Integrand size = 132, antiderivative size = 23 \[ \int \frac {-36-12 x^2 \log ^2(3)+\left (-36 x \log (3)-4 x^3 \log ^3(3)\right ) \log (4)+\left (-24-12 x^2 \log ^2(3)\right ) \log ^2(4)-12 x \log (3) \log ^3(4)-4 \log ^4(4)+\left (-12-4 x^2 \log ^2(3)-8 x \log (3) \log (4)-4 \log ^2(4)\right ) \log (x)+\left (24+4 x^2 \log ^2(3)+12 x \log (3) \log (4)+8 \log ^2(4)\right ) \log ^2(x)+4 \log ^3(x)-4 \log ^4(x)}{x^5} \, dx=\frac {\left (3+(x \log (3)+\log (4))^2-\log ^2(x)\right )^2}{x^4} \] Output:
((x*ln(3)+2*ln(2))^2+3-ln(x)^2)^2/x^4
Leaf count is larger than twice the leaf count of optimal. \(184\) vs. \(2(23)=46\).
Time = 0.04 (sec) , antiderivative size = 184, normalized size of antiderivative = 8.00 \[ \int \frac {-36-12 x^2 \log ^2(3)+\left (-36 x \log (3)-4 x^3 \log ^3(3)\right ) \log (4)+\left (-24-12 x^2 \log ^2(3)\right ) \log ^2(4)-12 x \log (3) \log ^3(4)-4 \log ^4(4)+\left (-12-4 x^2 \log ^2(3)-8 x \log (3) \log (4)-4 \log ^2(4)\right ) \log (x)+\left (24+4 x^2 \log ^2(3)+12 x \log (3) \log (4)+8 \log ^2(4)\right ) \log ^2(x)+4 \log ^3(x)-4 \log ^4(x)}{x^5} \, dx=\frac {9}{x^4}+\frac {6 \log ^2(3)}{x^2}+\frac {12 \log (3) \log (4)}{x^3}+\frac {4 \log ^3(3) \log (4)}{x}+\frac {6 \log ^2(4)}{x^4}+\frac {6 \log ^2(3) \log ^2(4)}{x^2}+\frac {\log ^4(4)}{x^4}+\frac {4 \log (4) \log (9)}{9 x^3}-\frac {8 \log (4) \log (27)}{27 x^3}+\frac {4 \log ^3(4) \log (27)}{3 x^3}+\frac {4 \log (4) \log (9) \log (x)}{3 x^3}-\frac {8 \log (4) \log (27) \log (x)}{9 x^3}-\frac {6 \log ^2(x)}{x^4}-\frac {2 \log ^2(3) \log ^2(x)}{x^2}-\frac {2 \log ^2(4) \log ^2(x)}{x^4}-\frac {4 \log (4) \log (27) \log ^2(x)}{3 x^3}+\frac {\log ^4(x)}{x^4} \] Input:
Integrate[(-36 - 12*x^2*Log[3]^2 + (-36*x*Log[3] - 4*x^3*Log[3]^3)*Log[4] + (-24 - 12*x^2*Log[3]^2)*Log[4]^2 - 12*x*Log[3]*Log[4]^3 - 4*Log[4]^4 + ( -12 - 4*x^2*Log[3]^2 - 8*x*Log[3]*Log[4] - 4*Log[4]^2)*Log[x] + (24 + 4*x^ 2*Log[3]^2 + 12*x*Log[3]*Log[4] + 8*Log[4]^2)*Log[x]^2 + 4*Log[x]^3 - 4*Lo g[x]^4)/x^5,x]
Output:
9/x^4 + (6*Log[3]^2)/x^2 + (12*Log[3]*Log[4])/x^3 + (4*Log[3]^3*Log[4])/x + (6*Log[4]^2)/x^4 + (6*Log[3]^2*Log[4]^2)/x^2 + Log[4]^4/x^4 + (4*Log[4]* Log[9])/(9*x^3) - (8*Log[4]*Log[27])/(27*x^3) + (4*Log[4]^3*Log[27])/(3*x^ 3) + (4*Log[4]*Log[9]*Log[x])/(3*x^3) - (8*Log[4]*Log[27]*Log[x])/(9*x^3) - (6*Log[x]^2)/x^4 - (2*Log[3]^2*Log[x]^2)/x^2 - (2*Log[4]^2*Log[x]^2)/x^4 - (4*Log[4]*Log[27]*Log[x]^2)/(3*x^3) + Log[x]^4/x^4
Leaf count is larger than twice the leaf count of optimal. \(153\) vs. \(2(23)=46\).
