Integrand size = 90, antiderivative size = 23 \[ \int \frac {-4 \log ^2(5) \log \left (x^2\right )+2 \log ^2(5) \log ^2\left (x^2\right )+(-34+3 x) \log ^4\left (x^2\right )}{x^3 \log ^4(5)+\left (-34 x^3+2 x^4\right ) \log ^2(5) \log ^2\left (x^2\right )+\left (289 x^3-34 x^4+x^5\right ) \log ^4\left (x^2\right )} \, dx=\frac {1}{x^2 \left (17-x-\frac {\log ^2(5)}{\log ^2\left (x^2\right )}\right )} \] Output:
1/x^2/(17-ln(5)^2/ln(x^2)^2-x)
Time = 0.35 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \frac {-4 \log ^2(5) \log \left (x^2\right )+2 \log ^2(5) \log ^2\left (x^2\right )+(-34+3 x) \log ^4\left (x^2\right )}{x^3 \log ^4(5)+\left (-34 x^3+2 x^4\right ) \log ^2(5) \log ^2\left (x^2\right )+\left (289 x^3-34 x^4+x^5\right ) \log ^4\left (x^2\right )} \, dx=-\frac {\log ^2\left (x^2\right )}{x^2 \left (\log ^2(5)+(-17+x) \log ^2\left (x^2\right )\right )} \] Input:
Integrate[(-4*Log[5]^2*Log[x^2] + 2*Log[5]^2*Log[x^2]^2 + (-34 + 3*x)*Log[ x^2]^4)/(x^3*Log[5]^4 + (-34*x^3 + 2*x^4)*Log[5]^2*Log[x^2]^2 + (289*x^3 - 34*x^4 + x^5)*Log[x^2]^4),x]
Output:
-(Log[x^2]^2/(x^2*(Log[5]^2 + (-17 + x)*Log[x^2]^2)))
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(3 x-34) \log ^4\left (x^2\right )+2 \log ^2(5) \log ^2\left (x^2\right )-4 \log ^2(5) \log \left (x^2\right )}{x^3 \log ^4(5)+\left (2 x^4-34 x^3\right ) \log ^2(5) \log ^2\left (x^2\right )+\left (x^5-34 x^4+289 x^3\right ) \log ^4\left (x^2\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {(3 x-34) \log ^4\left (x^2\right )+2 \log ^2(5) \log ^2\left (x^2\right )-4 \log ^2(5) \log \left (x^2\right )}{x^3 \left (x \log ^2\left (x^2\right )-17 \log ^2\left (x^2\right )+\log ^2(5)\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {3 x-34}{(x-17)^2 x^3}-\frac {2 (2 x-17) \log ^2(5)}{(x-17)^2 x^3 \left (x \log ^2\left (x^2\right )-17 \log ^2\left (x^2\right )+\log ^2(5)\right )}-\frac {\log ^2(5) \left (4 x^2 \log \left (x^2\right )-136 x \log \left (x^2\right )+1156 \log \left (x^2\right )-x \log ^2(5)\right )}{(x-17)^2 x^3 \left (x \log ^2\left (x^2\right )-17 \log ^2\left (x^2\right )+\log ^2(5)\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2}{289} \log ^2(5) \int \frac {1}{(x-17)^2 \left (x \log ^2\left (x^2\right )-17 \log ^2\left (x^2\right )+\log ^2(5)\right )}dx+\frac {2 \log ^2(5) \int \frac {1}{(x-17) \left (x \log ^2\left (x^2\right )-17 \log ^2\left (x^2\right )+\log ^2(5)\right )}dx}{4913}-\frac {2 \log ^2(5) \int \frac {1}{x \left (x \log ^2\left (x^2\right )-17 \log ^2\left (x^2\right )+\log ^2(5)\right )}dx}{4913}+\frac {1}{289} \log ^4(5) \int \frac {1}{(x-17)^2 \left (x \log ^2\left (x^2\right )-17 \log ^2\left (x^2\right )+\log ^2(5)\right )^2}dx-\frac {2 \log ^4(5) \int \frac {1}{(x-17) \left (x \log ^2\left (x^2\right )-17 \log ^2\left (x^2\right )+\log ^2(5)\right )^2}dx}{4913}+\frac {1}{289} \log ^4(5) \int \frac {1}{x^2 \left (x \log ^2\left (x^2\right )-17 \log ^2\left (x^2\right )+\log ^2(5)\right )^2}dx+\frac {2 \log ^4(5) \int \frac {1}{x \left (x \log ^2\left (x^2\right )-17 \log ^2\left (x^2\right )+\log ^2(5)\right )^2}dx}{4913}-4 \log ^2(5) \int \frac {\log \left (x^2\right )}{x^3 \left (x \log ^2\left (x^2\right )-17 \log ^2\left (x^2\right )+\log ^2(5)\right )^2}dx+\frac {2}{17} \log ^2(5) \int \frac {1}{x^3 \left (x \log ^2\left (x^2\right )-17 \log ^2\left (x^2\right )+\log ^2(5)\right )}dx+\frac {1}{(17-x) x^2}\) |
Input:
Int[(-4*Log[5]^2*Log[x^2] + 2*Log[5]^2*Log[x^2]^2 + (-34 + 3*x)*Log[x^2]^4 )/(x^3*Log[5]^4 + (-34*x^3 + 2*x^4)*Log[5]^2*Log[x^2]^2 + (289*x^3 - 34*x^ 4 + x^5)*Log[x^2]^4),x]
Output:
$Aborted
Time = 0.52 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.52
method | result | size |
parallelrisch | \(-\frac {\ln \left (x^{2}\right )^{2}}{x^{2} \left (x \ln \left (x^{2}\right )^{2}+\ln \left (5\right )^{2}-17 \ln \left (x^{2}\right )^{2}\right )}\) | \(35\) |
risch | \(-\frac {1}{x^{2} \left (x -17\right )}+\frac {\ln \left (5\right )^{2}}{x^{2} \left (x -17\right ) \left (x \ln \left (x^{2}\right )^{2}+\ln \left (5\right )^{2}-17 \ln \left (x^{2}\right )^{2}\right )}\) | \(48\) |
Input:
int(((3*x-34)*ln(x^2)^4+2*ln(5)^2*ln(x^2)^2-4*ln(5)^2*ln(x^2))/((x^5-34*x^ 4+289*x^3)*ln(x^2)^4+(2*x^4-34*x^3)*ln(5)^2*ln(x^2)^2+x^3*ln(5)^4),x,metho d=_RETURNVERBOSE)
Output:
-1/x^2*ln(x^2)^2/(x*ln(x^2)^2+ln(5)^2-17*ln(x^2)^2)
Time = 0.11 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.52 \[ \int \frac {-4 \log ^2(5) \log \left (x^2\right )+2 \log ^2(5) \log ^2\left (x^2\right )+(-34+3 x) \log ^4\left (x^2\right )}{x^3 \log ^4(5)+\left (-34 x^3+2 x^4\right ) \log ^2(5) \log ^2\left (x^2\right )+\left (289 x^3-34 x^4+x^5\right ) \log ^4\left (x^2\right )} \, dx=-\frac {\log \left (x^{2}\right )^{2}}{x^{2} \log \left (5\right )^{2} + {\left (x^{3} - 17 \, x^{2}\right )} \log \left (x^{2}\right )^{2}} \] Input:
integrate(((3*x-34)*log(x^2)^4+2*log(5)^2*log(x^2)^2-4*log(5)^2*log(x^2))/ ((x^5-34*x^4+289*x^3)*log(x^2)^4+(2*x^4-34*x^3)*log(5)^2*log(x^2)^2+x^3*lo g(5)^4),x, algorithm="fricas")
Output:
-log(x^2)^2/(x^2*log(5)^2 + (x^3 - 17*x^2)*log(x^2)^2)
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (19) = 38\).
Time = 0.12 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.30 \[ \int \frac {-4 \log ^2(5) \log \left (x^2\right )+2 \log ^2(5) \log ^2\left (x^2\right )+(-34+3 x) \log ^4\left (x^2\right )}{x^3 \log ^4(5)+\left (-34 x^3+2 x^4\right ) \log ^2(5) \log ^2\left (x^2\right )+\left (289 x^3-34 x^4+x^5\right ) \log ^4\left (x^2\right )} \, dx=\frac {\log {\left (5 \right )}^{2}}{x^{3} \log {\left (5 \right )}^{2} - 17 x^{2} \log {\left (5 \right )}^{2} + \left (x^{4} - 34 x^{3} + 289 x^{2}\right ) \log {\left (x^{2} \right )}^{2}} - \frac {1}{x^{3} - 17 x^{2}} \] Input:
integrate(((3*x-34)*ln(x**2)**4+2*ln(5)**2*ln(x**2)**2-4*ln(5)**2*ln(x**2) )/((x**5-34*x**4+289*x**3)*ln(x**2)**4+(2*x**4-34*x**3)*ln(5)**2*ln(x**2)* *2+x**3*ln(5)**4),x)
Output:
log(5)**2/(x**3*log(5)**2 - 17*x**2*log(5)**2 + (x**4 - 34*x**3 + 289*x**2 )*log(x**2)**2) - 1/(x**3 - 17*x**2)
Time = 0.15 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39 \[ \int \frac {-4 \log ^2(5) \log \left (x^2\right )+2 \log ^2(5) \log ^2\left (x^2\right )+(-34+3 x) \log ^4\left (x^2\right )}{x^3 \log ^4(5)+\left (-34 x^3+2 x^4\right ) \log ^2(5) \log ^2\left (x^2\right )+\left (289 x^3-34 x^4+x^5\right ) \log ^4\left (x^2\right )} \, dx=-\frac {4 \, \log \left (x\right )^{2}}{x^{2} \log \left (5\right )^{2} + 4 \, {\left (x^{3} - 17 \, x^{2}\right )} \log \left (x\right )^{2}} \] Input:
integrate(((3*x-34)*log(x^2)^4+2*log(5)^2*log(x^2)^2-4*log(5)^2*log(x^2))/ ((x^5-34*x^4+289*x^3)*log(x^2)^4+(2*x^4-34*x^3)*log(5)^2*log(x^2)^2+x^3*lo g(5)^4),x, algorithm="maxima")
Output:
-4*log(x)^2/(x^2*log(5)^2 + 4*(x^3 - 17*x^2)*log(x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (21) = 42\).
