Integrand size = 127, antiderivative size = 28 \[ \int \frac {2+4 x-4 e^6 x+e^{12} x+\left (-4 x+2 e^6 x\right ) \log \left (\frac {4}{x}\right )+x \log ^2\left (\frac {4}{x}\right )+\left (4 x-2 e^6 x-2 x \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (8 x^2-6 e^6 x^2+e^{12} x^2+\left (-6 x^2+2 e^6 x^2\right ) \log \left (\frac {4}{x}\right )+x^2 \log ^2\left (\frac {4}{x}\right )\right ) \log (\log (2))}{x^2} \, dx=-\frac {2}{x}+\left (-2+e^6+\log \left (\frac {4}{x}\right )\right )^2 (\log (x)+x \log (\log (2))) \] Output:
(ln(x)+x*ln(ln(2)))*(ln(4/x)+exp(3)^2-2)^2-2/x
Leaf count is larger than twice the leaf count of optimal. \(90\) vs. \(2(28)=56\).
Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.21 \[ \int \frac {2+4 x-4 e^6 x+e^{12} x+\left (-4 x+2 e^6 x\right ) \log \left (\frac {4}{x}\right )+x \log ^2\left (\frac {4}{x}\right )+\left (4 x-2 e^6 x-2 x \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (8 x^2-6 e^6 x^2+e^{12} x^2+\left (-6 x^2+2 e^6 x^2\right ) \log \left (\frac {4}{x}\right )+x^2 \log ^2\left (\frac {4}{x}\right )\right ) \log (\log (2))}{x^2} \, dx=-\frac {2}{x}+\left (-2+e^6\right )^2 \log (x)-\left (-2+e^6\right ) \log ^2(x)+4 x \log (\log (2))-4 e^6 x \log (\log (2))+e^{12} x \log (\log (2))+2 \left (-2+e^6\right ) x \log \left (\frac {4}{x}\right ) \log (\log (2))+\log ^2\left (\frac {4}{x}\right ) \left (2-e^6+\log (x)+x \log (\log (2))\right ) \] Input:
Integrate[(2 + 4*x - 4*E^6*x + E^12*x + (-4*x + 2*E^6*x)*Log[4/x] + x*Log[ 4/x]^2 + (4*x - 2*E^6*x - 2*x*Log[4/x])*Log[x] + (8*x^2 - 6*E^6*x^2 + E^12 *x^2 + (-6*x^2 + 2*E^6*x^2)*Log[4/x] + x^2*Log[4/x]^2)*Log[Log[2]])/x^2,x]
Output:
-2/x + (-2 + E^6)^2*Log[x] - (-2 + E^6)*Log[x]^2 + 4*x*Log[Log[2]] - 4*E^6 *x*Log[Log[2]] + E^12*x*Log[Log[2]] + 2*(-2 + E^6)*x*Log[4/x]*Log[Log[2]] + Log[4/x]^2*(2 - E^6 + Log[x] + x*Log[Log[2]])
Leaf count is larger than twice the leaf count of optimal. \(174\) vs. \(2(28)=56\).
Time = 0.67 (sec) , antiderivative size = 174, normalized size of antiderivative = 6.21, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {6, 6, 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.
