Integrand size = 93, antiderivative size = 24 \[ \int \frac {-2268-486 x+\left (270 x+162 x^2\right ) \log (x)+\left (-756 x+162 x^2\right ) \log ^2(x)+\left (576 x+306 x^2\right ) \log ^3(x)+\left (252+306 x-54 x^2\right ) \log ^4(x)-54 x \log ^5(x)}{\left (2744 x+1764 x^2+378 x^3+27 x^4\right ) \log ^3(x)} \, dx=5+\frac {\left (x-\frac {3}{\log (x)}+\log (x)\right )^2}{\left (-\frac {14}{3}-x\right )^2} \] Output:
5+(x-3/ln(x)+ln(x))^2/(-14/3-x)^2
Time = 0.24 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71 \[ \int \frac {-2268-486 x+\left (270 x+162 x^2\right ) \log (x)+\left (-756 x+162 x^2\right ) \log ^2(x)+\left (576 x+306 x^2\right ) \log ^3(x)+\left (252+306 x-54 x^2\right ) \log ^4(x)-54 x \log ^5(x)}{\left (2744 x+1764 x^2+378 x^3+27 x^4\right ) \log ^3(x)} \, dx=\frac {\left (-9-14 \log (x)+3 \log ^2(x)\right ) \left (-9+2 (7+3 x) \log (x)+3 \log ^2(x)\right )}{(14+3 x)^2 \log ^2(x)} \] Input:
Integrate[(-2268 - 486*x + (270*x + 162*x^2)*Log[x] + (-756*x + 162*x^2)*L og[x]^2 + (576*x + 306*x^2)*Log[x]^3 + (252 + 306*x - 54*x^2)*Log[x]^4 - 5 4*x*Log[x]^5)/((2744*x + 1764*x^2 + 378*x^3 + 27*x^4)*Log[x]^3),x]
Output:
((-9 - 14*Log[x] + 3*Log[x]^2)*(-9 + 2*(7 + 3*x)*Log[x] + 3*Log[x]^2))/((1 4 + 3*x)^2*Log[x]^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.
\(\displaystyle \int \frac {\left (-54 x^2+306 x+252\right ) \log ^4(x)+\left (306 x^2+576 x\right ) \log ^3(x)+\left (162 x^2-756 x\right ) \log ^2(x)+\left (162 x^2+270 x\right ) \log (x)-486 x-54 x \log ^5(x)-2268}{\left (27 x^4+378 x^3+1764 x^2+2744 x\right ) \log ^3(x)} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {\left (-54 x^2+306 x+252\right ) \log ^4(x)+\left (306 x^2+576 x\right ) \log ^3(x)+\left (162 x^2-756 x\right ) \log ^2(x)+\left (162 x^2+270 x\right ) \log (x)-486 x-54 x \log ^5(x)-2268}{x \left (27 x^3+378 x^2+1764 x+2744\right ) \log ^3(x)}dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {\left (-54 x^2+306 x+252\right ) \log ^4(x)+\left (306 x^2+576 x\right ) \log ^3(x)+\left (162 x^2-756 x\right ) \log ^2(x)+\left (162 x^2+270 x\right ) \log (x)-486 x-54 x \log ^5(x)-2268}{x (3 x+14)^3 \log ^3(x)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {18 \left (3 x^2-17 x-14\right ) \log (x)}{x (3 x+14)^3}+\frac {18 (17 x+32)}{(3 x+14)^3}-\frac {162}{x (3 x+14)^2 \log ^3(x)}-\frac {54 \log ^2(x)}{(3 x+14)^3}+\frac {54 (3 x+5)}{(3 x+14)^3 \log ^2(x)}+\frac {54 (3 x-14)}{(3 x+14)^3 \log (x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -162 \int \frac {1}{x (3 x+14)^2 \log ^3(x)}dx+54 \int \frac {3 x+5}{(3 x+14)^3 \log ^2(x)}dx+54 \int \frac {3 x-14}{(3 x+14)^3 \log (x)}dx-\frac {9}{98} \operatorname {PolyLog}\left (2,-\frac {14}{3 x}\right )-\frac {9}{98} \operatorname {PolyLog}\left (2,-\frac {3 x}{14}\right )+\frac {9 (17 x+32)^2}{142 (3 x+14)^2}+\frac {6}{3 x+14}+\frac {9 \log ^2(x)}{(3 x+14)^2}+\frac {9 \log ^2(x)}{196}+\frac {9}{98} \log \left (\frac {14}{3 x}+1\right ) \log (x)-\frac {9}{98} \log \left (\frac {3 x}{14}+1\right ) \log (x)-\frac {9 x \log (x)}{7 (3 x+14)}-\frac {84 \log (x)}{(3 x+14)^2}+\frac {3 \log (x)}{7}\) |
Input:
Int[(-2268 - 486*x + (270*x + 162*x^2)*Log[x] + (-756*x + 162*x^2)*Log[x]^ 2 + (576*x + 306*x^2)*Log[x]^3 + (252 + 306*x - 54*x^2)*Log[x]^4 - 54*x*Lo g[x]^5)/((2744*x + 1764*x^2 + 378*x^3 + 27*x^4)*Log[x]^3),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(45\) vs. \(2(22)=44\).
