Integrand size = 44, antiderivative size = 21 \[ \int \frac {(-8-6 x) \log ^2(5) \log (2 x-x \log (3))-3 x \log ^2(5) \log ^2(2 x-x \log (3))}{x} \, dx=(-4-3 x) \log ^2(5) \log ^2(x (2-\log (3))) \] Output:
ln(5)^2*ln(x*(2-ln(3)))^2*(-4-3*x)
Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.71 \[ \int \frac {(-8-6 x) \log ^2(5) \log (2 x-x \log (3))-3 x \log ^2(5) \log ^2(2 x-x \log (3))}{x} \, dx=-4 \log ^2(5) \log ^2(x (2-\log (3)))-3 x \log ^2(5) \log ^2(x (2-\log (3))) \] Input:
Integrate[((-8 - 6*x)*Log[5]^2*Log[2*x - x*Log[3]] - 3*x*Log[5]^2*Log[2*x - x*Log[3]]^2)/x,x]
Output:
-4*Log[5]^2*Log[x*(2 - Log[3])]^2 - 3*x*Log[5]^2*Log[x*(2 - Log[3])]^2
Time = 0.22 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.71, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {2010, 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.
\(\displaystyle \int \frac {(-6 x-8) \log ^2(5) \log (2 x-x \log (3))-3 x \log ^2(5) \log ^2(2 x-x \log (3))}{x} \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (\frac {2 (-3 x-4) \log ^2(5) \log (x (2-\log (3)))}{x}-3 \log ^2(5) \log ^2(x (2-\log (3)))\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -3 x \log ^2(5) \log ^2(x (2-\log (3)))-4 \log ^2(5) \log ^2(x (2-\log (3)))\) |
Input:
Int[((-8 - 6*x)*Log[5]^2*Log[2*x - x*Log[3]] - 3*x*Log[5]^2*Log[2*x - x*Lo g[3]]^2)/x,x]
Output:
-4*Log[5]^2*Log[x*(2 - Log[3])]^2 - 3*x*Log[5]^2*Log[x*(2 - Log[3])]^2
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Time = 0.47 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33
method | result | size |
risch | \(\left (-3 x \ln \left (5\right )^{2}-4 \ln \left (5\right )^{2}\right ) \ln \left (-x \ln \left (3\right )+2 x \right )^{2}\) | \(28\) |
norman | \(-4 \ln \left (5\right )^{2} \ln \left (-x \ln \left (3\right )+2 x \right )^{2}-3 x \ln \left (5\right )^{2} \ln \left (-x \ln \left (3\right )+2 x \right )^{2}\) | \(39\) |
parallelrisch | \(-4 \ln \left (5\right )^{2} \ln \left (-x \ln \left (3\right )+2 x \right )^{2}-3 x \ln \left (5\right )^{2} \ln \left (-x \ln \left (3\right )+2 x \right )^{2}\) | \(39\) |
parts | \(-\frac {3 \ln \left (5\right )^{2} \left (x \left (2-\ln \left (3\right )\right ) \ln \left (x \left (2-\ln \left (3\right )\right )\right )^{2}-2 x \left (2-\ln \left (3\right )\right ) \ln \left (x \left (2-\ln \left (3\right )\right )\right )+2 x \left (2-\ln \left (3\right )\right )\right )}{2-\ln \left (3\right )}-2 \ln \left (5\right )^{2} \left (-\frac {4 \ln \left (x \left (2-\ln \left (3\right )\right )\right )^{2}}{-2+\ln \left (3\right )}+\frac {2 \ln \left (3\right ) \ln \left (x \left (2-\ln \left (3\right )\right )\right )^{2}}{-2+\ln \left (3\right )}-\frac {3 \left (x \left (2-\ln \left (3\right )\right ) \ln \left (x \left (2-\ln \left (3\right )\right )\right )-x \left (2-\ln \left (3\right )\right )\right )}{-2+\ln \left (3\right )}\right )\) | \(145\) |
