Integrand size = 17, antiderivative size = 20 \[ \int \frac {1}{\left (1+x^2\right ) \sqrt [3]{x+x^3}} \, dx=\frac {3 \left (x+x^3\right )^{2/3}}{2 \left (1+x^2\right )} \]
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Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2081, 270} \[ \int \frac {1}{\left (1+x^2\right ) \sqrt [3]{x+x^3}} \, dx=\frac {3 x}{2 \sqrt [3]{x^3+x}} \]
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Rule 270
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \int \frac {1}{\sqrt [3]{x} \left (1+x^2\right )^{4/3}} \, dx}{\sqrt [3]{x+x^3}} \\ & = \frac {3 x}{2 \sqrt [3]{x+x^3}} \\ \end{align*}
Time = 0.61 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\left (1+x^2\right ) \sqrt [3]{x+x^3}} \, dx=\frac {3 x}{2 \sqrt [3]{x+x^3}} \]
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Time = 0.92 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.55
method | result | size |
gosper | \(\frac {3 x}{2 \left (x^{3}+x \right )^{\frac {1}{3}}}\) | \(11\) |
meijerg | \(\frac {3 x^{\frac {2}{3}}}{2 \left (x^{2}+1\right )^{\frac {1}{3}}}\) | \(13\) |
risch | \(\frac {3 x}{2 {\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{3}}}\) | \(13\) |
pseudoelliptic | \(\frac {3 x}{2 {\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{3}}}\) | \(13\) |
trager | \(\frac {3 \left (x^{3}+x \right )^{\frac {2}{3}}}{2 \left (x^{2}+1\right )}\) | \(17\) |
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Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (1+x^2\right ) \sqrt [3]{x+x^3}} \, dx=\frac {3 \, {\left (x^{3} + x\right )}^{\frac {2}{3}}}{2 \, {\left (x^{2} + 1\right )}} \]
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\[ \int \frac {1}{\left (1+x^2\right ) \sqrt [3]{x+x^3}} \, dx=\int \frac {1}{\sqrt [3]{x \left (x^{2} + 1\right )} \left (x^{2} + 1\right )}\, dx \]
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\[ \int \frac {1}{\left (1+x^2\right ) \sqrt [3]{x+x^3}} \, dx=\int { \frac {1}{{\left (x^{3} + x\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )}} \,d x } \]
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Time = 0.29 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.45 \[ \int \frac {1}{\left (1+x^2\right ) \sqrt [3]{x+x^3}} \, dx=\frac {3}{2 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}}} \]
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Time = 5.06 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (1+x^2\right ) \sqrt [3]{x+x^3}} \, dx=\frac {3\,{\left (x^3+x\right )}^{2/3}}{2\,\left (x^2+1\right )} \]
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