Time = 9.22 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.92
method | result | size |
default | \(\frac {81-250 \ln \left (x \right )^{2}-84 x \ln \left (x \right )^{2}+9 \ln \left (x \right )^{4}-54 x \ln \left (x \right )+18 x \ln \left (x \right )^{3}}{\left (3 x +14\right )^{2} \ln \left (x \right )^{2}}\) | \(46\) |
parallelrisch | \(\frac {729+162 x \ln \left (x \right )^{3}+81 \ln \left (x \right )^{4}-756 x \ln \left (x \right )^{2}-486 x \ln \left (x \right )-2250 \ln \left (x \right )^{2}}{9 \ln \left (x \right )^{2} \left (9 x^{2}+84 x +196\right )}\) | \(52\) |
risch | \(\frac {9 \ln \left (x \right )^{2}}{9 x^{2}+84 x +196}+\frac {18 x \ln \left (x \right )}{9 x^{2}+84 x +196}-\frac {2 \left (42 x +125\right )}{9 x^{2}+84 x +196}-\frac {27 \left (2 x \ln \left (x \right )-3\right )}{\left (9 x^{2}+84 x +196\right ) \ln \left (x \right )^{2}}\) | \(81\) |
Input:
int((-54*x*ln(x)^5+(-54*x^2+306*x+252)*ln(x)^4+(306*x^2+576*x)*ln(x)^3+(16 2*x^2-756*x)*ln(x)^2+(162*x^2+270*x)*ln(x)-486*x-2268)/(27*x^4+378*x^3+176 4*x^2+2744*x)/ln(x)^3,x,method=_RETURNVERBOSE)
Output:
(81-250*ln(x)^2-84*x*ln(x)^2+9*ln(x)^4-54*x*ln(x)+18*x*ln(x)^3)/(3*x+14)^2 /ln(x)^2
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (23) = 46\).
Time = 0.07 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.00 \[ \int \frac {-2268-486 x+\left (270 x+162 x^2\right ) \log (x)+\left (-756 x+162 x^2\right ) \log ^2(x)+\left (576 x+306 x^2\right ) \log ^3(x)+\left (252+306 x-54 x^2\right ) \log ^4(x)-54 x \log ^5(x)}{\left (2744 x+1764 x^2+378 x^3+27 x^4\right ) \log ^3(x)} \, dx=\frac {18 \, x \log \left (x\right )^{3} + 9 \, \log \left (x\right )^{4} - 2 \, {\left (42 \, x + 125\right )} \log \left (x\right )^{2} - 54 \, x \log \left (x\right ) + 81}{{\left (9 \, x^{2} + 84 \, x + 196\right )} \log \left (x\right )^{2}} \] Input:
integrate((-54*x*log(x)^5+(-54*x^2+306*x+252)*log(x)^4+(306*x^2+576*x)*log (x)^3+(162*x^2-756*x)*log(x)^2+(162*x^2+270*x)*log(x)-486*x-2268)/(27*x^4+ 378*x^3+1764*x^2+2744*x)/log(x)^3,x, algorithm="fricas")
Output:
(18*x*log(x)^3 + 9*log(x)^4 - 2*(42*x + 125)*log(x)^2 - 54*x*log(x) + 81)/ ((9*x^2 + 84*x + 196)*log(x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (20) = 40\).
