )^{2/3}+i \sqrt [3]{2} \sqrt {3} x \left (-2+\sqrt [3]{2} \sqrt [3]{-1+x^2}\right )\right )\right )}{36\ 2^{2/3}} \]
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Timed out.
hanged
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Exception generated. \[ \int \frac {1+2 x}{\sqrt [3]{-1+x^2} \left (3+x^2\right )} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1+2 x}{\sqrt [3]{-1+x^2} \left (3+x^2\right )} \, dx=\int \frac {2 x + 1}{\sqrt [3]{\left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )}\, dx \]
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\[ \int \frac {1+2 x}{\sqrt [3]{-1+x^2} \left (3+x^2\right )} \, dx=\int { \frac {2 \, x + 1}{{\left (x^{2} + 3\right )} {\left (x^{2} - 1\right )}^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {1+2 x}{\sqrt [3]{-1+x^2} \left (3+x^2\right )} \, dx=\int { \frac {2 \, x + 1}{{\left (x^{2} + 3\right )} {\left (x^{2} - 1\right )}^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {1+2 x}{\sqrt [3]{-1+x^2} \left (3+x^2\right )} \, dx=\int \frac {2\,x+1}{{\left (x^2-1\right )}^{1/3}\,\left (x^2+3\right )} \,d x \]
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