Integrand size = 41, antiderivative size = 21 \[ \int \left (\left (4-2 e^2\right ) \log (x)+\left (8-e^2\right ) \log ^2(x)+\left (6 \log (x)+3 \log ^2(x)\right ) \log \left (9 x^2\right )\right ) \, dx=x \log ^2(x) \left (2-e^2+3 \log \left (9 x^2\right )\right ) \] Output:
ln(x)^2*x*(2+3*ln(9*x^2)-exp(2))
Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \left (\left (4-2 e^2\right ) \log (x)+\left (8-e^2\right ) \log ^2(x)+\left (6 \log (x)+3 \log ^2(x)\right ) \log \left (9 x^2\right )\right ) \, dx=x \log ^2(x) \left (2-e^2+3 \log \left (9 x^2\right )\right ) \] Input:
Integrate[(4 - 2*E^2)*Log[x] + (8 - E^2)*Log[x]^2 + (6*Log[x] + 3*Log[x]^2 )*Log[9*x^2],x]
Output:
x*Log[x]^2*(2 - E^2 + 3*Log[9*x^2])
Leaf count is larger than twice the leaf count of optimal. \(86\) vs. \(2(21)=42\).
Time = 0.32 (sec) , antiderivative size = 86, normalized size of antiderivative = 4.10, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {2009}
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.
\(\displaystyle \int \left (\left (3 \log ^2(x)+6 \log (x)\right ) \log \left (9 x^2\right )+\left (8-e^2\right ) \log ^2(x)+\left (4-2 e^2\right ) \log (x)\right ) \, dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 x \log \left (9 x^2\right ) \log ^2(x)+2 \left (8-e^2\right ) x-2 \left (2-e^2\right ) x-12 x+\left (8-e^2\right ) x \log ^2(x)-6 x \log ^2(x)-2 \left (8-e^2\right ) x \log (x)+2 \left (2-e^2\right ) x \log (x)+12 x \log (x)\) |
Input:
Int[(4 - 2*E^2)*Log[x] + (8 - E^2)*Log[x]^2 + (6*Log[x] + 3*Log[x]^2)*Log[ 9*x^2],x]
Output:
-12*x - 2*(2 - E^2)*x + 2*(8 - E^2)*x + 12*x*Log[x] + 2*(2 - E^2)*x*Log[x] - 2*(8 - E^2)*x*Log[x] - 6*x*Log[x]^2 + (8 - E^2)*x*Log[x]^2 + 3*x*Log[x] ^2*Log[9*x^2]
Time = 0.47 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29
method | result | size |
norman | \(\left (2-{\mathrm e}^{2}\right ) x \ln \left (x \right )^{2}+3 x \ln \left (x \right )^{2} \ln \left (9 x^{2}\right )\) | \(27\) |
parallelrisch | \(-x \,{\mathrm e}^{2} \ln \left (x \right )^{2}+3 x \ln \left (x \right )^{2} \ln \left (9 x^{2}\right )+2 x \ln \left (x \right )^{2}\) | \(31\) |
risch | \(6 x \ln \left (x \right )^{3}-\frac {3 i \operatorname {csgn}\left (i x^{2}\right )^{3} \ln \left (x \right )^{2} \pi x}{2}+3 i \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right ) \ln \left (x \right )^{2} \pi x -\frac {3 i \operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2} \ln \left (x \right )^{2} \pi x}{2}+6 x \ln \left (3\right ) \ln \left (x \right )^{2}+2 x \ln \left (x \right )^{2}-x \,{\mathrm e}^{2} \ln \left (x \right )^{2}\) | \(98\) |
default | \(\left (8-{\mathrm e}^{2}\right ) \left (x \ln \left (x \right )^{2}-2 x \ln \left (x \right )+2 x \right )+3 \left (\ln \left (x^{2}\right )-2 \ln \left (x \right )\right ) \left (x \ln \left (x \right )^{2}-2 x \ln \left (x \right )+2 x \right )+6 x \ln \left (x \right )^{3}-6 x \ln \left (x \right )^{2}+12 x \ln \left (x \right )-12 x +6 x \ln \left (3\right ) \ln \left (x \right )^{2}+6 \left (\ln \left (x^{2}\right )-2 \ln \left (x \right )\right ) \left (x \ln \left (x \right )-x \right )+\left (-2 \,{\mathrm e}^{2}+4\right ) \left (x \ln \left (x \right )-x \right )\) | \(115\) |