Time = 0.17 (sec) , antiderivative size = 73, normalized size of antiderivative = 3.04 \[ \int \frac {-2268-486 x+\left (270 x+162 x^2\right ) \log (x)+\left (-756 x+162 x^2\right ) \log ^2(x)+\left (576 x+306 x^2\right ) \log ^3(x)+\left (252+306 x-54 x^2\right ) \log ^4(x)-54 x \log ^5(x)}{\left (2744 x+1764 x^2+378 x^3+27 x^4\right ) \log ^3(x)} \, dx=\frac {18 x \log {\left (x \right )}}{9 x^{2} + 84 x + 196} + \frac {- 84 x - 250}{9 x^{2} + 84 x + 196} + \frac {- 54 x \log {\left (x \right )} + 81}{\left (9 x^{2} + 84 x + 196\right ) \log {\left (x \right )}^{2}} + \frac {9 \log {\left (x \right )}^{2}}{9 x^{2} + 84 x + 196} \] Input:
integrate((-54*x*ln(x)**5+(-54*x**2+306*x+252)*ln(x)**4+(306*x**2+576*x)*l n(x)**3+(162*x**2-756*x)*ln(x)**2+(162*x**2+270*x)*ln(x)-486*x-2268)/(27*x **4+378*x**3+1764*x**2+2744*x)/ln(x)**3,x)
Output:
18*x*log(x)/(9*x**2 + 84*x + 196) + (-84*x - 250)/(9*x**2 + 84*x + 196) + (-54*x*log(x) + 81)/((9*x**2 + 84*x + 196)*log(x)**2) + 9*log(x)**2/(9*x** 2 + 84*x + 196)
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (23) = 46\).
Time = 0.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.00 \[ \int \frac {-2268-486 x+\left (270 x+162 x^2\right ) \log (x)+\left (-756 x+162 x^2\right ) \log ^2(x)+\left (576 x+306 x^2\right ) \log ^3(x)+\left (252+306 x-54 x^2\right ) \log ^4(x)-54 x \log ^5(x)}{\left (2744 x+1764 x^2+378 x^3+27 x^4\right ) \log ^3(x)} \, dx=\frac {18 \, x \log \left (x\right )^{3} + 9 \, \log \left (x\right )^{4} - 2 \, {\left (42 \, x + 125\right )} \log \left (x\right )^{2} - 54 \, x \log \left (x\right ) + 81}{{\left (9 \, x^{2} + 84 \, x + 196\right )} \log \left (x\right )^{2}} \] Input:
integrate((-54*x*log(x)^5+(-54*x^2+306*x+252)*log(x)^4+(306*x^2+576*x)*log (x)^3+(162*x^2-756*x)*log(x)^2+(162*x^2+270*x)*log(x)-486*x-2268)/(27*x^4+ 378*x^3+1764*x^2+2744*x)/log(x)^3,x, algorithm="maxima")
Output:
(18*x*log(x)^3 + 9*log(x)^4 - 2*(42*x + 125)*log(x)^2 - 54*x*log(x) + 81)/ ((9*x^2 + 84*x + 196)*log(x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (23) = 46\).