| \(\displaystyle \int \frac {\log (\log (2)) \left (e^{12} x^2-6 e^6 x^2+8 x^2+x^2 \log ^2\left (\frac {4}{x}\right )+\left (2 e^6 x^2-6 x^2\right ) \log \left (\frac {4}{x}\right )\right )+e^{12} x-4 e^6 x+4 x+x \log ^2\left (\frac {4}{x}\right )+\left (2 e^6 x-4 x\right ) \log \left (\frac {4}{x}\right )+\left (-2 e^6 x+4 x-2 x \log \left (\frac {4}{x}\right )\right ) \log (x)+2}{x^2} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {\log (\log (2)) \left (e^{12} x^2-6 e^6 x^2+8 x^2+x^2 \log ^2\left (\frac {4}{x}\right )+\left (2 e^6 x^2-6 x^2\right ) \log \left (\frac {4}{x}\right )\right )+\left (4-4 e^6\right ) x+e^{12} x+x \log ^2\left (\frac {4}{x}\right )+\left (2 e^6 x-4 x\right ) \log \left (\frac {4}{x}\right )+\left (-2 e^6 x+4 x-2 x \log \left (\frac {4}{x}\right )\right ) \log (x)+2}{x^2}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {\log (\log (2)) \left (e^{12} x^2-6 e^6 x^2+8 x^2+x^2 \log ^2\left (\frac {4}{x}\right )+\left (2 e^6 x^2-6 x^2\right ) \log \left (\frac {4}{x}\right )\right )+\left (4-4 e^6+e^{12}\right ) x+x \log ^2\left (\frac {4}{x}\right )+\left (2 e^6 x-4 x\right ) \log \left (\frac {4}{x}\right )+\left (-2 e^6 x+4 x-2 x \log \left (\frac {4}{x}\right )\right ) \log (x)+2}{x^2}dx\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \int \left (\frac {x^2 \log (\log (2)) \log ^2\left (\frac {4}{x}\right )-6 \left (1-\frac {e^6}{3}\right ) x^2 \log (\log (2)) \log \left (\frac {4}{x}\right )+8 \left (1+\frac {1}{8} e^6 \left (e^6-6\right )\right ) x^2 \log (\log (2))+4 \left (1+\frac {1}{4} e^6 \left (e^6-4\right )\right ) x+x \log ^2\left (\frac {4}{x}\right )-4 \left (1-\frac {e^6}{2}\right ) x \log \left (\frac {4}{x}\right )+2}{x^2}+\frac {2 \left (2 \left (1-\frac {e^6}{2}\right )-\log \left (\frac {4}{x}\right )\right ) \log (x)}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2}{x}-\frac {1}{3} \log ^3\left (\frac {4}{x}\right )+\left (2-e^6\right ) \log ^2\left (\frac {4}{x}\right )+x \log (\log (2)) \log ^2\left (\frac {4}{x}\right )-\frac {1}{3} \left (-\log \left (\frac {4}{x}\right )-e^6+2\right )^3+\log (x) \left (-\log \left (\frac {4}{x}\right )-e^6+2\right )^2+\left (2-e^6\right )^2 \log (x)-2 \left (3-e^6\right ) x \log (\log (2))+\left (2-e^3\right ) \left (2+e^3\right ) \left (2-e^6\right ) x \log (\log (2))+2 x \log (\log (2))-2 \left (3-e^6\right ) x \log (\log (2)) \log \left (\frac {4}{x}\right )+2 x \log (\log (2)) \log \left (\frac {4}{x}\right )\) |
Input:
Int[(2 + 4*x - 4*E^6*x + E^12*x + (-4*x + 2*E^6*x)*Log[4/x] + x*Log[4/x]^2 + (4*x - 2*E^6*x - 2*x*Log[4/x])*Log[x] + (8*x^2 - 6*E^6*x^2 + E^12*x^2 + (-6*x^2 + 2*E^6*x^2)*Log[4/x] + x^2*Log[4/x]^2)*Log[Log[2]])/x^2,x]
Output:
-2/x - (2 - E^6 - Log[4/x])^3/3 + (2 - E^6)*Log[4/x]^2 - Log[4/x]^3/3 + (2 - E^6)^2*Log[x] + (2 - E^6 - Log[4/x])^2*Log[x] + 2*x*Log[Log[2]] + (2 - E^3)*(2 + E^3)*(2 - E^6)*x*Log[Log[2]] - 2*(3 - E^6)*x*Log[Log[2]] + 2*x*L og[4/x]*Log[Log[2]] - 2*(3 - E^6)*x*Log[4/x]*Log[Log[2]] + x*Log[4/x]^2*Lo g[Log[2]]
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
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. \(144\) vs. \(2(29)=58\).