Time = 0.66 (sec) , antiderivative size = 153, normalized size of antiderivative = 6.65, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.015, 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.
| \(\displaystyle \int \frac {\log (4) \left (-4 x^3 \log ^3(3)-36 x \log (3)\right )+\left (4 x^2 \log ^2(3)+12 x \log (3) \log (4)+24+8 \log ^2(4)\right ) \log ^2(x)+\left (-4 x^2 \log ^2(3)-8 x \log (3) \log (4)-12-4 \log ^2(4)\right ) \log (x)+\log ^2(4) \left (-12 x^2 \log ^2(3)-24\right )-12 x^2 \log ^2(3)-4 \log ^4(x)+4 \log ^3(x)-12 x \log (3) \log ^3(4)-36-4 \log ^4(4)}{x^5} \, dx\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \int \left (-\frac {4 \log ^4(x)}{x^5}+\frac {4 \log ^3(x)}{x^5}+\frac {4 \left (x^2 \log ^2(3)+x \log (4) \log (27)+2 \left (3+\log ^2(4)\right )\right ) \log ^2(x)}{x^5}-\frac {4 \left (x^2 \log ^2(3)+x \log (4) \log (9)+3+\log ^2(4)\right ) \log (x)}{x^5}+\frac {4 \left (x^3 \left (-\log ^3(3)\right ) \log (4)-3 x^2 \log ^2(3) \left (1+\log ^2(4)\right )-3 x \log (3) \log (4) \left (3+\log ^2(4)\right )-\left (3+\log ^2(4)\right )^2\right )}{x^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\log ^4(x)}{x^4}-\frac {2 \left (3+\log ^2(4)\right ) \log ^2(x)}{x^4}+\frac {\left (3+\log ^2(4)\right )^2}{x^4}-\frac {4 \log (4) \log (27) \log ^2(x)}{3 x^3}+\frac {4 \log (3) \log (4) \left (3+\log ^2(4)\right )}{x^3}-\frac {8 \log (4) \log (27) \log (x)}{9 x^3}+\frac {4 \log (4) \log (9) \log (x)}{3 x^3}-\frac {8 \log (4) \log (27)}{27 x^3}+\frac {4 \log (4) \log (9)}{9 x^3}-\frac {2 \log ^2(3) \log ^2(x)}{x^2}+\frac {6 \log ^2(3) \left (1+\log ^2(4)\right )}{x^2}+\frac {4 \log ^3(3) \log (4)}{x}\) |
Input:
Int[(-36 - 12*x^2*Log[3]^2 + (-36*x*Log[3] - 4*x^3*Log[3]^3)*Log[4] + (-24 - 12*x^2*Log[3]^2)*Log[4]^2 - 12*x*Log[3]*Log[4]^3 - 4*Log[4]^4 + (-12 - 4*x^2*Log[3]^2 - 8*x*Log[3]*Log[4] - 4*Log[4]^2)*Log[x] + (24 + 4*x^2*Log[ 3]^2 + 12*x*Log[3]*Log[4] + 8*Log[4]^2)*Log[x]^2 + 4*Log[x]^3 - 4*Log[x]^4 )/x^5,x]
Output:
(4*Log[3]^3*Log[4])/x + (6*Log[3]^2*(1 + Log[4]^2))/x^2 + (4*Log[3]*Log[4] *(3 + Log[4]^2))/x^3 + (3 + Log[4]^2)^2/x^4 + (4*Log[4]*Log[9])/(9*x^3) - (8*Log[4]*Log[27])/(27*x^3) + (4*Log[4]*Log[9]*Log[x])/(3*x^3) - (8*Log[4] *Log[27]*Log[x])/(9*x^3) - (2*Log[3]^2*Log[x]^2)/x^2 - (2*(3 + Log[4]^2)*L og[x]^2)/x^4 - (4*Log[4]*Log[27]*Log[x]^2)/(3*x^3) + Log[x]^4/x^4
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]]
Leaf count of result is larger than twice the leaf count of optimal. \(108\) vs. \(2(25)=50\).
