Integrand size = 49, antiderivative size = 22 \[ \int \frac {\left (2508+4 x^2+32 x^3\right ) \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{e^{12} \left (-50787 x+81 x^3+324 x^4\right )} \, dx=\frac {\log ^4\left (-\frac {627}{x}+x+4 x^2\right )}{81 e^{12}} \] Output:
1/81*ln(x+4*x^2-627/x)^4/exp(3)^4
Time = 0.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\left (2508+4 x^2+32 x^3\right ) \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{e^{12} \left (-50787 x+81 x^3+324 x^4\right )} \, dx=\frac {\log ^4\left (-\frac {627}{x}+x+4 x^2\right )}{81 e^{12}} \] Input:
Integrate[((2508 + 4*x^2 + 32*x^3)*Log[(-627 + x^2 + 4*x^3)/x]^3)/(E^12*(- 50787*x + 81*x^3 + 324*x^4)),x]
Output:
Log[-627/x + x + 4*x^2]^4/(81*E^12)
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.
\(\displaystyle \int \frac {\left (32 x^3+4 x^2+2508\right ) \log ^3\left (\frac {4 x^3+x^2-627}{x}\right )}{e^{12} \left (324 x^4+81 x^3-50787 x\right )} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int -\frac {4 \left (8 x^3+x^2+627\right ) \log ^3\left (-\frac {-4 x^3-x^2+627}{x}\right )}{81 \left (-4 x^4-x^3+627 x\right )}dx}{e^{12}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {4 \int \frac {\left (8 x^3+x^2+627\right ) \log ^3\left (-\frac {-4 x^3-x^2+627}{x}\right )}{-4 x^4-x^3+627 x}dx}{81 e^{12}}\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle -\frac {4 \int \frac {\left (8 x^3+x^2+627\right ) \log ^3\left (-\frac {-4 x^3-x^2+627}{x}\right )}{x \left (-4 x^3-x^2+627\right )}dx}{81 e^{12}}\) |
\(\Big \downarrow \) 3008 |
\(\displaystyle -\frac {4 \int \left (\frac {\log ^3\left (-\frac {-4 x^3-x^2+627}{x}\right )}{x}-\frac {2 x (6 x+1) \log ^3\left (-\frac {-4 x^3-x^2+627}{x}\right )}{4 x^3+x^2-627}\right )dx}{81 e^{12}}\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle -\frac {4 \int \left (\frac {\log ^3\left (\frac {4 x^3+x^2-627}{x}\right )}{x}-\frac {2 x (6 x+1) \log ^3\left (\frac {4 x^3+x^2-627}{x}\right )}{4 x^3+x^2-627}\right )dx}{81 e^{12}}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {4 \int \left (\frac {\log ^3\left (\frac {4 x^3+x^2-627}{x}\right )}{x}+\frac {2 x (6 x+1) \log ^3\left (\frac {4 x^3+x^2-627}{x}\right )}{-4 x^3-x^2+627}\right )dx}{81 e^{12}}\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle -\frac {4 \int \left (\frac {\log ^3\left (\frac {4 x^3+x^2-627}{x}\right )}{x}+\frac {2 x (6 x+1) \log ^3\left (\frac {4 x^3+x^2-627}{x}\right )}{-4 x^3-x^2+627}\right )dx}{81 e^{12}}\) |
Input:
Int[((2508 + 4*x^2 + 32*x^3)*Log[(-627 + x^2 + 4*x^3)/x]^3)/(E^12*(-50787* x + 81*x^3 + 324*x^4)),x]
Output:
$Aborted
Time = 0.72 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00
method | result | size |
risch | \(\frac {{\mathrm e}^{-12} \ln \left (\frac {4 x^{3}+x^{2}-627}{x}\right )^{4}}{81}\) | \(22\) |
default | \(\frac {{\mathrm e}^{-12} \ln \left (\frac {4 x^{3}+x^{2}-627}{x}\right )^{4}}{81}\) | \(24\) |
norman | \(\frac {{\mathrm e}^{-12} \ln \left (\frac {4 x^{3}+x^{2}-627}{x}\right )^{4}}{81}\) | \(24\) |
Input:
int((32*x^3+4*x^2+2508)*ln((4*x^3+x^2-627)/x)^3/(324*x^4+81*x^3-50787*x)/e xp(3)^4,x,method=_RETURNVERBOSE)
Output:
1/81*exp(-12)*ln((4*x^3+x^2-627)/x)^4
Time = 0.07 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {\left (2508+4 x^2+32 x^3\right ) \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{e^{12} \left (-50787 x+81 x^3+324 x^4\right )} \, dx=\frac {1}{81} \, e^{\left (-12\right )} \log \left (\frac {4 \, x^{3} + x^{2} - 627}{x}\right )^{4} \] Input:
integrate((32*x^3+4*x^2+2508)*log((4*x^3+x^2-627)/x)^3/(324*x^4+81*x^3-507 87*x)/exp(3)^4,x, algorithm="fricas")
Output:
1/81*e^(-12)*log((4*x^3 + x^2 - 627)/x)^4
Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {\left (2508+4 x^2+32 x^3\right ) \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{e^{12} \left (-50787 x+81 x^3+324 x^4\right )} \, dx=\frac {\log {\left (\frac {4 x^{3} + x^{2} - 627}{x} \right )}^{4}}{81 e^{12}} \] Input:
integrate((32*x**3+4*x**2+2508)*ln((4*x**3+x**2-627)/x)**3/(324*x**4+81*x* *3-50787*x)/exp(3)**4,x)
Output:
exp(-12)*log((4*x**3 + x**2 - 627)/x)**4/81
Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (19) = 38\).