Time = 0.70 (sec) , antiderivative size = 145, normalized size of antiderivative = 5.18
| method | result | size |
| parallelrisch | \(\frac {-2+\ln \left (x \right ) {\mathrm e}^{12} x +2 \,{\mathrm e}^{6} x \ln \left (x \right ) \ln \left (\frac {4}{x}\right )+4 x^{2} \ln \left (\ln \left (2\right )\right )+2 \,{\mathrm e}^{6} \ln \left (\frac {4}{x}\right ) \ln \left (\ln \left (2\right )\right ) x^{2}+4 x \ln \left (x \right )-4 \ln \left (\frac {4}{x}\right ) \ln \left (\ln \left (2\right )\right ) x^{2}-4 \ln \left (x \right ) \ln \left (\frac {4}{x}\right ) x +{\mathrm e}^{12} \ln \left (\ln \left (2\right )\right ) x^{2}+\ln \left (\frac {4}{x}\right )^{2} \ln \left (\ln \left (2\right )\right ) x^{2}-4 \,{\mathrm e}^{6} \ln \left (\ln \left (2\right )\right ) x^{2}-4 \ln \left (x \right ) {\mathrm e}^{6} x +\ln \left (\frac {4}{x}\right )^{2} \ln \left (x \right ) x}{x}\) | \(145\) |
| risch | \(\ln \left (x \right )^{3}+\left (x \ln \left (\ln \left (2\right )\right )+4-2 \,{\mathrm e}^{6}-4 \ln \left (2\right )\right ) \ln \left (x \right )^{2}-x \left (2 \,{\mathrm e}^{6} \ln \left (\ln \left (2\right )\right )+4 \ln \left (\ln \left (2\right )\right ) \ln \left (2\right )-4 \ln \left (\ln \left (2\right )\right )\right ) \ln \left (x \right )+\frac {-8+4 \ln \left (x \right ) {\mathrm e}^{12} x +16 x^{2} \ln \left (\ln \left (2\right )\right )+16 \ln \left (x \right ) {\mathrm e}^{6} \ln \left (2\right ) x +16 \,{\mathrm e}^{6} \ln \left (\ln \left (2\right )\right ) x^{2} \ln \left (2\right )+16 x \ln \left (x \right )-32 x \ln \left (2\right ) \ln \left (x \right )-32 x^{2} \ln \left (\ln \left (2\right )\right ) \ln \left (2\right )+16 x^{2} \ln \left (\ln \left (2\right )\right ) \ln \left (2\right )^{2}+4 \,{\mathrm e}^{12} \ln \left (\ln \left (2\right )\right ) x^{2}-16 \,{\mathrm e}^{6} \ln \left (\ln \left (2\right )\right ) x^{2}-16 \ln \left (x \right ) {\mathrm e}^{6} x +16 x \ln \left (x \right ) \ln \left (2\right )^{2}}{4 x}\) | \(164\) |
| parts | \(\ln \left (\frac {1}{x}\right )^{2} \ln \left (x \right )+\frac {\ln \left (\frac {1}{x}\right )^{3}}{3}+\left (-{\mathrm e}^{6}+2\right ) \ln \left (x \right )^{2}-2 \ln \left (2\right ) \ln \left (x \right )^{2}+x \ln \left (\ln \left (2\right )\right ) {\mathrm e}^{12}-6 x \,{\mathrm e}^{6} \ln \left (\ln \left (2\right )\right )+8 x \ln \left (\ln \left (2\right )\right )+\left ({\mathrm e}^{12}-4 \,{\mathrm e}^{6}+4\right ) \ln \left (x \right )-\frac {2}{x}-4 \ln \left (\ln \left (2\right )\right ) \left (-\frac {x \ln \left (\frac {4}{x}\right )^{2}}{4}-\frac {x \ln \left (\frac {4}{x}\right )}{2}-\frac {x}{2}\right )-\frac {\ln \left (\frac {4}{x}\right )^{3}}{3}-8 \,{\mathrm e}^{6} \ln \left (\ln \left (2\right )\right ) \left (-\frac {x \ln \left (\frac {4}{x}\right )}{4}-\frac {x}{4}\right )-{\mathrm e}^{6} \ln \left (\frac {4}{x}\right )^{2}+24 \ln \left (\ln \left (2\right )\right ) \left (-\frac {x \ln \left (\frac {4}{x}\right )}{4}-\frac {x}{4}\right )+2 \ln \left (\frac {4}{x}\right )^{2}\) | \(180\) |
