Integrand size = 56, antiderivative size = 24 \[ \int \frac {10+x+(-20-2 x) \log (x)-2 x \log \left (\frac {1}{3} (20+2 x)\right )+(20+2 x) \log ^2\left (\frac {1}{3} (20+2 x)\right )}{450 x^3+45 x^4} \, dx=\frac {\log (x)-\log ^2\left (\frac {1}{3} (20+2 x)\right )}{45 x^2} \]
[Out]
Time = 0.33 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21, number of steps used = 18, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.196, Rules used = {1607, 6874, 2458, 12, 2389, 2379, 2438, 2351, 31, 2445, 2340} \[ \int \frac {10+x+(-20-2 x) \log (x)-2 x \log \left (\frac {1}{3} (20+2 x)\right )+(20+2 x) \log ^2\left (\frac {1}{3} (20+2 x)\right )}{450 x^3+45 x^4} \, dx=\frac {\log (x)}{45 x^2}-\frac {\log ^2\left (\frac {2 x}{3}+\frac {20}{3}\right )}{45 x^2} \]
[In]
[Out]
Rule 12
Rule 31
Rule 1607
Rule 2340
Rule 2351
Rule 2379
Rule 2389
Rule 2438
Rule 2445
Rule 2458
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {10+x+(-20-2 x) \log (x)-2 x \log \left (\frac {1}{3} (20+2 x)\right )+(20+2 x) \log ^2\left (\frac {1}{3} (20+2 x)\right )}{x^3 (450+45 x)} \, dx \\ & = \int \left (\frac {2 \log \left (\frac {20}{3}+\frac {2 x}{3}\right )}{45 (-10-x) x^2}+\frac {2 \log ^2\left (\frac {20}{3}+\frac {2 x}{3}\right )}{45 x^3}+\frac {1-2 \log (x)}{45 x^3}\right ) \, dx \\ & = \frac {1}{45} \int \frac {1-2 \log (x)}{x^3} \, dx+\frac {2}{45} \int \frac {\log \left (\frac {20}{3}+\frac {2 x}{3}\right )}{(-10-x) x^2} \, dx+\frac {2}{45} \int \frac {\log ^2\left (\frac {20}{3}+\frac {2 x}{3}\right )}{x^3} \, dx \\ & = -\frac {\log ^2\left (\frac {20}{3}+\frac {2 x}{3}\right )}{45 x^2}+\frac {\log (x)}{45 x^2}+\frac {4}{135} \int \frac {\log \left (\frac {20}{3}+\frac {2 x}{3}\right )}{\left (\frac {20}{3}+\frac {2 x}{3}\right ) x^2} \, dx+\frac {1}{15} \text {Subst}\left (\int -\frac {2 \log (x)}{3 x \left (-10+\frac {3 x}{2}\right )^2} \, dx,x,\frac {20}{3}+\frac {2 x}{3}\right ) \\ & = -\frac {\log ^2\left (\frac {20}{3}+\frac {2 x}{3}\right )}{45 x^2}+\frac {\log (x)}{45 x^2} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {10+x+(-20-2 x) \log (x)-2 x \log \left (\frac {1}{3} (20+2 x)\right )+(20+2 x) \log ^2\left (\frac {1}{3} (20+2 x)\right )}{450 x^3+45 x^4} \, dx=\frac {1}{45} \left (-\frac {\log ^2\left (\frac {20}{3}+\frac {2 x}{3}\right )}{x^2}+\frac {\log (x)}{x^2}\right ) \]
[In]
[Out]
Time = 0.34 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88
method | result | size |
parallelrisch | \(-\frac {100 \ln \left (\frac {2 x}{3}+\frac {20}{3}\right )^{2}-100 \ln \left (x \right )}{4500 x^{2}}\) | \(21\) |
risch | \(-\frac {\ln \left (\frac {2 x}{3}+\frac {20}{3}\right )^{2}}{45 x^{2}}+\frac {\ln \left (x \right )}{45 x^{2}}\) | \(22\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.75 \[ \int \frac {10+x+(-20-2 x) \log (x)-2 x \log \left (\frac {1}{3} (20+2 x)\right )+(20+2 x) \log ^2\left (\frac {1}{3} (20+2 x)\right )}{450 x^3+45 x^4} \, dx=-\frac {\log \left (\frac {2}{3} \, x + \frac {20}{3}\right )^{2} - \log \left (x\right )}{45 \, x^{2}} \]
[In]
[Out]
Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {10+x+(-20-2 x) \log (x)-2 x \log \left (\frac {1}{3} (20+2 x)\right )+(20+2 x) \log ^2\left (\frac {1}{3} (20+2 x)\right )}{450 x^3+45 x^4} \, dx=\frac {\log {\left (x \right )}}{45 x^{2}} - \frac {\log {\left (\frac {2 x}{3} + \frac {20}{3} \right )}^{2}}{45 x^{2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (18) = 36\).
Time = 0.34 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.38 \[ \int \frac {10+x+(-20-2 x) \log (x)-2 x \log \left (\frac {1}{3} (20+2 x)\right )+(20+2 x) \log ^2\left (\frac {1}{3} (20+2 x)\right )}{450 x^3+45 x^4} \, dx=-\frac {100 \, \log \left (3\right )^{2} - 200 \, \log \left (3\right ) \log \left (2\right ) + 100 \, \log \left (2\right )^{2} - {\left (x^{2} + 200 \, \log \left (3\right ) - 200 \, \log \left (2\right )\right )} \log \left (x + 10\right ) + 100 \, \log \left (x + 10\right )^{2} + {\left (x^{2} - 100\right )} \log \left (x\right ) + 10 \, x - 50}{4500 \, x^{2}} + \frac {x - 5}{450 \, x^{2}} - \frac {1}{4500} \, \log \left (x + 10\right ) + \frac {1}{4500} \, \log \left (x\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (18) = 36\).
Time = 0.28 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.08 \[ \int \frac {10+x+(-20-2 x) \log (x)-2 x \log \left (\frac {1}{3} (20+2 x)\right )+(20+2 x) \log ^2\left (\frac {1}{3} (20+2 x)\right )}{450 x^3+45 x^4} \, dx=\frac {2 \, \log \left (3\right ) \log \left (2 \, x + 20\right )}{45 \, x^{2}} - \frac {\log \left (2 \, x + 20\right )^{2}}{45 \, x^{2}} - \frac {\log \left (3\right )^{2} - \log \left (3\right )}{45 \, x^{2}} + \frac {\log \left (\frac {1}{3} \, x\right )}{45 \, x^{2}} \]
[In]
[Out]
Time = 11.34 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.75 \[ \int \frac {10+x+(-20-2 x) \log (x)-2 x \log \left (\frac {1}{3} (20+2 x)\right )+(20+2 x) \log ^2\left (\frac {1}{3} (20+2 x)\right )}{450 x^3+45 x^4} \, dx=\frac {\ln \left (x\right )-{\ln \left (\frac {2\,x}{3}+\frac {20}{3}\right )}^2}{45\,x^2} \]
[In]
[Out]