Integrand size = 101, antiderivative size = 25 \[ \int \frac {\left (7 x-7 x^3\right ) \log (x)+\left (28 x-28 x^2\right ) \log ^2(x)+\left (2-2 x^2+4 x^2 \log (x)+8 x \log ^2(x)\right ) \log \left (\frac {1-x^2+(4-4 x) \log (x)}{4 \log (x)}\right )}{\left (-x+x^3\right ) \log (x)+\left (-4 x+4 x^2\right ) \log ^2(x)} \, dx=-7 x+\log ^2\left (1-x-\frac {-1+x^2}{4 \log (x)}\right ) \] Output:
ln(1-x-1/4*(x^2-1)/ln(x))^2-7*x
Time = 0.05 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {\left (7 x-7 x^3\right ) \log (x)+\left (28 x-28 x^2\right ) \log ^2(x)+\left (2-2 x^2+4 x^2 \log (x)+8 x \log ^2(x)\right ) \log \left (\frac {1-x^2+(4-4 x) \log (x)}{4 \log (x)}\right )}{\left (-x+x^3\right ) \log (x)+\left (-4 x+4 x^2\right ) \log ^2(x)} \, dx=-7 x+\log ^2\left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right ) \] Input:
Integrate[((7*x - 7*x^3)*Log[x] + (28*x - 28*x^2)*Log[x]^2 + (2 - 2*x^2 + 4*x^2*Log[x] + 8*x*Log[x]^2)*Log[(1 - x^2 + (4 - 4*x)*Log[x])/(4*Log[x])]) /((-x + x^3)*Log[x] + (-4*x + 4*x^2)*Log[x]^2),x]
Output:
-7*x + Log[-1/4*((-1 + x)*(1 + x + 4*Log[x]))/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.
\(\displaystyle \int \frac {\left (7 x-7 x^3\right ) \log (x)+\left (28 x-28 x^2\right ) \log ^2(x)+\left (-2 x^2+4 x^2 \log (x)+8 x \log ^2(x)+2\right ) \log \left (\frac {-x^2+(4-4 x) \log (x)+1}{4 \log (x)}\right )}{\left (x^3-x\right ) \log (x)+\left (4 x^2-4 x\right ) \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-\left (7 x-7 x^3\right ) \log (x)-\left (\left (28 x-28 x^2\right ) \log ^2(x)\right )-\left (-2 x^2+4 x^2 \log (x)+8 x \log ^2(x)+2\right ) \log \left (\frac {-x^2+(4-4 x) \log (x)+1}{4 \log (x)}\right )}{(1-x) x \log (x) (x+4 \log (x)+1)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 \left (-x^2+2 x^2 \log (x)+4 x \log ^2(x)+1\right ) \log \left (-\frac {(x-1) (x+4 \log (x)+1)}{4 \log (x)}\right )}{(x-1) x \log (x) (x+4 \log (x)+1)}-7\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 \int \frac {\log \left (-\frac {(x-1) (x+4 \log (x)+1)}{4 \log (x)}\right )}{x+4 \log (x)+1}dx+4 \int \frac {\log \left (-\frac {(x-1) (x+4 \log (x)+1)}{4 \log (x)}\right )}{(x-1) (x+4 \log (x)+1)}dx-2 \int \frac {\log \left (-\frac {(x-1) (x+4 \log (x)+1)}{4 \log (x)}\right )}{\log (x) (x+4 \log (x)+1)}dx-2 \int \frac {\log \left (-\frac {(x-1) (x+4 \log (x)+1)}{4 \log (x)}\right )}{x \log (x) (x+4 \log (x)+1)}dx+8 \int \frac {\log (x) \log \left (-\frac {(x-1) (x+4 \log (x)+1)}{4 \log (x)}\right )}{(x-1) (x+4 \log (x)+1)}dx-7 x\) |
Input:
Int[((7*x - 7*x^3)*Log[x] + (28*x - 28*x^2)*Log[x]^2 + (2 - 2*x^2 + 4*x^2* Log[x] + 8*x*Log[x]^2)*Log[(1 - x^2 + (4 - 4*x)*Log[x])/(4*Log[x])])/((-x + x^3)*Log[x] + (-4*x + 4*x^2)*Log[x]^2),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(50\) vs. \(2(23)=46\).
