Integrand size = 129, antiderivative size = 22 \[ \int \frac {6 x^4+24 x^9+12 x^3 \log ^2(2)}{x^3+3 x^4+3 x^5+x^6-3 x^8-6 x^9-3 x^{10}+3 x^{13}+3 x^{14}-x^{18}+\left (3 x^2+6 x^3+3 x^4-6 x^7-6 x^8+3 x^{12}\right ) \log ^2(2)+\left (3 x+3 x^2-3 x^6\right ) \log ^4(2)+\log ^6(2)} \, dx=\frac {3}{\left (1+\frac {1}{x}-x^4+\frac {\log ^2(2)}{x^2}\right )^2} \] Output:
3/(ln(2)^2/x^2+1-x^4+1/x)^2
Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {6 x^4+24 x^9+12 x^3 \log ^2(2)}{x^3+3 x^4+3 x^5+x^6-3 x^8-6 x^9-3 x^{10}+3 x^{13}+3 x^{14}-x^{18}+\left (3 x^2+6 x^3+3 x^4-6 x^7-6 x^8+3 x^{12}\right ) \log ^2(2)+\left (3 x+3 x^2-3 x^6\right ) \log ^4(2)+\log ^6(2)} \, dx=\frac {3 x^4}{\left (x+x^2-x^6+\log ^2(2)\right )^2} \] Input:
Integrate[(6*x^4 + 24*x^9 + 12*x^3*Log[2]^2)/(x^3 + 3*x^4 + 3*x^5 + x^6 - 3*x^8 - 6*x^9 - 3*x^10 + 3*x^13 + 3*x^14 - x^18 + (3*x^2 + 6*x^3 + 3*x^4 - 6*x^7 - 6*x^8 + 3*x^12)*Log[2]^2 + (3*x + 3*x^2 - 3*x^6)*Log[2]^4 + Log[2 ]^6),x]
Output:
(3*x^4)/(x + x^2 - x^6 + Log[2]^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.
\(\displaystyle \int \frac {24 x^9+6 x^4+12 x^3 \log ^2(2)}{-x^{18}+3 x^{14}+3 x^{13}-3 x^{10}-6 x^9-3 x^8+x^6+3 x^5+3 x^4+x^3+\left (-3 x^6+3 x^2+3 x\right ) \log ^4(2)+\left (3 x^{12}-6 x^8-6 x^7+3 x^4+6 x^3+3 x^2\right ) \log ^2(2)+\log ^6(2)} \, dx\) |
\(\Big \downarrow \) 2028 |
\(\displaystyle \int \frac {x^3 \left (24 x^6+6 x+12 \log ^2(2)\right )}{-x^{18}+3 x^{14}+3 x^{13}-3 x^{10}-6 x^9-3 x^8+x^6+3 x^5+3 x^4+x^3+\left (-3 x^6+3 x^2+3 x\right ) \log ^4(2)+\left (3 x^{12}-6 x^8-6 x^7+3 x^4+6 x^3+3 x^2\right ) \log ^2(2)+\log ^6(2)}dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle \int \left (-\frac {6 x^3 \left (4 x^2+5 x+6 \log ^2(2)\right )}{\left (x^6-x^2-x-\log ^2(2)\right )^3}-\frac {24 x^3}{\left (x^6-x^2-x-\log ^2(2)\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -4 \int \frac {1}{\left (x^6-x^2-x-\log ^2(2)\right )^3}dx-8 \int \frac {x}{\left (x^6-x^2-x-\log ^2(2)\right )^3}dx-30 \int \frac {x^4}{\left (x^6-x^2-x-\log ^2(2)\right )^3}dx-36 \log ^2(2) \int \frac {x^3}{\left (x^6-x^2-x-\log ^2(2)\right )^3}dx-24 \int \frac {x^3}{\left (x^6-x^2-x-\log ^2(2)\right )^2}dx+\frac {2}{\left (-x^6+x^2+x+\log ^2(2)\right )^2}\) |
Input:
Int[(6*x^4 + 24*x^9 + 12*x^3*Log[2]^2)/(x^3 + 3*x^4 + 3*x^5 + x^6 - 3*x^8 - 6*x^9 - 3*x^10 + 3*x^13 + 3*x^14 - x^18 + (3*x^2 + 6*x^3 + 3*x^4 - 6*x^7 - 6*x^8 + 3*x^12)*Log[2]^2 + (3*x + 3*x^2 - 3*x^6)*Log[2]^4 + Log[2]^6),x ]
Output:
$Aborted
Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00
method | result | size |
default | \(\frac {3 x^{4}}{\left (-x^{6}+\ln \left (2\right )^{2}+x^{2}+x \right )^{2}}\) | \(22\) |
norman | \(\frac {3 x^{4}}{\left (-x^{6}+\ln \left (2\right )^{2}+x^{2}+x \right )^{2}}\) | \(22\) |
gosper | \(\frac {3 x^{4}}{x^{12}-2 \ln \left (2\right )^{2} x^{6}-2 x^{8}-2 x^{7}+\ln \left (2\right )^{4}+2 x^{2} \ln \left (2\right )^{2}+x^{4}+2 x \ln \left (2\right )^{2}+2 x^{3}+x^{2}}\) | \(62\) |
risch | \(\frac {3 x^{4}}{x^{12}-2 \ln \left (2\right )^{2} x^{6}-2 x^{8}-2 x^{7}+\ln \left (2\right )^{4}+2 x^{2} \ln \left (2\right )^{2}+x^{4}+2 x \ln \left (2\right )^{2}+2 x^{3}+x^{2}}\) | \(62\) |
parallelrisch | \(\frac {3 x^{4}}{x^{12}-2 \ln \left (2\right )^{2} x^{6}-2 x^{8}-2 x^{7}+\ln \left (2\right )^{4}+2 x^{2} \ln \left (2\right )^{2}+x^{4}+2 x \ln \left (2\right )^{2}+2 x^{3}+x^{2}}\) | \(62\) |
Input:
int((12*x^3*ln(2)^2+24*x^9+6*x^4)/(ln(2)^6+(-3*x^6+3*x^2+3*x)*ln(2)^4+(3*x ^12-6*x^8-6*x^7+3*x^4+6*x^3+3*x^2)*ln(2)^2-x^18+3*x^14+3*x^13-3*x^10-6*x^9 -3*x^8+x^6+3*x^5+3*x^4+x^3),x,method=_RETURNVERBOSE)
Output:
3*x^4/(-x^6+ln(2)^2+x^2+x)^2
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (23) = 46\).