| default | \(4 \ln \left (2\right )^{2} \left (\ln \left (x \right )+x \ln \left (\ln \left (2\right )\right )\right )-2 \,{\mathrm e}^{6} \ln \left (\ln \left (2\right )\right ) \left (-x \ln \left (\frac {1}{x}\right )-x \right )-{\mathrm e}^{6} \ln \left (\frac {1}{x}\right )^{2}+6 \ln \left (\ln \left (2\right )\right ) \left (-x \ln \left (\frac {1}{x}\right )-x \right )+2 \ln \left (\frac {1}{x}\right )^{2}+\ln \left (\frac {1}{x}\right )^{2} \ln \left (x \right )+\left (-{\mathrm e}^{6}+2\right ) \ln \left (x \right )^{2}-\ln \left (\ln \left (2\right )\right ) \left (-x \ln \left (\frac {1}{x}\right )^{2}-2 x \ln \left (\frac {1}{x}\right )-2 x \right )+\left ({\mathrm e}^{12}-6 \,{\mathrm e}^{6}+8\right ) \ln \left (\ln \left (2\right )\right ) x +\left ({\mathrm e}^{12}-4 \,{\mathrm e}^{6}+4\right ) \ln \left (x \right )-\frac {2}{x}+\ln \left (2\right ) \left (-2 \ln \left (\frac {1}{x}\right )^{2}-4 \ln \left (\ln \left (2\right )\right ) \left (-x \ln \left (\frac {1}{x}\right )-x \right )+4 \left ({\mathrm e}^{6}-3\right ) \ln \left (\ln \left (2\right )\right ) x +\left (4 \,{\mathrm e}^{6}-8\right ) \ln \left (x \right )-2 \ln \left (x \right )^{2}\right )\) | \(209\) |
Input:
int(((x^2*ln(4/x)^2+(2*x^2*exp(3)^2-6*x^2)*ln(4/x)+x^2*exp(3)^4-6*x^2*exp( 3)^2+8*x^2)*ln(ln(2))+(-2*x*ln(4/x)-2*x*exp(3)^2+4*x)*ln(x)+x*ln(4/x)^2+(2 *x*exp(3)^2-4*x)*ln(4/x)+x*exp(3)^4-4*x*exp(3)^2+4*x+2)/x^2,x,method=_RETU RNVERBOSE)
Output:
1/x*(-2+ln(x)*exp(3)^4*x+2*exp(3)^2*x*ln(x)*ln(4/x)+4*x^2*ln(ln(2))+2*exp( 3)^2*ln(4/x)*ln(ln(2))*x^2+4*x*ln(x)-4*ln(4/x)*ln(ln(2))*x^2-4*ln(x)*ln(4/ x)*x+exp(3)^4*ln(ln(2))*x^2+ln(4/x)^2*ln(ln(2))*x^2-4*exp(3)^2*ln(ln(2))*x ^2-4*ln(x)*exp(3)^2*x+ln(4/x)^2*ln(x)*x)
Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (27) = 54\).
Time = 0.10 (sec) , antiderivative size = 128, normalized size of antiderivative = 4.57 \[ \int \frac {2+4 x-4 e^6 x+e^{12} x+\left (-4 x+2 e^6 x\right ) \log \left (\frac {4}{x}\right )+x \log ^2\left (\frac {4}{x}\right )+\left (4 x-2 e^6 x-2 x \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (8 x^2-6 e^6 x^2+e^{12} x^2+\left (-6 x^2+2 e^6 x^2\right ) \log \left (\frac {4}{x}\right )+x^2 \log ^2\left (\frac {4}{x}\right )\right ) \log (\log (2))}{x^2} \, dx=-\frac {x \log \left (\frac {4}{x}\right )^{3} + 2 \, {\left (x e^{6} - x \log \left (2\right ) - 2 \, x\right )} \log \left (\frac {4}{x}\right )^{2} + {\left (x e^{12} - 4 \, x e^{6} - 4 \, {\left (x e^{6} - 2 \, x\right )} \log \left (2\right ) + 4 \, x\right )} \log \left (\frac {4}{x}\right ) - {\left (x^{2} \log \left (\frac {4}{x}\right )^{2} + x^{2} e^{12} - 4 \, x^{2} e^{6} + 4 \, x^{2} + 2 \, {\left (x^{2} e^{6} - 2 \, x^{2}\right )} \log \left (\frac {4}{x}\right )\right )} \log \left (\log \left (2\right )\right ) + 2}{x} \] Input:
integrate(((x^2*log(4/x)^2+(2*x^2*exp(3)^2-6*x^2)*log(4/x)+x^2*exp(3)^4-6* x^2*exp(3)^2+8*x^2)*log(log(2))+(-2*x*log(4/x)-2*x*exp(3)^2+4*x)*log(x)+x* log(4/x)^2+(2*x*exp(3)^2-4*x)*log(4/x)+x*exp(3)^4-4*x*exp(3)^2+4*x+2)/x^2, x, algorithm="fricas")
Output:
-(x*log(4/x)^3 + 2*(x*e^6 - x*log(2) - 2*x)*log(4/x)^2 + (x*e^12 - 4*x*e^6 - 4*(x*e^6 - 2*x)*log(2) + 4*x)*log(4/x) - (x^2*log(4/x)^2 + x^2*e^12 - 4 *x^2*e^6 + 4*x^2 + 2*(x^2*e^6 - 2*x^2)*log(4/x))*log(log(2)) + 2)/x
Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (24) = 48\).