Time = 1.53 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.04
method | result | size |
default | \(-7 x -4 \ln \left (2\right ) \left (-\ln \left (\ln \left (x \right )\right )+\ln \left (-1+x \right )+\ln \left (x +4 \ln \left (x \right )+1\right )\right )+\ln \left (\frac {-4 x \ln \left (x \right )-x^{2}+4 \ln \left (x \right )+1}{\ln \left (x \right )}\right )^{2}\) | \(51\) |
Input:
int(((8*x*ln(x)^2+4*x^2*ln(x)-2*x^2+2)*ln(1/4*((4-4*x)*ln(x)-x^2+1)/ln(x)) +(-28*x^2+28*x)*ln(x)^2+(-7*x^3+7*x)*ln(x))/((4*x^2-4*x)*ln(x)^2+(x^3-x)*l n(x)),x,method=_RETURNVERBOSE)
Output:
-7*x-4*ln(2)*(-ln(ln(x))+ln(-1+x)+ln(x+4*ln(x)+1))+ln((-4*x*ln(x)-x^2+4*ln (x)+1)/ln(x))^2
Time = 0.09 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {\left (7 x-7 x^3\right ) \log (x)+\left (28 x-28 x^2\right ) \log ^2(x)+\left (2-2 x^2+4 x^2 \log (x)+8 x \log ^2(x)\right ) \log \left (\frac {1-x^2+(4-4 x) \log (x)}{4 \log (x)}\right )}{\left (-x+x^3\right ) \log (x)+\left (-4 x+4 x^2\right ) \log ^2(x)} \, dx=\log \left (-\frac {x^{2} + 4 \, {\left (x - 1\right )} \log \left (x\right ) - 1}{4 \, \log \left (x\right )}\right )^{2} - 7 \, x \] Input:
integrate(((8*x*log(x)^2+4*x^2*log(x)-2*x^2+2)*log(1/4*((4-4*x)*log(x)-x^2 +1)/log(x))+(-28*x^2+28*x)*log(x)^2+(-7*x^3+7*x)*log(x))/((4*x^2-4*x)*log( x)^2+(x^3-x)*log(x)),x, algorithm="fricas")
Output:
log(-1/4*(x^2 + 4*(x - 1)*log(x) - 1)/log(x))^2 - 7*x
Time = 0.21 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\left (7 x-7 x^3\right ) \log (x)+\left (28 x-28 x^2\right ) \log ^2(x)+\left (2-2 x^2+4 x^2 \log (x)+8 x \log ^2(x)\right ) \log \left (\frac {1-x^2+(4-4 x) \log (x)}{4 \log (x)}\right )}{\left (-x+x^3\right ) \log (x)+\left (-4 x+4 x^2\right ) \log ^2(x)} \, dx=- 7 x + \log {\left (\frac {- \frac {x^{2}}{4} + \frac {\left (4 - 4 x\right ) \log {\left (x \right )}}{4} + \frac {1}{4}}{\log {\left (x \right )}} \right )}^{2} \] Input:
integrate(((8*x*ln(x)**2+4*x**2*ln(x)-2*x**2+2)*ln(1/4*((4-4*x)*ln(x)-x**2 +1)/ln(x))+(-28*x**2+28*x)*ln(x)**2+(-7*x**3+7*x)*ln(x))/((4*x**2-4*x)*ln( x)**2+(x**3-x)*ln(x)),x)
Output:
-7*x + log((-x**2/4 + (4 - 4*x)*log(x)/4 + 1/4)/log(x))**2
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (23) = 46\).
Time = 0.16 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.04 \[ \int \frac {\left (7 x-7 x^3\right ) \log (x)+\left (28 x-28 x^2\right ) \log ^2(x)+\left (2-2 x^2+4 x^2 \log (x)+8 x \log ^2(x)\right ) \log \left (\frac {1-x^2+(4-4 x) \log (x)}{4 \log (x)}\right )}{\left (-x+x^3\right ) \log (x)+\left (-4 x+4 x^2\right ) \log ^2(x)} \, dx=-2 \, {\left (2 \, \log \left (2\right ) - \log \left (-x + 1\right ) + \log \left (\log \left (x\right )\right )\right )} \log \left (x + 4 \, \log \left (x\right ) + 1\right ) + \log \left (x + 4 \, \log \left (x\right ) + 1\right )^{2} - 2 \, {\left (2 \, \log \left (2\right ) + \log \left (\log \left (x\right )\right )\right )} \log \left (-x + 1\right ) + \log \left (-x + 1\right )^{2} + 4 \, \log \left (2\right ) \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2} - 7 \, x \] Input:
integrate(((8*x*log(x)^2+4*x^2*log(x)-2*x^2+2)*log(1/4*((4-4*x)*log(x)-x^2 +1)/log(x))+(-28*x^2+28*x)*log(x)^2+(-7*x^3+7*x)*log(x))/((4*x^2-4*x)*log( x)^2+(x^3-x)*log(x)),x, algorithm="maxima")
Output:
-2*(2*log(2) - log(-x + 1) + log(log(x)))*log(x + 4*log(x) + 1) + log(x + 4*log(x) + 1)^2 - 2*(2*log(2) + log(log(x)))*log(-x + 1) + log(-x + 1)^2 + 4*log(2)*log(log(x)) + log(log(x))^2 - 7*x
Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (23) = 46\).