Time = 11.91 (sec) , antiderivative size = 109, normalized size of antiderivative = 4.74
| method | result | size |
| risch | \(\frac {\ln \left (x \right )^{4}}{x^{4}}-\frac {2 \left (x^{2} \ln \left (3\right )^{2}+4 x \ln \left (2\right ) \ln \left (3\right )+4 \ln \left (2\right )^{2}+3\right ) \ln \left (x \right )^{2}}{x^{4}}+\frac {8 \ln \left (3\right )^{3} \ln \left (2\right ) x^{3}+24 x^{2} \ln \left (3\right )^{2} \ln \left (2\right )^{2}+32 x \ln \left (3\right ) \ln \left (2\right )^{3}+6 x^{2} \ln \left (3\right )^{2}+16 \ln \left (2\right )^{4}+24 x \ln \left (2\right ) \ln \left (3\right )+24 \ln \left (2\right )^{2}+9}{x^{4}}\) | \(109\) |
| parallelrisch | \(\frac {8 \ln \left (3\right )^{3} \ln \left (2\right ) x^{3}+24 x^{2} \ln \left (3\right )^{2} \ln \left (2\right )^{2}-2 \ln \left (x \right )^{2} \ln \left (3\right )^{2} x^{2}+32 x \ln \left (3\right ) \ln \left (2\right )^{3}-8 \ln \left (x \right )^{2} \ln \left (3\right ) \ln \left (2\right ) x +9+6 x^{2} \ln \left (3\right )^{2}+16 \ln \left (2\right )^{4}-8 \ln \left (2\right )^{2} \ln \left (x \right )^{2}+\ln \left (x \right )^{4}+24 x \ln \left (2\right ) \ln \left (3\right )+24 \ln \left (2\right )^{2}-6 \ln \left (x \right )^{2}}{x^{4}}\) | \(112\) |
| parts | \(\frac {\ln \left (x \right )^{4}}{x^{4}}+\frac {6 \ln \left (3\right )^{2} \left (4 \ln \left (2\right )^{2}+1\right )}{x^{2}}+\frac {16 \ln \left (2\right )^{4}+24 \ln \left (2\right )^{2}+9}{x^{4}}+\frac {8 \ln \left (3\right )^{3} \ln \left (2\right )}{x}+\frac {8 \ln \left (2\right ) \ln \left (3\right ) \left (4 \ln \left (2\right )^{2}+3\right )}{x^{3}}-4 \ln \left (3\right )^{2} \left (-\frac {\ln \left (x \right )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )-16 \ln \left (2\right ) \ln \left (3\right ) \left (-\frac {\ln \left (x \right )}{3 x^{3}}-\frac {1}{9 x^{3}}\right )-16 \ln \left (2\right )^{2} \left (-\frac {\ln \left (x \right )}{4 x^{4}}-\frac {1}{16 x^{4}}\right )+4 \ln \left (3\right )^{2} \left (-\frac {\ln \left (x \right )^{2}}{2 x^{2}}-\frac {\ln \left (x \right )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )+24 \ln \left (2\right ) \ln \left (3\right ) \left (-\frac {\ln \left (x \right )^{2}}{3 x^{3}}-\frac {2 \ln \left (x \right )}{9 x^{3}}-\frac {2}{27 x^{3}}\right )+32 \ln \left (2\right )^{2} \left (-\frac {\ln \left (x \right )^{2}}{4 x^{4}}-\frac {\ln \left (x \right )}{8 x^{4}}-\frac {1}{32 x^{4}}\right )-\frac {6 \ln \left (x \right )^{2}}{x^{4}}\) | \(223\) |
| default | \(\frac {9}{x^{4}}+\frac {24 \ln \left (2\right )^{2}}{x^{4}}-16 \ln \left (2\right )^{2} \left (-\frac {\ln \left (x \right )}{4 x^{4}}-\frac {1}{16 x^{4}}\right )+\frac {16 \ln \left (2\right )^{4}}{x^{4}}+32 \ln \left (2\right )^{2} \left (-\frac {\ln \left (x \right )^{2}}{4 x^{4}}-\frac {\ln \left (x \right )}{8 x^{4}}-\frac {1}{32 x^{4}}\right )-4 \ln \left (3\right )^{2} \left (-\frac {\ln \left (x \right )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )+\frac {\ln \left (x \right )^{4}}{x^{4}}-\frac {6 \ln \left (x \right )^{2}}{x^{4}}+4 \ln \left (3\right )^{2} \left (-\frac {\ln \left (x \right )^{2}}{2 x^{2}}-\frac {\ln \left (x \right )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )+\frac {6 \ln \left (3\right )^{2}}{x^{2}}+\frac {24 \ln \left (2\right ) \ln \left (3\right )}{x^{3}}-16 \ln \left (2\right ) \ln \left (3\right ) \left (-\frac {\ln \left (x \right )}{3 x^{3}}-\frac {1}{9 x^{3}}\right )+\frac {32 \ln \left (3\right ) \ln \left (2\right )^{3}}{x^{3}}+24 \ln \left (2\right ) \ln \left (3\right ) \left (-\frac {\ln \left (x \right )^{2}}{3 x^{3}}-\frac {2 \ln \left (x \right )}{9 x^{3}}-\frac {2}{27 x^{3}}\right )+\frac {24 \ln \left (3\right )^{2} \ln \left (2\right )^{2}}{x^{2}}+\frac {8 \ln \left (3\right )^{3} \ln \left (2\right )}{x}\) | \(236\) |
Input:
int((-4*ln(x)^4+4*ln(x)^3+(32*ln(2)^2+24*x*ln(2)*ln(3)+4*x^2*ln(3)^2+24)*l n(x)^2+(-16*ln(2)^2-16*x*ln(2)*ln(3)-4*x^2*ln(3)^2-12)*ln(x)-64*ln(2)^4-96 *x*ln(3)*ln(2)^3+4*(-12*x^2*ln(3)^2-24)*ln(2)^2+2*(-4*x^3*ln(3)^3-36*x*ln( 3))*ln(2)-12*x^2*ln(3)^2-36)/x^5,x,method=_RETURNVERBOSE)
Output:
1/x^4*ln(x)^4-2*(x^2*ln(3)^2+4*x*ln(2)*ln(3)+4*ln(2)^2+3)/x^4*ln(x)^2+(8*l n(3)^3*ln(2)*x^3+24*x^2*ln(3)^2*ln(2)^2+32*x*ln(3)*ln(2)^3+6*x^2*ln(3)^2+1 6*ln(2)^4+24*x*ln(2)*ln(3)+24*ln(2)^2+9)/x^4
Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (25) = 50\).
Time = 0.11 (sec) , antiderivative size = 98, normalized size of antiderivative = 4.26 \[ \int \frac {-36-12 x^2 \log ^2(3)+\left (-36 x \log (3)-4 x^3 \log ^3(3)\right ) \log (4)+\left (-24-12 x^2 \log ^2(3)\right ) \log ^2(4)-12 x \log (3) \log ^3(4)-4 \log ^4(4)+\left (-12-4 x^2 \log ^2(3)-8 x \log (3) \log (4)-4 \log ^2(4)\right ) \log (x)+\left (24+4 x^2 \log ^2(3)+12 x \log (3) \log (4)+8 \log ^2(4)\right ) \log ^2(x)+4 \log ^3(x)-4 \log ^4(x)}{x^5} \, dx=\frac {8 \, x^{3} \log \left (3\right )^{3} \log \left (2\right ) + 16 \, \log \left (2\right )^{4} + \log \left (x\right )^{4} + 6 \, {\left (4 \, x^{2} \log \left (2\right )^{2} + x^{2}\right )} \log \left (3\right )^{2} - 2 \, {\left (x^{2} \log \left (3\right )^{2} + 4 \, x \log \left (3\right ) \log \left (2\right ) + 4 \, \log \left (2\right )^{2} + 3\right )} \log \left (x\right )^{2} + 8 \, {\left (4 \, x \log \left (2\right )^{3} + 3 \, x \log \left (2\right )\right )} \log \left (3\right ) + 24 \, \log \left (2\right )^{2} + 9}{x^{4}} \] Input:
integrate((-4*log(x)^4+4*log(x)^3+(32*log(2)^2+24*x*log(2)*log(3)+4*x^2*lo g(3)^2+24)*log(x)^2+(-16*log(2)^2-16*x*log(2)*log(3)-4*x^2*log(3)^2-12)*lo g(x)-64*log(2)^4-96*x*log(3)*log(2)^3+4*(-12*x^2*log(3)^2-24)*log(2)^2+2*( -4*x^3*log(3)^3-36*x*log(3))*log(2)-12*x^2*log(3)^2-36)/x^5,x, algorithm=" fricas")
Output:
(8*x^3*log(3)^3*log(2) + 16*log(2)^4 + log(x)^4 + 6*(4*x^2*log(2)^2 + x^2) *log(3)^2 - 2*(x^2*log(3)^2 + 4*x*log(3)*log(2) + 4*log(2)^2 + 3)*log(x)^2 + 8*(4*x*log(2)^3 + 3*x*log(2))*log(3) + 24*log(2)^2 + 9)/x^4
Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (22) = 44\).