Time = 0.14 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.71 \[ \int \frac {-2268-486 x+\left (270 x+162 x^2\right ) \log (x)+\left (-756 x+162 x^2\right ) \log ^2(x)+\left (576 x+306 x^2\right ) \log ^3(x)+\left (252+306 x-54 x^2\right ) \log ^4(x)-54 x \log ^5(x)}{\left (2744 x+1764 x^2+378 x^3+27 x^4\right ) \log ^3(x)} \, dx=\frac {18 \, x \log \left (x\right )}{9 \, x^{2} + 84 \, x + 196} + \frac {9 \, \log \left (x\right )^{2}}{9 \, x^{2} + 84 \, x + 196} - \frac {27 \, {\left (2 \, x \log \left (x\right ) - 3\right )}}{9 \, x^{2} \log \left (x\right )^{2} + 84 \, x \log \left (x\right )^{2} + 196 \, \log \left (x\right )^{2}} - \frac {2 \, {\left (42 \, x + 125\right )}}{9 \, x^{2} + 84 \, x + 196} \] Input:
integrate((-54*x*log(x)^5+(-54*x^2+306*x+252)*log(x)^4+(306*x^2+576*x)*log (x)^3+(162*x^2-756*x)*log(x)^2+(162*x^2+270*x)*log(x)-486*x-2268)/(27*x^4+ 378*x^3+1764*x^2+2744*x)/log(x)^3,x, algorithm="giac")
Output:
18*x*log(x)/(9*x^2 + 84*x + 196) + 9*log(x)^2/(9*x^2 + 84*x + 196) - 27*(2 *x*log(x) - 3)/(9*x^2*log(x)^2 + 84*x*log(x)^2 + 196*log(x)^2) - 2*(42*x + 125)/(9*x^2 + 84*x + 196)
Time = 1.86 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.04 \[ \int \frac {-2268-486 x+\left (270 x+162 x^2\right ) \log (x)+\left (-756 x+162 x^2\right ) \log ^2(x)+\left (576 x+306 x^2\right ) \log ^3(x)+\left (252+306 x-54 x^2\right ) \log ^4(x)-54 x \log ^5(x)}{\left (2744 x+1764 x^2+378 x^3+27 x^4\right ) \log ^3(x)} \, dx=\frac {9\,\left (125\,x^2\,{\ln \left (x\right )}^2+196\,x\,{\ln \left (x\right )}^3+252\,x\,{\ln \left (x\right )}^2-588\,x\,\ln \left (x\right )+98\,{\ln \left (x\right )}^4+882\right )}{98\,{\ln \left (x\right )}^2\,{\left (3\,x+14\right )}^2} \] Input:
int(-(486*x - log(x)^4*(306*x - 54*x^2 + 252) - log(x)^3*(576*x + 306*x^2) + log(x)^2*(756*x - 162*x^2) + 54*x*log(x)^5 - log(x)*(270*x + 162*x^2) + 2268)/(log(x)^3*(2744*x + 1764*x^2 + 378*x^3 + 27*x^4)),x)
Output:
(9*(252*x*log(x)^2 + 196*x*log(x)^3 + 98*log(x)^4 + 125*x^2*log(x)^2 - 588 *x*log(x) + 882))/(98*log(x)^2*(3*x + 14)^2)
Time = 0.19 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.08 \[ \int \frac {-2268-486 x+\left (270 x+162 x^2\right ) \log (x)+\left (-756 x+162 x^2\right ) \log ^2(x)+\left (576 x+306 x^2\right ) \log ^3(x)+\left (252+306 x-54 x^2\right ) \log ^4(x)-54 x \log ^5(x)}{\left (2744 x+1764 x^2+378 x^3+27 x^4\right ) \log ^3(x)} \, dx=\frac {9 \mathrm {log}\left (x \right )^{4}+18 \mathrm {log}\left (x \right )^{3} x +9 \mathrm {log}\left (x \right )^{2} x^{2}-54 \mathrm {log}\left (x \right )^{2}-54 \,\mathrm {log}\left (x \right ) x +81}{\mathrm {log}\left (x \right )^{2} \left (9 x^{2}+84 x +196\right )} \] Input:
int((-54*x*log(x)^5+(-54*x^2+306*x+252)*log(x)^4+(306*x^2+576*x)*log(x)^3+ (162*x^2-756*x)*log(x)^2+(162*x^2+270*x)*log(x)-486*x-2268)/(27*x^4+378*x^ 3+1764*x^2+2744*x)/log(x)^3,x)
Output:
(9*(log(x)**4 + 2*log(x)**3*x + log(x)**2*x**2 - 6*log(x)**2 - 6*log(x)*x + 9))/(log(x)**2*(9*x**2 + 84*x + 196))