Time = 0.28 (sec) , antiderivative size = 73, normalized size of antiderivative = 3.17 \[ \int \frac {-4 \log ^2(5) \log \left (x^2\right )+2 \log ^2(5) \log ^2\left (x^2\right )+(-34+3 x) \log ^4\left (x^2\right )}{x^3 \log ^4(5)+\left (-34 x^3+2 x^4\right ) \log ^2(5) \log ^2\left (x^2\right )+\left (289 x^3-34 x^4+x^5\right ) \log ^4\left (x^2\right )} \, dx=\frac {\log \left (5\right )^{2}}{x^{4} \log \left (x^{2}\right )^{2} + x^{3} \log \left (5\right )^{2} - 34 \, x^{3} \log \left (x^{2}\right )^{2} - 17 \, x^{2} \log \left (5\right )^{2} + 289 \, x^{2} \log \left (x^{2}\right )^{2}} - \frac {1}{289 \, {\left (x - 17\right )}} + \frac {x + 17}{289 \, x^{2}} \] Input:
integrate(((3*x-34)*log(x^2)^4+2*log(5)^2*log(x^2)^2-4*log(5)^2*log(x^2))/ ((x^5-34*x^4+289*x^3)*log(x^2)^4+(2*x^4-34*x^3)*log(5)^2*log(x^2)^2+x^3*lo g(5)^4),x, algorithm="giac")
Output:
log(5)^2/(x^4*log(x^2)^2 + x^3*log(5)^2 - 34*x^3*log(x^2)^2 - 17*x^2*log(5 )^2 + 289*x^2*log(x^2)^2) - 1/289/(x - 17) + 1/289*(x + 17)/x^2
Time = 4.00 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48 \[ \int \frac {-4 \log ^2(5) \log \left (x^2\right )+2 \log ^2(5) \log ^2\left (x^2\right )+(-34+3 x) \log ^4\left (x^2\right )}{x^3 \log ^4(5)+\left (-34 x^3+2 x^4\right ) \log ^2(5) \log ^2\left (x^2\right )+\left (289 x^3-34 x^4+x^5\right ) \log ^4\left (x^2\right )} \, dx=-\frac {{\ln \left (x^2\right )}^2}{x^2\,\left (x\,{\ln \left (x^2\right )}^2-17\,{\ln \left (x^2\right )}^2+{\ln \left (5\right )}^2\right )} \] Input:
int((log(x^2)^4*(3*x - 34) - 4*log(x^2)*log(5)^2 + 2*log(x^2)^2*log(5)^2)/ (x^3*log(5)^4 + log(x^2)^4*(289*x^3 - 34*x^4 + x^5) - log(x^2)^2*log(5)^2* (34*x^3 - 2*x^4)),x)
Output:
-log(x^2)^2/(x^2*(x*log(x^2)^2 - 17*log(x^2)^2 + log(5)^2))
Time = 0.15 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48 \[ \int \frac {-4 \log ^2(5) \log \left (x^2\right )+2 \log ^2(5) \log ^2\left (x^2\right )+(-34+3 x) \log ^4\left (x^2\right )}{x^3 \log ^4(5)+\left (-34 x^3+2 x^4\right ) \log ^2(5) \log ^2\left (x^2\right )+\left (289 x^3-34 x^4+x^5\right ) \log ^4\left (x^2\right )} \, dx=-\frac {\mathrm {log}\left (x^{2}\right )^{2}}{x^{2} \left (\mathrm {log}\left (x^{2}\right )^{2} x -17 \mathrm {log}\left (x^{2}\right )^{2}+\mathrm {log}\left (5\right )^{2}\right )} \] Input:
int(((3*x-34)*log(x^2)^4+2*log(5)^2*log(x^2)^2-4*log(5)^2*log(x^2))/((x^5- 34*x^4+289*x^3)*log(x^2)^4+(2*x^4-34*x^3)*log(5)^2*log(x^2)^2+x^3*log(5)^4 ),x)
Output:
( - log(x**2)**2)/(x**2*(log(x**2)**2*x - 17*log(x**2)**2 + log(5)**2))