Time = 1.60 (sec) , antiderivative size = 124, normalized size of antiderivative = 5.39 \[ \int \frac {-36-12 x^2 \log ^2(3)+\left (-36 x \log (3)-4 x^3 \log ^3(3)\right ) \log (4)+\left (-24-12 x^2 \log ^2(3)\right ) \log ^2(4)-12 x \log (3) \log ^3(4)-4 \log ^4(4)+\left (-12-4 x^2 \log ^2(3)-8 x \log (3) \log (4)-4 \log ^2(4)\right ) \log (x)+\left (24+4 x^2 \log ^2(3)+12 x \log (3) \log (4)+8 \log ^2(4)\right ) \log ^2(x)+4 \log ^3(x)-4 \log ^4(x)}{x^5} \, dx=\frac {\left (- 2 x^{2} \log {\left (3 \right )}^{2} - 8 x \log {\left (2 \right )} \log {\left (3 \right )} - 6 - 8 \log {\left (2 \right )}^{2}\right ) \log {\left (x \right )}^{2}}{x^{4}} - \frac {- 8 x^{3} \log {\left (2 \right )} \log {\left (3 \right )}^{3} + x^{2} \left (- 24 \log {\left (2 \right )}^{2} \log {\left (3 \right )}^{2} - 6 \log {\left (3 \right )}^{2}\right ) + x \left (- 24 \log {\left (2 \right )} \log {\left (3 \right )} - 32 \log {\left (2 \right )}^{3} \log {\left (3 \right )}\right ) - 24 \log {\left (2 \right )}^{2} - 9 - 16 \log {\left (2 \right )}^{4}}{x^{4}} + \frac {\log {\left (x \right )}^{4}}{x^{4}} \] Input:
integrate((-4*ln(x)**4+4*ln(x)**3+(32*ln(2)**2+24*x*ln(2)*ln(3)+4*x**2*ln( 3)**2+24)*ln(x)**2+(-16*ln(2)**2-16*x*ln(2)*ln(3)-4*x**2*ln(3)**2-12)*ln(x )-64*ln(2)**4-96*x*ln(3)*ln(2)**3+4*(-12*x**2*ln(3)**2-24)*ln(2)**2+2*(-4* x**3*ln(3)**3-36*x*ln(3))*ln(2)-12*x**2*ln(3)**2-36)/x**5,x)
Output:
(-2*x**2*log(3)**2 - 8*x*log(2)*log(3) - 6 - 8*log(2)**2)*log(x)**2/x**4 - (-8*x**3*log(2)*log(3)**3 + x**2*(-24*log(2)**2*log(3)**2 - 6*log(3)**2) + x*(-24*log(2)*log(3) - 32*log(2)**3*log(3)) - 24*log(2)**2 - 9 - 16*log( 2)**4)/x**4 + log(x)**4/x**4
Leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (25) = 50\).