derivativedivides | \(\frac {3 \ln \left (5\right )^{2} \left (x \left (2-\ln \left (3\right )\right ) \ln \left (x \left (2-\ln \left (3\right )\right )\right )^{2}-2 x \left (2-\ln \left (3\right )\right ) \ln \left (x \left (2-\ln \left (3\right )\right )\right )+2 x \left (2-\ln \left (3\right )\right )\right )}{-2+\ln \left (3\right )}-\frac {4 \ln \left (5\right )^{2} \ln \left (3\right ) \ln \left (x \left (2-\ln \left (3\right )\right )\right )^{2}}{-2+\ln \left (3\right )}+\frac {6 \ln \left (5\right )^{2} \left (x \left (2-\ln \left (3\right )\right ) \ln \left (x \left (2-\ln \left (3\right )\right )\right )-x \left (2-\ln \left (3\right )\right )\right )}{-2+\ln \left (3\right )}+\frac {8 \ln \left (5\right )^{2} \ln \left (x \left (2-\ln \left (3\right )\right )\right )^{2}}{-2+\ln \left (3\right )}\) | \(148\) |
default | \(\frac {3 \ln \left (5\right )^{2} \left (x \left (2-\ln \left (3\right )\right ) \ln \left (x \left (2-\ln \left (3\right )\right )\right )^{2}-2 x \left (2-\ln \left (3\right )\right ) \ln \left (x \left (2-\ln \left (3\right )\right )\right )+2 x \left (2-\ln \left (3\right )\right )\right )}{-2+\ln \left (3\right )}-\frac {4 \ln \left (5\right )^{2} \ln \left (3\right ) \ln \left (x \left (2-\ln \left (3\right )\right )\right )^{2}}{-2+\ln \left (3\right )}+\frac {6 \ln \left (5\right )^{2} \left (x \left (2-\ln \left (3\right )\right ) \ln \left (x \left (2-\ln \left (3\right )\right )\right )-x \left (2-\ln \left (3\right )\right )\right )}{-2+\ln \left (3\right )}+\frac {8 \ln \left (5\right )^{2} \ln \left (x \left (2-\ln \left (3\right )\right )\right )^{2}}{-2+\ln \left (3\right )}\) | \(148\) |
orering | \(\frac {\left (x +\frac {4}{3}\right ) \left (-3 x \ln \left (5\right )^{2} \ln \left (-x \ln \left (3\right )+2 x \right )^{2}+\left (-6 x -8\right ) \ln \left (5\right )^{2} \ln \left (-x \ln \left (3\right )+2 x \right )\right )}{x}+\frac {64 x \left (1+3 x \right ) \left (\frac {-3 \ln \left (5\right )^{2} \ln \left (-x \ln \left (3\right )+2 x \right )^{2}-\frac {6 x \ln \left (5\right )^{2} \ln \left (-x \ln \left (3\right )+2 x \right ) \left (2-\ln \left (3\right )\right )}{-x \ln \left (3\right )+2 x}-6 \ln \left (5\right )^{2} \ln \left (-x \ln \left (3\right )+2 x \right )+\frac {\left (-6 x -8\right ) \ln \left (5\right )^{2} \left (2-\ln \left (3\right )\right )}{-x \ln \left (3\right )+2 x}}{x}-\frac {-3 x \ln \left (5\right )^{2} \ln \left (-x \ln \left (3\right )+2 x \right )^{2}+\left (-6 x -8\right ) \ln \left (5\right )^{2} \ln \left (-x \ln \left (3\right )+2 x \right )}{x^{2}}\right )}{9 x^{2}-48 x +16}+\frac {\left (27 x^{3}+288 x +64\right ) x^{2} \left (\frac {-\frac {12 \ln \left (5\right )^{2} \ln \left (-x \ln \left (3\right )+2 x \right ) \left (2-\ln \left (3\right )\right )}{-x \ln \left (3\right )+2 x}-\frac {6 x \ln \left (5\right )^{2} \left (2-\ln \left (3\right )\right )^{2}}{\left (-x \ln \left (3\right )+2 x \right )^{2}}+\frac {6 x \ln \left (5\right )^{2} \ln \left (-x \ln \left (3\right )+2 x \right ) \left (2-\ln \left (3\right )\right )^{2}}{\left (-x \ln \left (3\right )+2 x \right )^{2}}-\frac {12 \ln \left (5\right )^{2} \left (2-\ln \left (3\right )\right )}{-x \ln \left (3\right )+2 x}-\frac {\left (-6 x -8\right ) \ln \left (5\right )^{2} \left (2-\ln \left (3\right )\right )^{2}}{\left (-x \ln \left (3\right )+2 x \right )^{2}}}{x}-\frac {2 \left (-3 \ln \left (5\right )^{2} \ln \left (-x \ln \left (3\right )+2 x \right )^{2}-\frac {6 x \ln \left (5\right )^{2} \ln \left (-x \ln \left (3\right )+2 x \right ) \left (2-\ln \left (3\right )\right )}{-x \ln \left (3\right )+2 x}-6 \ln \left (5\right )^{2} \ln \left (-x \ln \left (3\right )+2 x \right )+\frac {\left (-6 x -8\right ) \ln \left (5\right )^{2} \left (2-\ln \left (3\right )\right )}{-x \ln \left (3\right )+2 x}\right )}{x^{2}}+\frac {-6 x \ln \left (5\right )^{2} \ln \left (-x \ln \left (3\right )+2 x \right )^{2}+2 \left (-6 x -8\right ) \ln \left (5\right )^{2} \ln \left (-x \ln \left (3\right )+2 x \right )}{x^{3}}\right )}{27 x^{2}-144 x +48}\) | \(542\) |
Input:
int((-3*x*ln(5)^2*ln(-x*ln(3)+2*x)^2+(-6*x-8)*ln(5)^2*ln(-x*ln(3)+2*x))/x, x,method=_RETURNVERBOSE)
Output:
(-3*x*ln(5)^2-4*ln(5)^2)*ln(-x*ln(3)+2*x)^2
Time = 0.09 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {(-8-6 x) \log ^2(5) \log (2 x-x \log (3))-3 x \log ^2(5) \log ^2(2 x-x \log (3))}{x} \, dx=-{\left (3 \, x + 4\right )} \log \left (5\right )^{2} \log \left (-x \log \left (3\right ) + 2 \, x\right )^{2} \] Input:
integrate((-3*x*log(5)^2*log(-x*log(3)+2*x)^2+(-6*x-8)*log(5)^2*log(-x*log (3)+2*x))/x,x, algorithm="fricas")
Output:
-(3*x + 4)*log(5)^2*log(-x*log(3) + 2*x)^2
Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29 \[ \int \frac {(-8-6 x) \log ^2(5) \log (2 x-x \log (3))-3 x \log ^2(5) \log ^2(2 x-x \log (3))}{x} \, dx=\left (- 3 x \log {\left (5 \right )}^{2} - 4 \log {\left (5 \right )}^{2}\right ) \log {\left (- x \log {\left (3 \right )} + 2 x \right )}^{2} \] Input:
integrate((-3*x*ln(5)**2*ln(-x*ln(3)+2*x)**2+(-6*x-8)*ln(5)**2*ln(-x*ln(3) +2*x))/x,x)
Output:
(-3*x*log(5)**2 - 4*log(5)**2)*log(-x*log(3) + 2*x)**2
Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (21) = 42\).
