Integrand size = 18, antiderivative size = 123 \[ \int \frac {1}{(1+x) \sqrt [3]{3+2 x+x^2}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} \sqrt [3]{3+2 x+x^2}}{\sqrt {3}}\right )}{2 \sqrt [3]{2}}+\frac {\log \left (-2+2^{2/3} \sqrt [3]{3+2 x+x^2}\right )}{2 \sqrt [3]{2}}-\frac {\log \left (2+2^{2/3} \sqrt [3]{3+2 x+x^2}+\sqrt [3]{2} \left (3+2 x+x^2\right )^{2/3}\right )}{4 \sqrt [3]{2}} \]
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Time = 0.05 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.67, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {708, 272, 57, 631, 210, 31} \[ \int \frac {1}{(1+x) \sqrt [3]{3+2 x+x^2}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {2^{2/3} \sqrt [3]{(x+1)^2+2}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{2}}-\frac {\log (x+1)}{2 \sqrt [3]{2}}+\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{(x+1)^2+2}\right )}{4 \sqrt [3]{2}} \]
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Rule 31
Rule 57
Rule 210
Rule 272
Rule 631
Rule 708
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{x \sqrt [3]{2+x^2}} \, dx,x,1+x\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x \sqrt [3]{2+x}} \, dx,x,(1+x)^2\right ) \\ & = -\frac {\log (1+x)}{2 \sqrt [3]{2}}+\frac {3}{4} \text {Subst}\left (\int \frac {1}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{2+(1+x)^2}\right )-\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt [3]{2}-x} \, dx,x,\sqrt [3]{2+(1+x)^2}\right )}{4 \sqrt [3]{2}} \\ & = -\frac {\log (1+x)}{2 \sqrt [3]{2}}+\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{2+(1+x)^2}\right )}{4 \sqrt [3]{2}}-\frac {3 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2^{2/3} \sqrt [3]{2+(1+x)^2}\right )}{2 \sqrt [3]{2}} \\ & = \frac {\sqrt {3} \arctan \left (\frac {1+2^{2/3} \sqrt [3]{2+(1+x)^2}}{\sqrt {3}}\right )}{2 \sqrt [3]{2}}-\frac {\log (1+x)}{2 \sqrt [3]{2}}+\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{2+(1+x)^2}\right )}{4 \sqrt [3]{2}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(1+x) \sqrt [3]{3+2 x+x^2}} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {1+2^{2/3} \sqrt [3]{3+2 x+x^2}}{\sqrt {3}}\right )+2 \log \left (-2+2^{2/3} \sqrt [3]{3+2 x+x^2}\right )-\log \left (2+2^{2/3} \sqrt [3]{3+2 x+x^2}+\sqrt [3]{2} \left (3+2 x+x^2\right )^{2/3}\right )}{4 \sqrt [3]{2}} \]
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Time = 6.52 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.68
method | result | size |
pseudoelliptic | \(\frac {2^{\frac {2}{3}} \left (2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} \left (x^{2}+2 x +3\right )^{\frac {1}{3}}+1\right )}{3}\right )+2 \ln \left (\left (x^{2}+2 x +3\right )^{\frac {1}{3}}-2^{\frac {1}{3}}\right )-\ln \left (\left (x^{2}+2 x +3\right )^{\frac {2}{3}}+2^{\frac {1}{3}} \left (x^{2}+2 x +3\right )^{\frac {1}{3}}+2^{\frac {2}{3}}\right )\right )}{8}\) | \(84\) |
trager | \(\text {Expression too large to display}\) | \(805\) |
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Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(1+x) \sqrt [3]{3+2 x+x^2}} \, dx=\frac {1}{4} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {1}{6}} {\left (2^{\frac {5}{6}} + 2 \, \sqrt {2} {\left (x^{2} + 2 \, x + 3\right )}^{\frac {1}{3}}\right )}\right ) - \frac {1}{8} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (x^{2} + 2 \, x + 3\right )}^{\frac {1}{3}} + {\left (x^{2} + 2 \, x + 3\right )}^{\frac {2}{3}}\right ) + \frac {1}{4} \cdot 2^{\frac {2}{3}} \log \left (-2^{\frac {1}{3}} + {\left (x^{2} + 2 \, x + 3\right )}^{\frac {1}{3}}\right ) \]
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\[ \int \frac {1}{(1+x) \sqrt [3]{3+2 x+x^2}} \, dx=\int \frac {1}{\left (x + 1\right ) \sqrt [3]{x^{2} + 2 x + 3}}\, dx \]
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\[ \int \frac {1}{(1+x) \sqrt [3]{3+2 x+x^2}} \, dx=\int { \frac {1}{{\left (x^{2} + 2 \, x + 3\right )}^{\frac {1}{3}} {\left (x + 1\right )}} \,d x } \]
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\[ \int \frac {1}{(1+x) \sqrt [3]{3+2 x+x^2}} \, dx=\int { \frac {1}{{\left (x^{2} + 2 \, x + 3\right )}^{\frac {1}{3}} {\left (x + 1\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{(1+x) \sqrt [3]{3+2 x+x^2}} \, dx=\int \frac {1}{\left (x+1\right )\,{\left (x^2+2\,x+3\right )}^{1/3}} \,d x \]
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