Time = 0.04 (sec) , antiderivative size = 265, normalized size of antiderivative = 11.52 \[ \int \frac {-36-12 x^2 \log ^2(3)+\left (-36 x \log (3)-4 x^3 \log ^3(3)\right ) \log (4)+\left (-24-12 x^2 \log ^2(3)\right ) \log ^2(4)-12 x \log (3) \log ^3(4)-4 \log ^4(4)+\left (-12-4 x^2 \log ^2(3)-8 x \log (3) \log (4)-4 \log ^2(4)\right ) \log (x)+\left (24+4 x^2 \log ^2(3)+12 x \log (3) \log (4)+8 \log ^2(4)\right ) \log ^2(x)+4 \log ^3(x)-4 \log ^4(x)}{x^5} \, dx={\left (\frac {2 \, \log \left (x\right )}{x^{2}} + \frac {1}{x^{2}}\right )} \log \left (3\right )^{2} + \frac {16}{9} \, {\left (\frac {3 \, \log \left (x\right )}{x^{3}} + \frac {1}{x^{3}}\right )} \log \left (3\right ) \log \left (2\right ) + \frac {8 \, \log \left (3\right )^{3} \log \left (2\right )}{x} + {\left (\frac {4 \, \log \left (x\right )}{x^{4}} + \frac {1}{x^{4}}\right )} \log \left (2\right )^{2} + \frac {24 \, \log \left (3\right )^{2} \log \left (2\right )^{2}}{x^{2}} - \frac {{\left (2 \, \log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 1\right )} \log \left (3\right )^{2}}{x^{2}} + \frac {32 \, \log \left (3\right ) \log \left (2\right )^{3}}{x^{3}} + \frac {6 \, \log \left (3\right )^{2}}{x^{2}} - \frac {8 \, {\left (9 \, \log \left (x\right )^{2} + 6 \, \log \left (x\right ) + 2\right )} \log \left (3\right ) \log \left (2\right )}{9 \, x^{3}} + \frac {16 \, \log \left (2\right )^{4}}{x^{4}} + \frac {24 \, \log \left (3\right ) \log \left (2\right )}{x^{3}} - \frac {{\left (8 \, \log \left (x\right )^{2} + 4 \, \log \left (x\right ) + 1\right )} \log \left (2\right )^{2}}{x^{4}} + \frac {24 \, \log \left (2\right )^{2}}{x^{4}} + \frac {32 \, \log \left (x\right )^{4} + 32 \, \log \left (x\right )^{3} + 24 \, \log \left (x\right )^{2} + 12 \, \log \left (x\right ) + 3}{32 \, x^{4}} - \frac {32 \, \log \left (x\right )^{3} + 24 \, \log \left (x\right )^{2} + 12 \, \log \left (x\right ) + 3}{32 \, x^{4}} - \frac {3 \, {\left (8 \, \log \left (x\right )^{2} + 4 \, \log \left (x\right ) + 1\right )}}{4 \, x^{4}} + \frac {3 \, \log \left (x\right )}{x^{4}} + \frac {39}{4 \, x^{4}} \] Input:
integrate((-4*log(x)^4+4*log(x)^3+(32*log(2)^2+24*x*log(2)*log(3)+4*x^2*lo g(3)^2+24)*log(x)^2+(-16*log(2)^2-16*x*log(2)*log(3)-4*x^2*log(3)^2-12)*lo g(x)-64*log(2)^4-96*x*log(3)*log(2)^3+4*(-12*x^2*log(3)^2-24)*log(2)^2+2*( -4*x^3*log(3)^3-36*x*log(3))*log(2)-12*x^2*log(3)^2-36)/x^5,x, algorithm=" maxima")
Output:
(2*log(x)/x^2 + 1/x^2)*log(3)^2 + 16/9*(3*log(x)/x^3 + 1/x^3)*log(3)*log(2 ) + 8*log(3)^3*log(2)/x + (4*log(x)/x^4 + 1/x^4)*log(2)^2 + 24*log(3)^2*lo g(2)^2/x^2 - (2*log(x)^2 + 2*log(x) + 1)*log(3)^2/x^2 + 32*log(3)*log(2)^3 /x^3 + 6*log(3)^2/x^2 - 8/9*(9*log(x)^2 + 6*log(x) + 2)*log(3)*log(2)/x^3 + 16*log(2)^4/x^4 + 24*log(3)*log(2)/x^3 - (8*log(x)^2 + 4*log(x) + 1)*log (2)^2/x^4 + 24*log(2)^2/x^4 + 1/32*(32*log(x)^4 + 32*log(x)^3 + 24*log(x)^ 2 + 12*log(x) + 3)/x^4 - 1/32*(32*log(x)^3 + 24*log(x)^2 + 12*log(x) + 3)/ x^4 - 3/4*(8*log(x)^2 + 4*log(x) + 1)/x^4 + 3*log(x)/x^4 + 39/4/x^4
Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (25) = 50\).