Time = 0.11 (sec) , antiderivative size = 143, normalized size of antiderivative = 6.81 \[ \int \frac {(-8-6 x) \log ^2(5) \log (2 x-x \log (3))-3 x \log ^2(5) \log ^2(2 x-x \log (3))}{x} \, dx=-8 \, \log \left (5\right )^{2} \log \left (-x \log \left (3\right ) + 2 \, x\right ) \log \left (x\right ) + 4 \, {\left (2 \, \log \left (-x \log \left (3\right ) + 2 \, x\right ) \log \left (x\right ) - \log \left (x\right )^{2} - 2 \, \log \left (x\right ) \log \left (-\log \left (3\right ) + 2\right )\right )} \log \left (5\right )^{2} - \frac {3 \, {\left (x \log \left (3\right ) - 2 \, x\right )} {\left (\log \left (-x \log \left (3\right ) + 2 \, x\right )^{2} - 2 \, \log \left (-x \log \left (3\right ) + 2 \, x\right ) + 2\right )} \log \left (5\right )^{2}}{\log \left (3\right ) - 2} + \frac {6 \, {\left (x \log \left (3\right ) - {\left (x \log \left (3\right ) - 2 \, x\right )} \log \left (-x \log \left (3\right ) + 2 \, x\right ) - 2 \, x\right )} \log \left (5\right )^{2}}{\log \left (3\right ) - 2} \] Input:
integrate((-3*x*log(5)^2*log(-x*log(3)+2*x)^2+(-6*x-8)*log(5)^2*log(-x*log (3)+2*x))/x,x, algorithm="maxima")
Output:
-8*log(5)^2*log(-x*log(3) + 2*x)*log(x) + 4*(2*log(-x*log(3) + 2*x)*log(x) - log(x)^2 - 2*log(x)*log(-log(3) + 2))*log(5)^2 - 3*(x*log(3) - 2*x)*(lo g(-x*log(3) + 2*x)^2 - 2*log(-x*log(3) + 2*x) + 2)*log(5)^2/(log(3) - 2) + 6*(x*log(3) - (x*log(3) - 2*x)*log(-x*log(3) + 2*x) - 2*x)*log(5)^2/(log( 3) - 2)
Time = 0.11 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33 \[ \int \frac {(-8-6 x) \log ^2(5) \log (2 x-x \log (3))-3 x \log ^2(5) \log ^2(2 x-x \log (3))}{x} \, dx=-{\left (3 \, x \log \left (5\right )^{2} + 4 \, \log \left (5\right )^{2}\right )} \log \left (-x \log \left (3\right ) + 2 \, x\right )^{2} \] Input:
integrate((-3*x*log(5)^2*log(-x*log(3)+2*x)^2+(-6*x-8)*log(5)^2*log(-x*log (3)+2*x))/x,x, algorithm="giac")
Output:
-(3*x*log(5)^2 + 4*log(5)^2)*log(-x*log(3) + 2*x)^2
Time = 4.33 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {(-8-6 x) \log ^2(5) \log (2 x-x \log (3))-3 x \log ^2(5) \log ^2(2 x-x \log (3))}{x} \, dx=-{\ln \left (5\right )}^2\,{\ln \left (2\,x-x\,\ln \left (3\right )\right )}^2\,\left (3\,x+4\right ) \] Input:
int(-(log(5)^2*log(2*x - x*log(3))*(6*x + 8) + 3*x*log(5)^2*log(2*x - x*lo g(3))^2)/x,x)
Output:
-log(5)^2*log(2*x - x*log(3))^2*(3*x + 4)
Time = 0.17 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \frac {(-8-6 x) \log ^2(5) \log (2 x-x \log (3))-3 x \log ^2(5) \log ^2(2 x-x \log (3))}{x} \, dx=\mathrm {log}\left (-\mathrm {log}\left (3\right ) x +2 x \right )^{2} \mathrm {log}\left (5\right )^{2} \left (-3 x -4\right ) \] Input:
int((-3*x*log(5)^2*log(-x*log(3)+2*x)^2+(-6*x-8)*log(5)^2*log(-x*log(3)+2* x))/x,x)
Output:
log( - log(3)*x + 2*x)**2*log(5)**2*( - 3*x - 4)