Time = 0.18 (sec) , antiderivative size = 148, normalized size of antiderivative = 5.92 \[ \int \frac {\left (7 x-7 x^3\right ) \log (x)+\left (28 x-28 x^2\right ) \log ^2(x)+\left (2-2 x^2+4 x^2 \log (x)+8 x \log ^2(x)\right ) \log \left (\frac {1-x^2+(4-4 x) \log (x)}{4 \log (x)}\right )}{\left (-x+x^3\right ) \log (x)+\left (-4 x+4 x^2\right ) \log ^2(x)} \, dx=2 \, {\left (\log \left (x + 4 \, \log \left (x\right ) + 1\right ) + \log \left (x - 1\right ) - \log \left (\log \left (x\right )\right )\right )} \log \left (-x^{2} - 4 \, x \log \left (x\right ) + 4 \, \log \left (x\right ) + 1\right ) - 2 \, {\left (\log \left (x + 4 \, \log \left (x\right ) + 1\right ) + \log \left (x - 1\right )\right )} \log \left (x + 4 \, \log \left (x\right ) + 1\right ) - 4 \, \log \left (2\right ) \log \left (x + 4 \, \log \left (x\right ) + 1\right ) + 2 \, \log \left (x + 4 \, \log \left (x\right ) + 1\right )^{2} - 4 \, \log \left (2\right ) \log \left (x - 1\right ) - \log \left (x - 1\right )^{2} - 2 \, \log \left (x + 4 \, \log \left (x\right ) + 1\right ) \log \left (-x - 4 \, \log \left (x\right ) - 1\right ) + \log \left (-x - 4 \, \log \left (x\right ) - 1\right )^{2} + 4 \, \log \left (2\right ) \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2} - 7 \, x \] Input:
integrate(((8*x*log(x)^2+4*x^2*log(x)-2*x^2+2)*log(1/4*((4-4*x)*log(x)-x^2 +1)/log(x))+(-28*x^2+28*x)*log(x)^2+(-7*x^3+7*x)*log(x))/((4*x^2-4*x)*log( x)^2+(x^3-x)*log(x)),x, algorithm="giac")
Output:
2*(log(x + 4*log(x) + 1) + log(x - 1) - log(log(x)))*log(-x^2 - 4*x*log(x) + 4*log(x) + 1) - 2*(log(x + 4*log(x) + 1) + log(x - 1))*log(x + 4*log(x) + 1) - 4*log(2)*log(x + 4*log(x) + 1) + 2*log(x + 4*log(x) + 1)^2 - 4*log (2)*log(x - 1) - log(x - 1)^2 - 2*log(x + 4*log(x) + 1)*log(-x - 4*log(x) - 1) + log(-x - 4*log(x) - 1)^2 + 4*log(2)*log(log(x)) + log(log(x))^2 - 7 *x
Time = 9.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {\left (7 x-7 x^3\right ) \log (x)+\left (28 x-28 x^2\right ) \log ^2(x)+\left (2-2 x^2+4 x^2 \log (x)+8 x \log ^2(x)\right ) \log \left (\frac {1-x^2+(4-4 x) \log (x)}{4 \log (x)}\right )}{\left (-x+x^3\right ) \log (x)+\left (-4 x+4 x^2\right ) \log ^2(x)} \, dx={\ln \left (-\frac {\frac {\ln \left (x\right )\,\left (4\,x-4\right )}{4}+\frac {x^2}{4}-\frac {1}{4}}{\ln \left (x\right )}\right )}^2-7\,x \] Input:
int(-(log(x)^2*(28*x - 28*x^2) + log(-((log(x)*(4*x - 4))/4 + x^2/4 - 1/4) /log(x))*(8*x*log(x)^2 + 4*x^2*log(x) - 2*x^2 + 2) + log(x)*(7*x - 7*x^3)) /(log(x)^2*(4*x - 4*x^2) + log(x)*(x - x^3)),x)
Output:
log(-((log(x)*(4*x - 4))/4 + x^2/4 - 1/4)/log(x))^2 - 7*x
Time = 0.16 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {\left (7 x-7 x^3\right ) \log (x)+\left (28 x-28 x^2\right ) \log ^2(x)+\left (2-2 x^2+4 x^2 \log (x)+8 x \log ^2(x)\right ) \log \left (\frac {1-x^2+(4-4 x) \log (x)}{4 \log (x)}\right )}{\left (-x+x^3\right ) \log (x)+\left (-4 x+4 x^2\right ) \log ^2(x)} \, dx=\mathrm {log}\left (\frac {-4 \,\mathrm {log}\left (x \right ) x +4 \,\mathrm {log}\left (x \right )-x^{2}+1}{4 \,\mathrm {log}\left (x \right )}\right )^{2}-7 x \] Input:
int(((8*x*log(x)^2+4*x^2*log(x)-2*x^2+2)*log(1/4*((4-4*x)*log(x)-x^2+1)/lo g(x))+(-28*x^2+28*x)*log(x)^2+(-7*x^3+7*x)*log(x))/((4*x^2-4*x)*log(x)^2+( x^3-x)*log(x)),x)
Output:
log(( - 4*log(x)*x + 4*log(x) - x**2 + 1)/(4*log(x)))**2 - 7*x