Time = 0.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.45 \[ \int \frac {6 x^4+24 x^9+12 x^3 \log ^2(2)}{x^3+3 x^4+3 x^5+x^6-3 x^8-6 x^9-3 x^{10}+3 x^{13}+3 x^{14}-x^{18}+\left (3 x^2+6 x^3+3 x^4-6 x^7-6 x^8+3 x^{12}\right ) \log ^2(2)+\left (3 x+3 x^2-3 x^6\right ) \log ^4(2)+\log ^6(2)} \, dx=\frac {3 \, x^{4}}{x^{12} - 2 \, x^{8} - 2 \, x^{7} + x^{4} + \log \left (2\right )^{4} + 2 \, x^{3} - 2 \, {\left (x^{6} - x^{2} - x\right )} \log \left (2\right )^{2} + x^{2}} \] Input:
integrate((12*x^3*log(2)^2+24*x^9+6*x^4)/(log(2)^6+(-3*x^6+3*x^2+3*x)*log( 2)^4+(3*x^12-6*x^8-6*x^7+3*x^4+6*x^3+3*x^2)*log(2)^2-x^18+3*x^14+3*x^13-3* x^10-6*x^9-3*x^8+x^6+3*x^5+3*x^4+x^3),x, algorithm="fricas")
Output:
3*x^4/(x^12 - 2*x^8 - 2*x^7 + x^4 + log(2)^4 + 2*x^3 - 2*(x^6 - x^2 - x)*l og(2)^2 + x^2)
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (19) = 38\).
Time = 6.83 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.77 \[ \int \frac {6 x^4+24 x^9+12 x^3 \log ^2(2)}{x^3+3 x^4+3 x^5+x^6-3 x^8-6 x^9-3 x^{10}+3 x^{13}+3 x^{14}-x^{18}+\left (3 x^2+6 x^3+3 x^4-6 x^7-6 x^8+3 x^{12}\right ) \log ^2(2)+\left (3 x+3 x^2-3 x^6\right ) \log ^4(2)+\log ^6(2)} \, dx=\frac {3 x^{4}}{x^{12} - 2 x^{8} - 2 x^{7} - 2 x^{6} \log {\left (2 \right )}^{2} + x^{4} + 2 x^{3} + x^{2} \cdot \left (2 \log {\left (2 \right )}^{2} + 1\right ) + 2 x \log {\left (2 \right )}^{2} + \log {\left (2 \right )}^{4}} \] Input:
integrate((12*x**3*ln(2)**2+24*x**9+6*x**4)/(ln(2)**6+(-3*x**6+3*x**2+3*x) *ln(2)**4+(3*x**12-6*x**8-6*x**7+3*x**4+6*x**3+3*x**2)*ln(2)**2-x**18+3*x* *14+3*x**13-3*x**10-6*x**9-3*x**8+x**6+3*x**5+3*x**4+x**3),x)
Output:
3*x**4/(x**12 - 2*x**8 - 2*x**7 - 2*x**6*log(2)**2 + x**4 + 2*x**3 + x**2* (2*log(2)**2 + 1) + 2*x*log(2)**2 + log(2)**4)
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (23) = 46\).