Time = 0.05 (sec) , antiderivative size = 236, normalized size of antiderivative = 10.73 \[ \int \frac {\left (2508+4 x^2+32 x^3\right ) \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{e^{12} \left (-50787 x+81 x^3+324 x^4\right )} \, dx=\frac {4}{81} \, {\left (\log \left (4 \, x^{3} + x^{2} - 627\right ) - \log \left (x\right )\right )} e^{\left (-12\right )} \log \left (\frac {4 \, x^{3} + x^{2} - 627}{x}\right )^{3} - \frac {1}{81} \, {\left (\log \left (4 \, x^{3} + x^{2} - 627\right )^{4} - 4 \, \log \left (4 \, x^{3} + x^{2} - 627\right )^{3} \log \left (x\right ) + 6 \, \log \left (4 \, x^{3} + x^{2} - 627\right )^{2} \log \left (x\right )^{2} - 4 \, \log \left (4 \, x^{3} + x^{2} - 627\right ) \log \left (x\right )^{3} + \log \left (x\right )^{4} + 6 \, {\left (\log \left (4 \, x^{3} + x^{2} - 627\right )^{2} - 2 \, \log \left (4 \, x^{3} + x^{2} - 627\right ) \log \left (x\right ) + \log \left (x\right )^{2}\right )} \log \left (\frac {4 \, x^{3} + x^{2} - 627}{x}\right )^{2} - 4 \, {\left (\log \left (4 \, x^{3} + x^{2} - 627\right )^{3} - 3 \, \log \left (4 \, x^{3} + x^{2} - 627\right )^{2} \log \left (x\right ) + 3 \, \log \left (4 \, x^{3} + x^{2} - 627\right ) \log \left (x\right )^{2} - \log \left (x\right )^{3}\right )} \log \left (\frac {4 \, x^{3} + x^{2} - 627}{x}\right )\right )} e^{\left (-12\right )} \] Input:
integrate((32*x^3+4*x^2+2508)*log((4*x^3+x^2-627)/x)^3/(324*x^4+81*x^3-507 87*x)/exp(3)^4,x, algorithm="maxima")
Output:
4/81*(log(4*x^3 + x^2 - 627) - log(x))*e^(-12)*log((4*x^3 + x^2 - 627)/x)^ 3 - 1/81*(log(4*x^3 + x^2 - 627)^4 - 4*log(4*x^3 + x^2 - 627)^3*log(x) + 6 *log(4*x^3 + x^2 - 627)^2*log(x)^2 - 4*log(4*x^3 + x^2 - 627)*log(x)^3 + l og(x)^4 + 6*(log(4*x^3 + x^2 - 627)^2 - 2*log(4*x^3 + x^2 - 627)*log(x) + log(x)^2)*log((4*x^3 + x^2 - 627)/x)^2 - 4*(log(4*x^3 + x^2 - 627)^3 - 3*l og(4*x^3 + x^2 - 627)^2*log(x) + 3*log(4*x^3 + x^2 - 627)*log(x)^2 - log(x )^3)*log((4*x^3 + x^2 - 627)/x))*e^(-12)
\[ \int \frac {\left (2508+4 x^2+32 x^3\right ) \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{e^{12} \left (-50787 x+81 x^3+324 x^4\right )} \, dx=\int { \frac {4 \, {\left (8 \, x^{3} + x^{2} + 627\right )} e^{\left (-12\right )} \log \left (\frac {4 \, x^{3} + x^{2} - 627}{x}\right )^{3}}{81 \, {\left (4 \, x^{4} + x^{3} - 627 \, x\right )}} \,d x } \] Input:
integrate((32*x^3+4*x^2+2508)*log((4*x^3+x^2-627)/x)^3/(324*x^4+81*x^3-507 87*x)/exp(3)^4,x, algorithm="giac")
Output:
integrate(4/81*(8*x^3 + x^2 + 627)*e^(-12)*log((4*x^3 + x^2 - 627)/x)^3/(4 *x^4 + x^3 - 627*x), x)
Time = 2.37 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {\left (2508+4 x^2+32 x^3\right ) \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{e^{12} \left (-50787 x+81 x^3+324 x^4\right )} \, dx=\frac {{\ln \left (\frac {4\,x^3+x^2-627}{x}\right )}^4\,{\mathrm {e}}^{-12}}{81} \] Input:
int((log((x^2 + 4*x^3 - 627)/x)^3*exp(-12)*(4*x^2 + 32*x^3 + 2508))/(81*x^ 3 - 50787*x + 324*x^4),x)
Output:
(log((x^2 + 4*x^3 - 627)/x)^4*exp(-12))/81
Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\left (2508+4 x^2+32 x^3\right ) \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{e^{12} \left (-50787 x+81 x^3+324 x^4\right )} \, dx=\frac {\mathrm {log}\left (\frac {4 x^{3}+x^{2}-627}{x}\right )^{4}}{81 e^{12}} \] Input:
int((32*x^3+4*x^2+2508)*log((4*x^3+x^2-627)/x)^3/(324*x^4+81*x^3-50787*x)/ exp(3)^4,x)
Output:
log((4*x**3 + x**2 - 627)/x)**4/(81*e**12)