Time = 0.12 (sec) , antiderivative size = 108, normalized size of antiderivative = 4.70 \[ \int \frac {-36-12 x^2 \log ^2(3)+\left (-36 x \log (3)-4 x^3 \log ^3(3)\right ) \log (4)+\left (-24-12 x^2 \log ^2(3)\right ) \log ^2(4)-12 x \log (3) \log ^3(4)-4 \log ^4(4)+\left (-12-4 x^2 \log ^2(3)-8 x \log (3) \log (4)-4 \log ^2(4)\right ) \log (x)+\left (24+4 x^2 \log ^2(3)+12 x \log (3) \log (4)+8 \log ^2(4)\right ) \log ^2(x)+4 \log ^3(x)-4 \log ^4(x)}{x^5} \, dx=\frac {\log \left (x\right )^{4}}{x^{4}} - \frac {2 \, {\left (x^{2} \log \left (3\right )^{2} + 4 \, x \log \left (3\right ) \log \left (2\right ) + 4 \, \log \left (2\right )^{2} + 3\right )} \log \left (x\right )^{2}}{x^{4}} + \frac {8 \, x^{3} \log \left (3\right )^{3} \log \left (2\right ) + 24 \, x^{2} \log \left (3\right )^{2} \log \left (2\right )^{2} + 32 \, x \log \left (3\right ) \log \left (2\right )^{3} + 6 \, x^{2} \log \left (3\right )^{2} + 16 \, \log \left (2\right )^{4} + 24 \, x \log \left (3\right ) \log \left (2\right ) + 24 \, \log \left (2\right )^{2} + 9}{x^{4}} \] Input:
integrate((-4*log(x)^4+4*log(x)^3+(32*log(2)^2+24*x*log(2)*log(3)+4*x^2*lo g(3)^2+24)*log(x)^2+(-16*log(2)^2-16*x*log(2)*log(3)-4*x^2*log(3)^2-12)*lo g(x)-64*log(2)^4-96*x*log(3)*log(2)^3+4*(-12*x^2*log(3)^2-24)*log(2)^2+2*( -4*x^3*log(3)^3-36*x*log(3))*log(2)-12*x^2*log(3)^2-36)/x^5,x, algorithm=" giac")
Output:
log(x)^4/x^4 - 2*(x^2*log(3)^2 + 4*x*log(3)*log(2) + 4*log(2)^2 + 3)*log(x )^2/x^4 + (8*x^3*log(3)^3*log(2) + 24*x^2*log(3)^2*log(2)^2 + 32*x*log(3)* log(2)^3 + 6*x^2*log(3)^2 + 16*log(2)^4 + 24*x*log(3)*log(2) + 24*log(2)^2 + 9)/x^4
Time = 3.53 (sec) , antiderivative size = 107, normalized size of antiderivative = 4.65 \[ \int \frac {-36-12 x^2 \log ^2(3)+\left (-36 x \log (3)-4 x^3 \log ^3(3)\right ) \log (4)+\left (-24-12 x^2 \log ^2(3)\right ) \log ^2(4)-12 x \log (3) \log ^3(4)-4 \log ^4(4)+\left (-12-4 x^2 \log ^2(3)-8 x \log (3) \log (4)-4 \log ^2(4)\right ) \log (x)+\left (24+4 x^2 \log ^2(3)+12 x \log (3) \log (4)+8 \log ^2(4)\right ) \log ^2(x)+4 \log ^3(x)-4 \log ^4(x)}{x^5} \, dx=\frac {x\,\left ({\ln \left (x\right )}^4+\left (-8\,{\ln \left (2\right )}^2-6\right )\,{\ln \left (x\right )}^2+{\left (4\,{\ln \left (2\right )}^2+3\right )}^2\right )-x^3\,\left (2\,{\ln \left (3\right )}^2\,{\ln \left (x\right )}^2-6\,{\ln \left (3\right )}^2\,\left (4\,{\ln \left (2\right )}^2+1\right )\right )-x^2\,\left (8\,\ln \left (2\right )\,\ln \left (3\right )\,{\ln \left (x\right )}^2-8\,\ln \left (2\right )\,\ln \left (3\right )\,\left (4\,{\ln \left (2\right )}^2+3\right )\right )+8\,x^4\,\ln \left (2\right )\,{\ln \left (3\right )}^3}{x^5} \] Input:
int(-(12*x^2*log(3)^2 - 4*log(x)^3 + 4*log(x)^4 + 2*log(2)*(4*x^3*log(3)^3 + 36*x*log(3)) + log(x)*(4*x^2*log(3)^2 + 16*log(2)^2 + 16*x*log(2)*log(3 ) + 12) + 4*log(2)^2*(12*x^2*log(3)^2 + 24) + 64*log(2)^4 - log(x)^2*(4*x^ 2*log(3)^2 + 32*log(2)^2 + 24*x*log(2)*log(3) + 24) + 96*x*log(2)^3*log(3) + 36)/x^5,x)
Output:
(x*(log(x)^4 - log(x)^2*(8*log(2)^2 + 6) + (4*log(2)^2 + 3)^2) - x^3*(2*lo g(3)^2*log(x)^2 - 6*log(3)^2*(4*log(2)^2 + 1)) - x^2*(8*log(2)*log(3)*log( x)^2 - 8*log(2)*log(3)*(4*log(2)^2 + 3)) + 8*x^4*log(2)*log(3)^3)/x^5
Time = 0.16 (sec) , antiderivative size = 111, normalized size of antiderivative = 4.83 \[ \int \frac {-36-12 x^2 \log ^2(3)+\left (-36 x \log (3)-4 x^3 \log ^3(3)\right ) \log (4)+\left (-24-12 x^2 \log ^2(3)\right ) \log ^2(4)-12 x \log (3) \log ^3(4)-4 \log ^4(4)+\left (-12-4 x^2 \log ^2(3)-8 x \log (3) \log (4)-4 \log ^2(4)\right ) \log (x)+\left (24+4 x^2 \log ^2(3)+12 x \log (3) \log (4)+8 \log ^2(4)\right ) \log ^2(x)+4 \log ^3(x)-4 \log ^4(x)}{x^5} \, dx=\frac {\mathrm {log}\left (x \right )^{4}-2 \mathrm {log}\left (x \right )^{2} \mathrm {log}\left (3\right )^{2} x^{2}-8 \mathrm {log}\left (x \right )^{2} \mathrm {log}\left (3\right ) \mathrm {log}\left (2\right ) x -8 \mathrm {log}\left (x \right )^{2} \mathrm {log}\left (2\right )^{2}-6 \mathrm {log}\left (x \right )^{2}+8 \mathrm {log}\left (3\right )^{3} \mathrm {log}\left (2\right ) x^{3}+24 \mathrm {log}\left (3\right )^{2} \mathrm {log}\left (2\right )^{2} x^{2}+6 \mathrm {log}\left (3\right )^{2} x^{2}+32 \,\mathrm {log}\left (3\right ) \mathrm {log}\left (2\right )^{3} x +24 \,\mathrm {log}\left (3\right ) \mathrm {log}\left (2\right ) x +16 \mathrm {log}\left (2\right )^{4}+24 \mathrm {log}\left (2\right )^{2}+9}{x^{4}} \] Input:
int((-4*log(x)^4+4*log(x)^3+(32*log(2)^2+24*x*log(2)*log(3)+4*x^2*log(3)^2 +24)*log(x)^2+(-16*log(2)^2-16*x*log(2)*log(3)-4*x^2*log(3)^2-12)*log(x)-6 4*log(2)^4-96*x*log(3)*log(2)^3+4*(-12*x^2*log(3)^2-24)*log(2)^2+2*(-4*x^3 *log(3)^3-36*x*log(3))*log(2)-12*x^2*log(3)^2-36)/x^5,x)
Output:
(log(x)**4 - 2*log(x)**2*log(3)**2*x**2 - 8*log(x)**2*log(3)*log(2)*x - 8* log(x)**2*log(2)**2 - 6*log(x)**2 + 8*log(3)**3*log(2)*x**3 + 24*log(3)**2 *log(2)**2*x**2 + 6*log(3)**2*x**2 + 32*log(3)*log(2)**3*x + 24*log(3)*log (2)*x + 16*log(2)**4 + 24*log(2)**2 + 9)/x**4