Time = 0.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.77 \[ \int \frac {6 x^4+24 x^9+12 x^3 \log ^2(2)}{x^3+3 x^4+3 x^5+x^6-3 x^8-6 x^9-3 x^{10}+3 x^{13}+3 x^{14}-x^{18}+\left (3 x^2+6 x^3+3 x^4-6 x^7-6 x^8+3 x^{12}\right ) \log ^2(2)+\left (3 x+3 x^2-3 x^6\right ) \log ^4(2)+\log ^6(2)} \, dx=\frac {3 \, x^{4}}{x^{12} - 2 \, x^{8} - 2 \, x^{6} \log \left (2\right )^{2} - 2 \, x^{7} + x^{4} + \log \left (2\right )^{4} + {\left (2 \, \log \left (2\right )^{2} + 1\right )} x^{2} + 2 \, x^{3} + 2 \, x \log \left (2\right )^{2}} \] Input:
integrate((12*x^3*log(2)^2+24*x^9+6*x^4)/(log(2)^6+(-3*x^6+3*x^2+3*x)*log( 2)^4+(3*x^12-6*x^8-6*x^7+3*x^4+6*x^3+3*x^2)*log(2)^2-x^18+3*x^14+3*x^13-3* x^10-6*x^9-3*x^8+x^6+3*x^5+3*x^4+x^3),x, algorithm="maxima")
Output:
3*x^4/(x^12 - 2*x^8 - 2*x^6*log(2)^2 - 2*x^7 + x^4 + log(2)^4 + (2*log(2)^ 2 + 1)*x^2 + 2*x^3 + 2*x*log(2)^2)
Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {6 x^4+24 x^9+12 x^3 \log ^2(2)}{x^3+3 x^4+3 x^5+x^6-3 x^8-6 x^9-3 x^{10}+3 x^{13}+3 x^{14}-x^{18}+\left (3 x^2+6 x^3+3 x^4-6 x^7-6 x^8+3 x^{12}\right ) \log ^2(2)+\left (3 x+3 x^2-3 x^6\right ) \log ^4(2)+\log ^6(2)} \, dx=\frac {3 \, x^{4}}{{\left (x^{6} - x^{2} - \log \left (2\right )^{2} - x\right )}^{2}} \] Input:
integrate((12*x^3*log(2)^2+24*x^9+6*x^4)/(log(2)^6+(-3*x^6+3*x^2+3*x)*log( 2)^4+(3*x^12-6*x^8-6*x^7+3*x^4+6*x^3+3*x^2)*log(2)^2-x^18+3*x^14+3*x^13-3* x^10-6*x^9-3*x^8+x^6+3*x^5+3*x^4+x^3),x, algorithm="giac")
Output:
3*x^4/(x^6 - x^2 - log(2)^2 - x)^2
Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {6 x^4+24 x^9+12 x^3 \log ^2(2)}{x^3+3 x^4+3 x^5+x^6-3 x^8-6 x^9-3 x^{10}+3 x^{13}+3 x^{14}-x^{18}+\left (3 x^2+6 x^3+3 x^4-6 x^7-6 x^8+3 x^{12}\right ) \log ^2(2)+\left (3 x+3 x^2-3 x^6\right ) \log ^4(2)+\log ^6(2)} \, dx=\frac {3\,x^4}{{\left (-x^6+x^2+x+{\ln \left (2\right )}^2\right )}^2} \] Input:
int((12*x^3*log(2)^2 + 6*x^4 + 24*x^9)/(log(2)^6 + log(2)^4*(3*x + 3*x^2 - 3*x^6) + log(2)^2*(3*x^2 + 6*x^3 + 3*x^4 - 6*x^7 - 6*x^8 + 3*x^12) + x^3 + 3*x^4 + 3*x^5 + x^6 - 3*x^8 - 6*x^9 - 3*x^10 + 3*x^13 + 3*x^14 - x^18),x )
Output:
(3*x^4)/(x + log(2)^2 + x^2 - x^6)^2
Time = 0.29 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.77 \[ \int \frac {6 x^4+24 x^9+12 x^3 \log ^2(2)}{x^3+3 x^4+3 x^5+x^6-3 x^8-6 x^9-3 x^{10}+3 x^{13}+3 x^{14}-x^{18}+\left (3 x^2+6 x^3+3 x^4-6 x^7-6 x^8+3 x^{12}\right ) \log ^2(2)+\left (3 x+3 x^2-3 x^6\right ) \log ^4(2)+\log ^6(2)} \, dx=\frac {3 x^{4}}{\mathrm {log}\left (2\right )^{4}-2 \mathrm {log}\left (2\right )^{2} x^{6}+2 \mathrm {log}\left (2\right )^{2} x^{2}+2 \mathrm {log}\left (2\right )^{2} x +x^{12}-2 x^{8}-2 x^{7}+x^{4}+2 x^{3}+x^{2}} \] Input:
int((12*x^3*log(2)^2+24*x^9+6*x^4)/(log(2)^6+(-3*x^6+3*x^2+3*x)*log(2)^4+( 3*x^12-6*x^8-6*x^7+3*x^4+6*x^3+3*x^2)*log(2)^2-x^18+3*x^14+3*x^13-3*x^10-6 *x^9-3*x^8+x^6+3*x^5+3*x^4+x^3),x)
Output:
(3*x**4)/(log(2)**4 - 2*log(2)**2*x**6 + 2*log(2)**2*x**2 + 2*log(2)**2*x + x**12 - 2*x**8 - 2*x**7 + x**4 + 2*x**3 + x**2)