parts | \(2 x \ln \left (x \right )^{2}+6 \ln \left (3\right ) \left (x \ln \left (x \right )^{2}-2 x \ln \left (x \right )+2 x \right )-{\mathrm e}^{2} \left (x \ln \left (x \right )^{2}-2 x \ln \left (x \right )+2 x \right )+3 \left (\ln \left (x^{2}\right )-2 \ln \left (x \right )\right ) \left (x \ln \left (x \right )^{2}-2 x \ln \left (x \right )+2 x \right )+6 x \ln \left (x \right )^{3}+12 \ln \left (3\right ) \left (x \ln \left (x \right )-x \right )-2 \,{\mathrm e}^{2} \left (x \ln \left (x \right )-x \right )+6 \left (\ln \left (x^{2}\right )-2 \ln \left (x \right )\right ) \left (x \ln \left (x \right )-x \right )\) | \(123\) |
orering | \(\left (\left (3 \ln \left (x \right )^{2}+6 \ln \left (x \right )\right ) \ln \left (9 x^{2}\right )+\left (8-{\mathrm e}^{2}\right ) \ln \left (x \right )^{2}+\left (-2 \,{\mathrm e}^{2}+4\right ) \ln \left (x \right )\right ) x -x^{2} \left (\left (\frac {6 \ln \left (x \right )}{x}+\frac {6}{x}\right ) \ln \left (9 x^{2}\right )+\frac {6 \ln \left (x \right )^{2}+12 \ln \left (x \right )}{x}+\frac {2 \left (8-{\mathrm e}^{2}\right ) \ln \left (x \right )}{x}+\frac {-2 \,{\mathrm e}^{2}+4}{x}\right )-2 x^{3} \left (-\frac {6 \ln \left (x \right ) \ln \left (9 x^{2}\right )}{x^{2}}+\frac {\frac {24 \ln \left (x \right )}{x}+\frac {24}{x}}{x}-\frac {2 \left (3 \ln \left (x \right )^{2}+6 \ln \left (x \right )\right )}{x^{2}}+\frac {16-2 \,{\mathrm e}^{2}}{x^{2}}-\frac {2 \left (8-{\mathrm e}^{2}\right ) \ln \left (x \right )}{x^{2}}-\frac {-2 \,{\mathrm e}^{2}+4}{x^{2}}\right )-x^{4} \left (-\frac {6 \ln \left (9 x^{2}\right )}{x^{3}}+\frac {12 \ln \left (x \right ) \ln \left (9 x^{2}\right )}{x^{3}}-\frac {36 \ln \left (x \right )}{x^{3}}-\frac {6 \left (\frac {6 \ln \left (x \right )}{x}+\frac {6}{x}\right )}{x^{2}}+\frac {12 \ln \left (x \right )^{2}+24 \ln \left (x \right )}{x^{3}}-\frac {6 \left (8-{\mathrm e}^{2}\right )}{x^{3}}+\frac {4 \left (8-{\mathrm e}^{2}\right ) \ln \left (x \right )}{x^{3}}+\frac {-4 \,{\mathrm e}^{2}+8}{x^{3}}\right )\) | \(302\) |
Input:
int((3*ln(x)^2+6*ln(x))*ln(9*x^2)+(8-exp(2))*ln(x)^2+(-2*exp(2)+4)*ln(x),x ,method=_RETURNVERBOSE)
Output:
(2-exp(2))*x*ln(x)^2+3*x*ln(x)^2*ln(9*x^2)
Time = 0.11 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29 \[ \int \left (\left (4-2 e^2\right ) \log (x)+\left (8-e^2\right ) \log ^2(x)+\left (6 \log (x)+3 \log ^2(x)\right ) \log \left (9 x^2\right )\right ) \, dx=6 \, x \log \left (x\right )^{3} - {\left (x e^{2} - 6 \, x \log \left (3\right ) - 2 \, x\right )} \log \left (x\right )^{2} \] Input:
integrate((3*log(x)^2+6*log(x))*log(9*x^2)+(8-exp(2))*log(x)^2+(-2*exp(2)+ 4)*log(x),x, algorithm="fricas")
Output:
6*x*log(x)^3 - (x*e^2 - 6*x*log(3) - 2*x)*log(x)^2
Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29 \[ \int \left (\left (4-2 e^2\right ) \log (x)+\left (8-e^2\right ) \log ^2(x)+\left (6 \log (x)+3 \log ^2(x)\right ) \log \left (9 x^2\right )\right ) \, dx=6 x \log {\left (x \right )}^{3} + \left (- x e^{2} + 2 x + 6 x \log {\left (3 \right )}\right ) \log {\left (x \right )}^{2} \] Input:
integrate((3*ln(x)**2+6*ln(x))*ln(9*x**2)+(8-exp(2))*ln(x)**2+(-2*exp(2)+4 )*ln(x),x)
Output:
6*x*log(x)**3 + (-x*exp(2) + 2*x + 6*x*log(3))*log(x)**2
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (19) = 38\).
Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.62 \[ \int \left (\left (4-2 e^2\right ) \log (x)+\left (8-e^2\right ) \log ^2(x)+\left (6 \log (x)+3 \log ^2(x)\right ) \log \left (9 x^2\right )\right ) \, dx=-{\left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 2\right )} x {\left (e^{2} - 8\right )} - 6 \, x \log \left (x\right )^{2} - 2 \, {\left (x \log \left (x\right ) - x\right )} {\left (e^{2} - 2\right )} + 3 \, {\left ({\left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 2\right )} x + 2 \, x \log \left (x\right ) - 2 \, x\right )} \log \left (9 \, x^{2}\right ) + 12 \, x \log \left (x\right ) - 12 \, x \] Input:
integrate((3*log(x)^2+6*log(x))*log(9*x^2)+(8-exp(2))*log(x)^2+(-2*exp(2)+ 4)*log(x),x, algorithm="maxima")
Output:
-(log(x)^2 - 2*log(x) + 2)*x*(e^2 - 8) - 6*x*log(x)^2 - 2*(x*log(x) - x)*( e^2 - 2) + 3*((log(x)^2 - 2*log(x) + 2)*x + 2*x*log(x) - 2*x)*log(9*x^2) + 12*x*log(x) - 12*x
Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (19) = 38\).
Time = 0.13 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.95 \[ \int \left (\left (4-2 e^2\right ) \log (x)+\left (8-e^2\right ) \log ^2(x)+\left (6 \log (x)+3 \log ^2(x)\right ) \log \left (9 x^2\right )\right ) \, dx=6 \, x {\left (\log \left (3\right ) - 1\right )} \log \left (x\right )^{2} + 6 \, x \log \left (x\right )^{3} - 2 \, {\left (x \log \left (x\right ) - x\right )} {\left (e^{2} - 2\right )} - {\left (x \log \left (x\right )^{2} - 2 \, x \log \left (x\right ) + 2 \, x\right )} {\left (e^{2} - 8\right )} + 12 \, x \log \left (x\right ) - 12 \, x \] Input:
integrate((3*log(x)^2+6*log(x))*log(9*x^2)+(8-exp(2))*log(x)^2+(-2*exp(2)+ 4)*log(x),x, algorithm="giac")
Output:
6*x*(log(3) - 1)*log(x)^2 + 6*x*log(x)^3 - 2*(x*log(x) - x)*(e^2 - 2) - (x *log(x)^2 - 2*x*log(x) + 2*x)*(e^2 - 8) + 12*x*log(x) - 12*x
Time = 3.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \left (\left (4-2 e^2\right ) \log (x)+\left (8-e^2\right ) \log ^2(x)+\left (6 \log (x)+3 \log ^2(x)\right ) \log \left (9 x^2\right )\right ) \, dx=x\,{\ln \left (x\right )}^2\,\left (3\,\ln \left (9\,x^2\right )-{\mathrm {e}}^2+2\right ) \] Input:
int(log(9*x^2)*(6*log(x) + 3*log(x)^2) - log(x)*(2*exp(2) - 4) - log(x)^2* (exp(2) - 8),x)
Output:
x*log(x)^2*(3*log(9*x^2) - exp(2) + 2)
Time = 0.17 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \left (\left (4-2 e^2\right ) \log (x)+\left (8-e^2\right ) \log ^2(x)+\left (6 \log (x)+3 \log ^2(x)\right ) \log \left (9 x^2\right )\right ) \, dx=\mathrm {log}\left (x \right )^{2} x \left (3 \,\mathrm {log}\left (9 x^{2}\right )-e^{2}+2\right ) \] Input:
int((3*log(x)^2+6*log(x))*log(9*x^2)+(8-exp(2))*log(x)^2+(-2*exp(2)+4)*log (x),x)
Output:
log(x)**2*x*(3*log(9*x**2) - e**2 + 2)