Integrand size = 13, antiderivative size = 33 \[ \int \frac {1}{x^4 \left (a^3+x^3\right )} \, dx=-\frac {1}{3 a^3 x^3}-\frac {\log (x)}{a^6}+\frac {\log \left (a^3+x^3\right )}{3 a^6} \]
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Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 46} \[ \int \frac {1}{x^4 \left (a^3+x^3\right )} \, dx=-\frac {\log (x)}{a^6}-\frac {1}{3 a^3 x^3}+\frac {\log \left (a^3+x^3\right )}{3 a^6} \]
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Rule 46
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {1}{x^2 \left (a^3+x\right )} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (\frac {1}{a^3 x^2}-\frac {1}{a^6 x}+\frac {1}{a^6 \left (a^3+x\right )}\right ) \, dx,x,x^3\right ) \\ & = -\frac {1}{3 a^3 x^3}-\frac {\log (x)}{a^6}+\frac {\log \left (a^3+x^3\right )}{3 a^6} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^4 \left (a^3+x^3\right )} \, dx=-\frac {1}{3 a^3 x^3}-\frac {\log (x)}{a^6}+\frac {\log \left (a^3+x^3\right )}{3 a^6} \]
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Time = 0.21 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03
method | result | size |
risch | \(-\frac {1}{3 a^{3} x^{3}}-\frac {\ln \left (x \right )}{a^{6}}+\frac {\ln \left (-a^{3}-x^{3}\right )}{3 a^{6}}\) | \(34\) |
default | \(-\frac {1}{3 a^{3} x^{3}}-\frac {\ln \left (x \right )}{a^{6}}+\frac {\ln \left (a^{2}-a x +x^{2}\right )}{3 a^{6}}+\frac {\ln \left (a +x \right )}{3 a^{6}}\) | \(43\) |
norman | \(-\frac {1}{3 a^{3} x^{3}}-\frac {\ln \left (x \right )}{a^{6}}+\frac {\ln \left (a^{2}-a x +x^{2}\right )}{3 a^{6}}+\frac {\ln \left (a +x \right )}{3 a^{6}}\) | \(43\) |
parallelrisch | \(-\frac {3 x^{3} \ln \left (x \right )-\ln \left (a +x \right ) x^{3}-\ln \left (a^{2}-a x +x^{2}\right ) x^{3}+a^{3}}{3 a^{6} x^{3}}\) | \(46\) |
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Time = 0.23 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^4 \left (a^3+x^3\right )} \, dx=\frac {x^{3} \log \left (a^{3} + x^{3}\right ) - 3 \, x^{3} \log \left (x\right ) - a^{3}}{3 \, a^{6} x^{3}} \]
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Time = 0.13 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^4 \left (a^3+x^3\right )} \, dx=- \frac {1}{3 a^{3} x^{3}} - \frac {\log {\left (x \right )}}{a^{6}} + \frac {\log {\left (a^{3} + x^{3} \right )}}{3 a^{6}} \]
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none
Time = 0.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x^4 \left (a^3+x^3\right )} \, dx=\frac {\log \left (a^{3} + x^{3}\right )}{3 \, a^{6}} - \frac {\log \left (x^{3}\right )}{3 \, a^{6}} - \frac {1}{3 \, a^{3} x^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.21 \[ \int \frac {1}{x^4 \left (a^3+x^3\right )} \, dx=\frac {\log \left ({\left | a^{3} + x^{3} \right |}\right )}{3 \, a^{6}} - \frac {\log \left ({\left | x \right |}\right )}{a^{6}} - \frac {a^{3} - x^{3}}{3 \, a^{6} x^{3}} \]
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Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^4 \left (a^3+x^3\right )} \, dx=\frac {\ln \left (a^3+x^3\right )}{3\,a^6}-\frac {\ln \left (x\right )}{a^6}-\frac {1}{3\,a^3\,x^3} \]
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