Integrand size = 9, antiderivative size = 12 \[ \int \frac {1}{\frac {1}{\sqrt [3]{x}}+x} \, dx=\frac {3}{4} \log \left (1+x^{4/3}\right ) \]
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Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1607, 266} \[ \int \frac {1}{\frac {1}{\sqrt [3]{x}}+x} \, dx=\frac {3}{4} \log \left (x^{4/3}+1\right ) \]
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Rule 266
Rule 1607
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt [3]{x}}{1+x^{4/3}} \, dx \\ & = \frac {3}{4} \log \left (1+x^{4/3}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\frac {1}{\sqrt [3]{x}}+x} \, dx=\frac {3}{4} \log \left (1+x^{4/3}\right ) \]
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Time = 0.25 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(\frac {3 \ln \left (1+x^{\frac {4}{3}}\right )}{4}\) | \(9\) |
default | \(\frac {3 \ln \left (1+x^{\frac {4}{3}}\right )}{4}\) | \(9\) |
meijerg | \(\frac {3 \ln \left (1+x^{\frac {4}{3}}\right )}{4}\) | \(9\) |
trager | \(-\frac {\ln \left (\frac {3 x^{\frac {20}{3}}-x^{8}-6 x^{\frac {16}{3}}-6 x^{\frac {8}{3}}+7 x^{4}+3 x^{\frac {4}{3}}-1}{\left (x^{4}+1\right )^{3}}\right )}{4}\) | \(44\) |
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none
Time = 0.25 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\frac {1}{\sqrt [3]{x}}+x} \, dx=\frac {3}{4} \, \log \left (x^{\frac {4}{3}} + 1\right ) \]
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Time = 0.08 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\frac {1}{\sqrt [3]{x}}+x} \, dx=\frac {3 \log {\left (x^{\frac {4}{3}} + 1 \right )}}{4} \]
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Time = 0.27 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\frac {1}{\sqrt [3]{x}}+x} \, dx=\frac {3}{4} \, \log \left (x^{\frac {4}{3}} + 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (8) = 16\).
Time = 0.32 (sec) , antiderivative size = 32, normalized size of antiderivative = 2.67 \[ \int \frac {1}{\frac {1}{\sqrt [3]{x}}+x} \, dx=\frac {3}{4} \, \log \left (\sqrt {2} x^{\frac {1}{3}} + x^{\frac {2}{3}} + 1\right ) + \frac {3}{4} \, \log \left (-\sqrt {2} x^{\frac {1}{3}} + x^{\frac {2}{3}} + 1\right ) \]
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Time = 0.28 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\frac {1}{\sqrt [3]{x}}+x} \, dx=\frac {3\,\ln \left (x^{4/3}+1\right )}{4} \]
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