Time = 0.30 (sec) , antiderivative size = 144, normalized size of antiderivative = 5.14 \[ \int \frac {2+4 x-4 e^6 x+e^{12} x+\left (-4 x+2 e^6 x\right ) \log \left (\frac {4}{x}\right )+x \log ^2\left (\frac {4}{x}\right )+\left (4 x-2 e^6 x-2 x \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (8 x^2-6 e^6 x^2+e^{12} x^2+\left (-6 x^2+2 e^6 x^2\right ) \log \left (\frac {4}{x}\right )+x^2 \log ^2\left (\frac {4}{x}\right )\right ) \log (\log (2))}{x^2} \, dx=x \left (e^{12} \log {\left (\log {\left (2 \right )} \right )} + 4 e^{6} \log {\left (2 \right )} \log {\left (\log {\left (2 \right )} \right )} + 4 \log {\left (\log {\left (2 \right )} \right )} + 4 \log {\left (2 \right )}^{2} \log {\left (\log {\left (2 \right )} \right )} - 8 \log {\left (2 \right )} \log {\left (\log {\left (2 \right )} \right )} - 4 e^{6} \log {\left (\log {\left (2 \right )} \right )}\right ) + \left (4 x \log {\left (\log {\left (2 \right )} \right )} - 4 x \log {\left (2 \right )} \log {\left (\log {\left (2 \right )} \right )} - 2 x e^{6} \log {\left (\log {\left (2 \right )} \right )}\right ) \log {\left (x \right )} + \left (x \log {\left (\log {\left (2 \right )} \right )} - 2 e^{6} - 4 \log {\left (2 \right )} + 4\right ) \log {\left (x \right )}^{2} + \log {\left (x \right )}^{3} + \left (-2 + 2 \log {\left (2 \right )} + e^{6}\right )^{2} \log {\left (x \right )} - \frac {2}{x} \] Input:
integrate(((x**2*ln(4/x)**2+(2*x**2*exp(3)**2-6*x**2)*ln(4/x)+x**2*exp(3)* *4-6*x**2*exp(3)**2+8*x**2)*ln(ln(2))+(-2*x*ln(4/x)-2*x*exp(3)**2+4*x)*ln( x)+x*ln(4/x)**2+(2*x*exp(3)**2-4*x)*ln(4/x)+x*exp(3)**4-4*x*exp(3)**2+4*x+ 2)/x**2,x)
Output:
x*(exp(12)*log(log(2)) + 4*exp(6)*log(2)*log(log(2)) + 4*log(log(2)) + 4*l og(2)**2*log(log(2)) - 8*log(2)*log(log(2)) - 4*exp(6)*log(log(2))) + (4*x *log(log(2)) - 4*x*log(2)*log(log(2)) - 2*x*exp(6)*log(log(2)))*log(x) + ( x*log(log(2)) - 2*exp(6) - 4*log(2) + 4)*log(x)**2 + log(x)**3 + (-2 + 2*l og(2) + exp(6))**2*log(x) - 2/x
Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (27) = 54\).
