Integrand size = 20, antiderivative size = 16 \[ \int \frac {-15 x^2+2 \log (x)-4 \log ^2(x)}{x^5} \, dx=\frac {\frac {15 x^2}{2}+\log ^2(x)}{x^4} \]
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Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {14, 2341, 2342} \[ \int \frac {-15 x^2+2 \log (x)-4 \log ^2(x)}{x^5} \, dx=\frac {\log ^2(x)}{x^4}+\frac {15}{2 x^2} \]
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Rule 14
Rule 2341
Rule 2342
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {15}{x^3}+\frac {2 \log (x)}{x^5}-\frac {4 \log ^2(x)}{x^5}\right ) \, dx \\ & = \frac {15}{2 x^2}+2 \int \frac {\log (x)}{x^5} \, dx-4 \int \frac {\log ^2(x)}{x^5} \, dx \\ & = -\frac {1}{8 x^4}+\frac {15}{2 x^2}-\frac {\log (x)}{2 x^4}+\frac {\log ^2(x)}{x^4}-2 \int \frac {\log (x)}{x^5} \, dx \\ & = \frac {15}{2 x^2}+\frac {\log ^2(x)}{x^4} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {-15 x^2+2 \log (x)-4 \log ^2(x)}{x^5} \, dx=\frac {15}{2 x^2}+\frac {\log ^2(x)}{x^4} \]
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Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {\ln \left (x \right )^{2}}{x^{4}}+\frac {15}{2 x^{2}}\) | \(15\) |
norman | \(\frac {\ln \left (x \right )^{2}+\frac {15 x^{2}}{2}}{x^{4}}\) | \(15\) |
risch | \(\frac {\ln \left (x \right )^{2}}{x^{4}}+\frac {15}{2 x^{2}}\) | \(15\) |
parts | \(\frac {\ln \left (x \right )^{2}}{x^{4}}+\frac {15}{2 x^{2}}\) | \(15\) |
parallelrisch | \(\frac {15 x^{2}+2 \ln \left (x \right )^{2}}{2 x^{4}}\) | \(18\) |
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none
Time = 0.23 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \frac {-15 x^2+2 \log (x)-4 \log ^2(x)}{x^5} \, dx=\frac {15 \, x^{2} + 2 \, \log \left (x\right )^{2}}{2 \, x^{4}} \]
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Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {-15 x^2+2 \log (x)-4 \log ^2(x)}{x^5} \, dx=\frac {15}{2 x^{2}} + \frac {\log {\left (x \right )}^{2}}{x^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (17) = 34\).
Time = 0.20 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.19 \[ \int \frac {-15 x^2+2 \log (x)-4 \log ^2(x)}{x^5} \, dx=\frac {15}{2 \, x^{2}} + \frac {8 \, \log \left (x\right )^{2} + 4 \, \log \left (x\right ) + 1}{8 \, x^{4}} - \frac {\log \left (x\right )}{2 \, x^{4}} - \frac {1}{8 \, x^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {-15 x^2+2 \log (x)-4 \log ^2(x)}{x^5} \, dx=\frac {15}{2 \, x^{2}} + \frac {\log \left (x\right )^{2}}{x^{4}} \]
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Time = 7.75 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {-15 x^2+2 \log (x)-4 \log ^2(x)}{x^5} \, dx=\frac {\frac {15\,x^2}{2}+{\ln \left (x\right )}^2}{x^4} \]
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