Integrand size = 82, antiderivative size = 31 \[ \int \frac {-28 x+4 x^4+6 x^4 \log \left (7-x^3\right )+\left (-14 x+14 x^3+2 x^4-2 x^6\right ) \log ^2\left (7-x^3\right )}{14-2 x^3+\left (-7 x^2+x^5\right ) \log ^2\left (7-x^3\right )} \, dx=-1-x^2+\log \left (-2+x^2 \log ^2\left (9-\frac {2 x+x^4}{x}\right )\right ) \] Output:
ln(ln(9-(x^4+2*x)/x)^2*x^2-2)-x^2-1
Time = 5.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {-28 x+4 x^4+6 x^4 \log \left (7-x^3\right )+\left (-14 x+14 x^3+2 x^4-2 x^6\right ) \log ^2\left (7-x^3\right )}{14-2 x^3+\left (-7 x^2+x^5\right ) \log ^2\left (7-x^3\right )} \, dx=2 \left (-\frac {x^2}{2}+\frac {1}{2} \log \left (2-x^2 \log ^2\left (7-x^3\right )\right )\right ) \] Input:
Integrate[(-28*x + 4*x^4 + 6*x^4*Log[7 - x^3] + (-14*x + 14*x^3 + 2*x^4 - 2*x^6)*Log[7 - x^3]^2)/(14 - 2*x^3 + (-7*x^2 + x^5)*Log[7 - x^3]^2),x]
Output:
2*(-1/2*x^2 + Log[2 - x^2*Log[7 - x^3]^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.
\(\displaystyle \int \frac {4 x^4+6 x^4 \log \left (7-x^3\right )+\left (-2 x^6+2 x^4+14 x^3-14 x\right ) \log ^2\left (7-x^3\right )-28 x}{-2 x^3+\left (x^5-7 x^2\right ) \log ^2\left (7-x^3\right )+14} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {4 x^4+6 x^4 \log \left (7-x^3\right )+\left (-2 x^6+2 x^4+14 x^3-14 x\right ) \log ^2\left (7-x^3\right )-28 x}{\left (7-x^3\right ) \left (2-x^2 \log ^2\left (7-x^3\right )\right )}dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (\frac {2 \left (2 x^3+3 x^5 \log \left (7-x^3\right )-14\right )}{x \left (x^3-7\right ) \left (x^2 \log ^2\left (7-x^3\right )-2\right )}-\frac {2 \left (x^2-1\right )}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 \int \frac {1}{x \left (x^2 \log ^2\left (7-x^3\right )-2\right )}dx-2\ 7^{2/3} \int \frac {\log \left (7-x^3\right )}{\left (\sqrt [3]{7}-x\right ) \left (x^2 \log ^2\left (7-x^3\right )-2\right )}dx+6 \int \frac {x \log \left (7-x^3\right )}{x^2 \log ^2\left (7-x^3\right )-2}dx-2 (-7)^{2/3} \int \frac {\log \left (7-x^3\right )}{\left (\sqrt [3]{-1} x+\sqrt [3]{7}\right ) \left (x^2 \log ^2\left (7-x^3\right )-2\right )}dx+2 \sqrt [3]{-1} 7^{2/3} \int \frac {\log \left (7-x^3\right )}{\left (\sqrt [3]{7}-(-1)^{2/3} x\right ) \left (x^2 \log ^2\left (7-x^3\right )-2\right )}dx-x^2+2 \log (x)\) |
Input:
Int[(-28*x + 4*x^4 + 6*x^4*Log[7 - x^3] + (-14*x + 14*x^3 + 2*x^4 - 2*x^6) *Log[7 - x^3]^2)/(14 - 2*x^3 + (-7*x^2 + x^5)*Log[7 - x^3]^2),x]
Output:
$Aborted
Time = 0.68 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77
method | result | size |
norman | \(-x^{2}+\ln \left (\ln \left (-x^{3}+7\right )^{2} x^{2}-2\right )\) | \(24\) |
parallelrisch | \(-x^{2}+\ln \left (\ln \left (-x^{3}+7\right )^{2} x^{2}-2\right )\) | \(24\) |
risch | \(-x^{2}+2 \ln \left (x \right )+\ln \left (\ln \left (-x^{3}+7\right )^{2}-\frac {2}{x^{2}}\right )\) | \(28\) |
Input:
int(((-2*x^6+2*x^4+14*x^3-14*x)*ln(-x^3+7)^2+6*x^4*ln(-x^3+7)+4*x^4-28*x)/ ((x^5-7*x^2)*ln(-x^3+7)^2-2*x^3+14),x,method=_RETURNVERBOSE)
Output:
-x^2+ln(ln(-x^3+7)^2*x^2-2)
Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {-28 x+4 x^4+6 x^4 \log \left (7-x^3\right )+\left (-14 x+14 x^3+2 x^4-2 x^6\right ) \log ^2\left (7-x^3\right )}{14-2 x^3+\left (-7 x^2+x^5\right ) \log ^2\left (7-x^3\right )} \, dx=-x^{2} + 2 \, \log \left (x\right ) + \log \left (\frac {x^{2} \log \left (-x^{3} + 7\right )^{2} - 2}{x^{2}}\right ) \] Input:
integrate(((-2*x^6+2*x^4+14*x^3-14*x)*log(-x^3+7)^2+6*x^4*log(-x^3+7)+4*x^ 4-28*x)/((x^5-7*x^2)*log(-x^3+7)^2-2*x^3+14),x, algorithm="fricas")
Output:
-x^2 + 2*log(x) + log((x^2*log(-x^3 + 7)^2 - 2)/x^2)
Time = 0.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71 \[ \int \frac {-28 x+4 x^4+6 x^4 \log \left (7-x^3\right )+\left (-14 x+14 x^3+2 x^4-2 x^6\right ) \log ^2\left (7-x^3\right )}{14-2 x^3+\left (-7 x^2+x^5\right ) \log ^2\left (7-x^3\right )} \, dx=- x^{2} + 2 \log {\left (x \right )} + \log {\left (\log {\left (7 - x^{3} \right )}^{2} - \frac {2}{x^{2}} \right )} \] Input:
integrate(((-2*x**6+2*x**4+14*x**3-14*x)*ln(-x**3+7)**2+6*x**4*ln(-x**3+7) +4*x**4-28*x)/((x**5-7*x**2)*ln(-x**3+7)**2-2*x**3+14),x)
Output:
-x**2 + 2*log(x) + log(log(7 - x**3)**2 - 2/x**2)
Time = 0.15 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {-28 x+4 x^4+6 x^4 \log \left (7-x^3\right )+\left (-14 x+14 x^3+2 x^4-2 x^6\right ) \log ^2\left (7-x^3\right )}{14-2 x^3+\left (-7 x^2+x^5\right ) \log ^2\left (7-x^3\right )} \, dx=-x^{2} + 2 \, \log \left (x\right ) + \log \left (\frac {x^{2} \log \left (-x^{3} + 7\right )^{2} - 2}{x^{2}}\right ) \] Input:
integrate(((-2*x^6+2*x^4+14*x^3-14*x)*log(-x^3+7)^2+6*x^4*log(-x^3+7)+4*x^ 4-28*x)/((x^5-7*x^2)*log(-x^3+7)^2-2*x^3+14),x, algorithm="maxima")
Output:
-x^2 + 2*log(x) + log((x^2*log(-x^3 + 7)^2 - 2)/x^2)
Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int \frac {-28 x+4 x^4+6 x^4 \log \left (7-x^3\right )+\left (-14 x+14 x^3+2 x^4-2 x^6\right ) \log ^2\left (7-x^3\right )}{14-2 x^3+\left (-7 x^2+x^5\right ) \log ^2\left (7-x^3\right )} \, dx=-x^{2} + \log \left (x^{2} \log \left (-x^{3} + 7\right )^{2} - 2\right ) \] Input:
integrate(((-2*x^6+2*x^4+14*x^3-14*x)*log(-x^3+7)^2+6*x^4*log(-x^3+7)+4*x^ 4-28*x)/((x^5-7*x^2)*log(-x^3+7)^2-2*x^3+14),x, algorithm="giac")
Output:
-x^2 + log(x^2*log(-x^3 + 7)^2 - 2)
Time = 1.90 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {-28 x+4 x^4+6 x^4 \log \left (7-x^3\right )+\left (-14 x+14 x^3+2 x^4-2 x^6\right ) \log ^2\left (7-x^3\right )}{14-2 x^3+\left (-7 x^2+x^5\right ) \log ^2\left (7-x^3\right )} \, dx=\ln \left (\frac {x^2\,{\ln \left (7-x^3\right )}^2-2}{x^2}\right )+2\,\ln \left (x\right )-x^2 \] Input:
int((28*x - 6*x^4*log(7 - x^3) + log(7 - x^3)^2*(14*x - 14*x^3 - 2*x^4 + 2 *x^6) - 4*x^4)/(log(7 - x^3)^2*(7*x^2 - x^5) + 2*x^3 - 14),x)
Output:
log((x^2*log(7 - x^3)^2 - 2)/x^2) + 2*log(x) - x^2
Time = 0.21 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {-28 x+4 x^4+6 x^4 \log \left (7-x^3\right )+\left (-14 x+14 x^3+2 x^4-2 x^6\right ) \log ^2\left (7-x^3\right )}{14-2 x^3+\left (-7 x^2+x^5\right ) \log ^2\left (7-x^3\right )} \, dx=\mathrm {log}\left (-\sqrt {2}+\mathrm {log}\left (-x^{3}+7\right ) x \right )+\mathrm {log}\left (\sqrt {2}+\mathrm {log}\left (-x^{3}+7\right ) x \right )-x^{2} \] Input:
int(((-2*x^6+2*x^4+14*x^3-14*x)*log(-x^3+7)^2+6*x^4*log(-x^3+7)+4*x^4-28*x )/((x^5-7*x^2)*log(-x^3+7)^2-2*x^3+14),x)
Output:
log( - sqrt(2) + log( - x**3 + 7)*x) + log(sqrt(2) + log( - x**3 + 7)*x) - x**2