Time = 0.06 (sec) , antiderivative size = 151, normalized size of antiderivative = 5.39 \[ \int \frac {2+4 x-4 e^6 x+e^{12} x+\left (-4 x+2 e^6 x\right ) \log \left (\frac {4}{x}\right )+x \log ^2\left (\frac {4}{x}\right )+\left (4 x-2 e^6 x-2 x \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (8 x^2-6 e^6 x^2+e^{12} x^2+\left (-6 x^2+2 e^6 x^2\right ) \log \left (\frac {4}{x}\right )+x^2 \log ^2\left (\frac {4}{x}\right )\right ) \log (\log (2))}{x^2} \, dx=x \log \left (\frac {4}{x}\right )^{2} \log \left (\log \left (2\right )\right ) - e^{6} \log \left (x\right )^{2} - \frac {1}{3} \, \log \left (x\right )^{3} - \log \left (x\right )^{2} \log \left (\frac {4}{x}\right ) - e^{6} \log \left (\frac {4}{x}\right )^{2} - \frac {1}{3} \, \log \left (\frac {4}{x}\right )^{3} + x e^{12} \log \left (\log \left (2\right )\right ) + 2 \, {\left (x \log \left (\frac {4}{x}\right ) + x\right )} e^{6} \log \left (\log \left (2\right )\right ) - 6 \, x e^{6} \log \left (\log \left (2\right )\right ) + e^{12} \log \left (x\right ) - 4 \, e^{6} \log \left (x\right ) + 2 \, \log \left (x\right )^{2} + 2 \, \log \left (\frac {4}{x}\right )^{2} - 4 \, {\left (x \log \left (\frac {4}{x}\right ) + x\right )} \log \left (\log \left (2\right )\right ) + 8 \, x \log \left (\log \left (2\right )\right ) - \frac {2}{x} + 4 \, \log \left (x\right ) \] Input:
integrate(((x^2*log(4/x)^2+(2*x^2*exp(3)^2-6*x^2)*log(4/x)+x^2*exp(3)^4-6* x^2*exp(3)^2+8*x^2)*log(log(2))+(-2*x*log(4/x)-2*x*exp(3)^2+4*x)*log(x)+x* log(4/x)^2+(2*x*exp(3)^2-4*x)*log(4/x)+x*exp(3)^4-4*x*exp(3)^2+4*x+2)/x^2, x, algorithm="maxima")
Output:
x*log(4/x)^2*log(log(2)) - e^6*log(x)^2 - 1/3*log(x)^3 - log(x)^2*log(4/x) - e^6*log(4/x)^2 - 1/3*log(4/x)^3 + x*e^12*log(log(2)) + 2*(x*log(4/x) + x)*e^6*log(log(2)) - 6*x*e^6*log(log(2)) + e^12*log(x) - 4*e^6*log(x) + 2* log(x)^2 + 2*log(4/x)^2 - 4*(x*log(4/x) + x)*log(log(2)) + 8*x*log(log(2)) - 2/x + 4*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (27) = 54\).
Time = 0.12 (sec) , antiderivative size = 186, normalized size of antiderivative = 6.64 \[ \int \frac {2+4 x-4 e^6 x+e^{12} x+\left (-4 x+2 e^6 x\right ) \log \left (\frac {4}{x}\right )+x \log ^2\left (\frac {4}{x}\right )+\left (4 x-2 e^6 x-2 x \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (8 x^2-6 e^6 x^2+e^{12} x^2+\left (-6 x^2+2 e^6 x^2\right ) \log \left (\frac {4}{x}\right )+x^2 \log ^2\left (\frac {4}{x}\right )\right ) \log (\log (2))}{x^2} \, dx=\frac {4 \, x^{2} e^{6} \log \left (2\right ) \log \left (\log \left (2\right )\right ) + 4 \, x^{2} \log \left (2\right )^{2} \log \left (\log \left (2\right )\right ) - 2 \, x^{2} e^{6} \log \left (x\right ) \log \left (\log \left (2\right )\right ) - 4 \, x^{2} \log \left (2\right ) \log \left (x\right ) \log \left (\log \left (2\right )\right ) + x^{2} \log \left (x\right )^{2} \log \left (\log \left (2\right )\right ) + 4 \, x e^{6} \log \left (2\right ) \log \left (x\right ) + 4 \, x \log \left (2\right )^{2} \log \left (x\right ) - 2 \, x e^{6} \log \left (x\right )^{2} - 4 \, x \log \left (2\right ) \log \left (x\right )^{2} + x \log \left (x\right )^{3} + x^{2} e^{12} \log \left (\log \left (2\right )\right ) - 4 \, x^{2} e^{6} \log \left (\log \left (2\right )\right ) - 8 \, x^{2} \log \left (2\right ) \log \left (\log \left (2\right )\right ) + 4 \, x^{2} \log \left (x\right ) \log \left (\log \left (2\right )\right ) + x e^{12} \log \left (x\right ) - 4 \, x e^{6} \log \left (x\right ) - 8 \, x \log \left (2\right ) \log \left (x\right ) + 4 \, x \log \left (x\right )^{2} + 4 \, x^{2} \log \left (\log \left (2\right )\right ) + 4 \, x \log \left (x\right ) - 2}{x} \] Input:
integrate(((x^2*log(4/x)^2+(2*x^2*exp(3)^2-6*x^2)*log(4/x)+x^2*exp(3)^4-6* x^2*exp(3)^2+8*x^2)*log(log(2))+(-2*x*log(4/x)-2*x*exp(3)^2+4*x)*log(x)+x* log(4/x)^2+(2*x*exp(3)^2-4*x)*log(4/x)+x*exp(3)^4-4*x*exp(3)^2+4*x+2)/x^2, x, algorithm="giac")
Output:
(4*x^2*e^6*log(2)*log(log(2)) + 4*x^2*log(2)^2*log(log(2)) - 2*x^2*e^6*log (x)*log(log(2)) - 4*x^2*log(2)*log(x)*log(log(2)) + x^2*log(x)^2*log(log(2 )) + 4*x*e^6*log(2)*log(x) + 4*x*log(2)^2*log(x) - 2*x*e^6*log(x)^2 - 4*x* log(2)*log(x)^2 + x*log(x)^3 + x^2*e^12*log(log(2)) - 4*x^2*e^6*log(log(2) ) - 8*x^2*log(2)*log(log(2)) + 4*x^2*log(x)*log(log(2)) + x*e^12*log(x) - 4*x*e^6*log(x) - 8*x*log(2)*log(x) + 4*x*log(x)^2 + 4*x^2*log(log(2)) + 4* x*log(x) - 2)/x
Time = 3.12 (sec) , antiderivative size = 174, normalized size of antiderivative = 6.21 \[ \int \frac {2+4 x-4 e^6 x+e^{12} x+\left (-4 x+2 e^6 x\right ) \log \left (\frac {4}{x}\right )+x \log ^2\left (\frac {4}{x}\right )+\left (4 x-2 e^6 x-2 x \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (8 x^2-6 e^6 x^2+e^{12} x^2+\left (-6 x^2+2 e^6 x^2\right ) \log \left (\frac {4}{x}\right )+x^2 \log ^2\left (\frac {4}{x}\right )\right ) \log (\log (2))}{x^2} \, dx=4\,\ln \left (x\right )+4\,{\ln \left (2\right )}^2\,\ln \left (x\right )-4\,\ln \left (\frac {1}{x}\right )\,\ln \left (x\right )-4\,{\mathrm {e}}^6\,\ln \left (x\right )+{\mathrm {e}}^{12}\,\ln \left (x\right )+4\,x\,\ln \left (\ln \left (2\right )\right )-8\,\ln \left (2\right )\,\ln \left (x\right )-\frac {2}{x}+{\ln \left (\frac {1}{x}\right )}^2\,\ln \left (x\right )-4\,x\,{\mathrm {e}}^6\,\ln \left (\ln \left (2\right )\right )+4\,{\mathrm {e}}^6\,\ln \left (2\right )\,\ln \left (x\right )+x\,{\mathrm {e}}^{12}\,\ln \left (\ln \left (2\right )\right )-8\,x\,\ln \left (2\right )\,\ln \left (\ln \left (2\right )\right )+x\,{\ln \left (\frac {1}{x}\right )}^2\,\ln \left (\ln \left (2\right )\right )+4\,x\,{\ln \left (2\right )}^2\,\ln \left (\ln \left (2\right )\right )+2\,\ln \left (\frac {1}{x}\right )\,{\mathrm {e}}^6\,\ln \left (x\right )-4\,x\,\ln \left (\frac {1}{x}\right )\,\ln \left (\ln \left (2\right )\right )+4\,\ln \left (\frac {1}{x}\right )\,\ln \left (2\right )\,\ln \left (x\right )+2\,x\,\ln \left (\frac {1}{x}\right )\,{\mathrm {e}}^6\,\ln \left (\ln \left (2\right )\right )+4\,x\,\ln \left (\frac {1}{x}\right )\,\ln \left (2\right )\,\ln \left (\ln \left (2\right )\right )+4\,x\,{\mathrm {e}}^6\,\ln \left (2\right )\,\ln \left (\ln \left (2\right )\right ) \] Input:
int((4*x - log(x)*(2*x*exp(6) - 4*x + 2*x*log(4/x)) - 4*x*exp(6) + x*exp(1 2) + log(log(2))*(log(4/x)*(2*x^2*exp(6) - 6*x^2) + x^2*log(4/x)^2 - 6*x^2 *exp(6) + x^2*exp(12) + 8*x^2) - log(4/x)*(4*x - 2*x*exp(6)) + x*log(4/x)^ 2 + 2)/x^2,x)
Output:
4*log(x) + 4*log(2)^2*log(x) - 4*log(1/x)*log(x) - 4*exp(6)*log(x) + exp(1 2)*log(x) + 4*x*log(log(2)) - 8*log(2)*log(x) - 2/x + log(1/x)^2*log(x) - 4*x*exp(6)*log(log(2)) + 4*exp(6)*log(2)*log(x) + x*exp(12)*log(log(2)) - 8*x*log(2)*log(log(2)) + x*log(1/x)^2*log(log(2)) + 4*x*log(2)^2*log(log(2 )) + 2*log(1/x)*exp(6)*log(x) - 4*x*log(1/x)*log(log(2)) + 4*log(1/x)*log( 2)*log(x) + 2*x*log(1/x)*exp(6)*log(log(2)) + 4*x*log(1/x)*log(2)*log(log( 2)) + 4*x*exp(6)*log(2)*log(log(2))
\[ \int \frac {2+4 x-4 e^6 x+e^{12} x+\left (-4 x+2 e^6 x\right ) \log \left (\frac {4}{x}\right )+x \log ^2\left (\frac {4}{x}\right )+\left (4 x-2 e^6 x-2 x \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (8 x^2-6 e^6 x^2+e^{12} x^2+\left (-6 x^2+2 e^6 x^2\right ) \log \left (\frac {4}{x}\right )+x^2 \log ^2\left (\frac {4}{x}\right )\right ) \log (\log (2))}{x^2} \, dx=\frac {-6 \left (\int \frac {\mathrm {log}\left (\frac {4}{x}\right ) \mathrm {log}\left (x \right )}{x}d x \right ) x +3 \,\mathrm {log}\left (\mathrm {log}\left (2\right )\right ) \mathrm {log}\left (\frac {4}{x}\right )^{2} x^{2}+6 \,\mathrm {log}\left (\mathrm {log}\left (2\right )\right ) \mathrm {log}\left (\frac {4}{x}\right ) e^{6} x^{2}-12 \,\mathrm {log}\left (\mathrm {log}\left (2\right )\right ) \mathrm {log}\left (\frac {4}{x}\right ) x^{2}+3 \,\mathrm {log}\left (\mathrm {log}\left (2\right )\right ) e^{12} x^{2}-12 \,\mathrm {log}\left (\mathrm {log}\left (2\right )\right ) e^{6} x^{2}+12 \,\mathrm {log}\left (\mathrm {log}\left (2\right )\right ) x^{2}-\mathrm {log}\left (\frac {4}{x}\right )^{3} x -3 \mathrm {log}\left (\frac {4}{x}\right )^{2} e^{6} x +6 \mathrm {log}\left (\frac {4}{x}\right )^{2} x -3 \mathrm {log}\left (x \right )^{2} e^{6} x +6 \mathrm {log}\left (x \right )^{2} x +3 \,\mathrm {log}\left (x \right ) e^{12} x -12 \,\mathrm {log}\left (x \right ) e^{6} x +12 \,\mathrm {log}\left (x \right ) x -6}{3 x} \] Input:
int(((x^2*log(4/x)^2+(2*x^2*exp(3)^2-6*x^2)*log(4/x)+x^2*exp(3)^4-6*x^2*ex p(3)^2+8*x^2)*log(log(2))+(-2*x*log(4/x)-2*x*exp(3)^2+4*x)*log(x)+x*log(4/ x)^2+(2*x*exp(3)^2-4*x)*log(4/x)+x*exp(3)^4-4*x*exp(3)^2+4*x+2)/x^2,x)
Output:
( - 6*int((log(4/x)*log(x))/x,x)*x + 3*log(log(2))*log(4/x)**2*x**2 + 6*lo g(log(2))*log(4/x)*e**6*x**2 - 12*log(log(2))*log(4/x)*x**2 + 3*log(log(2) )*e**12*x**2 - 12*log(log(2))*e**6*x**2 + 12*log(log(2))*x**2 - log(4/x)** 3*x - 3*log(4/x)**2*e**6*x + 6*log(4/x)**2*x - 3*log(x)**2*e**6*x + 6*log( x)**2*x + 3*log(x)*e**12*x - 12*log(x)*e**6*x + 12*log(x)*x - 6)/(3*x)