Integrand size = 117, antiderivative size = 23 \[ \int \frac {-6+4 e^3-2 e^6+\left (4+2 e^6\right ) \log (x)-4 e^3 \log ^2(x)+2 \log ^3(x)+\left (6-4 e^3+2 e^6+\left (2-4 e^3\right ) \log (x)+2 \log ^2(x)\right ) \log \left (4+e^6+\left (-1-2 e^3\right ) \log (x)+\log ^2(x)\right )}{4 x+e^6 x+\left (-x-2 e^3 x\right ) \log (x)+x \log ^2(x)} \, dx=\left (-1+\log (x)+\log \left (4-\log (x)+\left (-e^3+\log (x)\right )^2\right )\right )^2 \] Output:
(ln((ln(x)-exp(3))^2-ln(x)+4)+ln(x)-1)^2
Time = 0.03 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {-6+4 e^3-2 e^6+\left (4+2 e^6\right ) \log (x)-4 e^3 \log ^2(x)+2 \log ^3(x)+\left (6-4 e^3+2 e^6+\left (2-4 e^3\right ) \log (x)+2 \log ^2(x)\right ) \log \left (4+e^6+\left (-1-2 e^3\right ) \log (x)+\log ^2(x)\right )}{4 x+e^6 x+\left (-x-2 e^3 x\right ) \log (x)+x \log ^2(x)} \, dx=\left (-1+\log (x)+\log \left (4+e^6-\left (1+2 e^3\right ) \log (x)+\log ^2(x)\right )\right )^2 \] Input:
Integrate[(-6 + 4*E^3 - 2*E^6 + (4 + 2*E^6)*Log[x] - 4*E^3*Log[x]^2 + 2*Lo g[x]^3 + (6 - 4*E^3 + 2*E^6 + (2 - 4*E^3)*Log[x] + 2*Log[x]^2)*Log[4 + E^6 + (-1 - 2*E^3)*Log[x] + Log[x]^2])/(4*x + E^6*x + (-x - 2*E^3*x)*Log[x] + x*Log[x]^2),x]
Output:
(-1 + Log[x] + Log[4 + E^6 - (1 + 2*E^3)*Log[x] + Log[x]^2])^2
Time = 0.92 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.085, Rules used = {6, 3039, 6, 6, 6, 6, 6, 27, 7239, 7237}
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.
| \(\displaystyle \int \frac {2 \log ^3(x)-4 e^3 \log ^2(x)+\left (2 \log ^2(x)+\left (2-4 e^3\right ) \log (x)+2 e^6-4 e^3+6\right ) \log \left (\log ^2(x)+\left (-1-2 e^3\right ) \log (x)+e^6+4\right )+\left (4+2 e^6\right ) \log (x)-2 e^6+4 e^3-6}{e^6 x+4 x+x \log ^2(x)+\left (-2 e^3 x-x\right ) \log (x)} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {2 \log ^3(x)-4 e^3 \log ^2(x)+\left (2 \log ^2(x)+\left (2-4 e^3\right ) \log (x)+2 e^6-4 e^3+6\right ) \log \left (\log ^2(x)+\left (-1-2 e^3\right ) \log (x)+e^6+4\right )+\left (4+2 e^6\right ) \log (x)-2 e^6+4 e^3-6}{\left (4+e^6\right ) x+x \log ^2(x)+\left (-2 e^3 x-x\right ) \log (x)}dx\) |
| \(\Big \downarrow \) 3039 |
| \(\displaystyle \int -\frac {2 \left (-\log ^3(x)-\log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right ) \log ^2(x)+2 e^3 \log ^2(x)+2 e^3 \log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right ) \log (x)-\log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right ) \log (x)-e^6 \log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right )+2 e^3 \log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right )-3 \log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right )-e^6 \log (x)-2 \log (x)+e^6-2 e^3+3\right )}{\log ^2(x)-2 e^3 \log (x)-\log (x)+e^6+4}d\log (x)\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int -\frac {2 \left (-\log ^3(x)-\log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right ) \log ^2(x)+2 e^3 \log ^2(x)+2 e^3 \log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right ) \log (x)-\log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right ) \log (x)-e^6 \log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right )+2 e^3 \log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right )-3 \log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right )-e^6 \log (x)-2 \log (x)+e^6-2 e^3+3\right )}{\log ^2(x)+\left (-1-2 e^3\right ) \log (x)+e^6+4}d\log (x)\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int -\frac {2 \left (-\log ^3(x)-\log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right ) \log ^2(x)+2 e^3 \log ^2(x)+2 e^3 \log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right ) \log (x)-\log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right ) \log (x)-e^6 \log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right )+2 e^3 \log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right )-3 \log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right )+\left (-2-e^6\right ) \log (x)+e^6-2 e^3+3\right )}{\log ^2(x)+\left (-1-2 e^3\right ) \log (x)+e^6+4}d\log (x)\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int -\frac {2 \left (-\log ^3(x)-\log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right ) \log ^2(x)+2 e^3 \log ^2(x)+2 e^3 \log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right ) \log (x)-\log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right ) \log (x)+\left (2 e^3-3\right ) \log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right )-e^6 \log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right )+\left (-2-e^6\right ) \log (x)+e^6-2 e^3+3\right )}{\log ^2(x)+\left (-1-2 e^3\right ) \log (x)+e^6+4}d\log (x)\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int -\frac {2 \left (-\log ^3(x)-\log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right ) \log ^2(x)+2 e^3 \log ^2(x)+2 e^3 \log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right ) \log (x)-\log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right ) \log (x)+\left (-3+2 e^3-e^6\right ) \log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right )+\left (-2-e^6\right ) \log (x)+e^6-2 e^3+3\right )}{\log ^2(x)+\left (-1-2 e^3\right ) \log (x)+e^6+4}d\log (x)\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int -\frac {2 \left (-\log ^3(x)-\log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right ) \log ^2(x)+2 e^3 \log ^2(x)+\left (2 e^3-1\right ) \log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right ) \log (x)+\left (-3+2 e^3-e^6\right ) \log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right )+\left (-2-e^6\right ) \log (x)+e^6-2 e^3+3\right )}{\log ^2(x)+\left (-1-2 e^3\right ) \log (x)+e^6+4}d\log (x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 \int \frac {-\log ^3(x)-\log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right ) \log ^2(x)+2 e^3 \log ^2(x)-\left (1-2 e^3\right ) \log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right ) \log (x)-\left (2+e^6\right ) \log (x)-\left (3-2 e^3+e^6\right ) \log \left (\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4\right )+e^6-2 e^3+3}{\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4}d\log (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\left (\log ^2(x)+\left (1-2 e^3\right ) \log (x)+e^6-2 e^3+3\right ) \left (-\log (x)-\log \left (\log ^2(x)-2 e^3 \log (x)-\log (x)+e^6+4\right )+1\right )}{\log ^2(x)-\left (1+2 e^3\right ) \log (x)+e^6+4}d\log (x)\) |
| \(\Big \downarrow \) 7237 |
| \(\displaystyle \left (-\log \left (\log ^2(x)-2 e^3 \log (x)-\log (x)+e^6+4\right )-\log (x)+1\right )^2\) |
Input:
Int[(-6 + 4*E^3 - 2*E^6 + (4 + 2*E^6)*Log[x] - 4*E^3*Log[x]^2 + 2*Log[x]^3 + (6 - 4*E^3 + 2*E^6 + (2 - 4*E^3)*Log[x] + 2*Log[x]^2)*Log[4 + E^6 + (-1 - 2*E^3)*Log[x] + Log[x]^2])/(4*x + E^6*x + (-x - 2*E^3*x)*Log[x] + x*Log [x]^2),x]
Output:
(1 - Log[x] - Log[4 + E^6 - Log[x] - 2*E^3*Log[x] + Log[x]^2])^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[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst [[3]] Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /; !FalseQ[lst]] /; NonsumQ[u]
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(71\) vs. \(2(22)=44\).
Time = 0.52 (sec) , antiderivative size = 72, normalized size of antiderivative = 3.13
| method | result | size |
| risch | \(\ln \left (\ln \left (x \right )^{2}+\left (-2 \,{\mathrm e}^{3}-1\right ) \ln \left (x \right )+{\mathrm e}^{6}+4\right )^{2}+2 \ln \left (x \right ) \ln \left (\ln \left (x \right )^{2}+\left (-2 \,{\mathrm e}^{3}-1\right ) \ln \left (x \right )+{\mathrm e}^{6}+4\right )+\ln \left (x \right )^{2}-2 \ln \left (x \right )-2 \ln \left (\ln \left (x \right )^{2}+\left (-2 \,{\mathrm e}^{3}-1\right ) \ln \left (x \right )+{\mathrm e}^{6}+4\right )\) | \(72\) |
| default | \(-2 \ln \left (x \right )+\ln \left (x \right )^{2}+\left (2 \,{\mathrm e}^{3}-1\right ) \ln \left (\ln \left (x \right )^{2}-2 \,{\mathrm e}^{3} \ln \left (x \right )+{\mathrm e}^{6}-\ln \left (x \right )+4\right )-\frac {4 \left (-2 \,{\mathrm e}^{6}-7+2 \,{\mathrm e}^{3}-\frac {\left (2 \,{\mathrm e}^{3}-1\right ) \left (-2 \,{\mathrm e}^{3}-1\right )}{2}\right ) \operatorname {arctanh}\left (\frac {2 \ln \left (x \right )-2 \,{\mathrm e}^{3}-1}{\sqrt {4 \,{\mathrm e}^{3}-15}}\right )}{\sqrt {4 \,{\mathrm e}^{3}-15}}+2 \ln \left (x \right ) \ln \left (\ln \left (x \right )^{2}-2 \,{\mathrm e}^{3} \ln \left (x \right )+{\mathrm e}^{6}-\ln \left (x \right )+4\right )-\left (2 \,{\mathrm e}^{3}+1\right ) \ln \left (\ln \left (x \right )^{2}-2 \,{\mathrm e}^{3} \ln \left (x \right )+{\mathrm e}^{6}-\ln \left (x \right )+4\right )+\frac {4 \left (-2 \,{\mathrm e}^{6}-8-\frac {\left (2 \,{\mathrm e}^{3}+1\right ) \left (-2 \,{\mathrm e}^{3}-1\right )}{2}\right ) \operatorname {arctanh}\left (\frac {2 \ln \left (x \right )-2 \,{\mathrm e}^{3}-1}{\sqrt {4 \,{\mathrm e}^{3}-15}}\right )}{\sqrt {4 \,{\mathrm e}^{3}-15}}+2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{2}+\left (-2 \,{\mathrm e}^{3}-1\right ) \textit {\_Z} +{\mathrm e}^{6}+4\right )}{\sum }\left (\ln \left (\ln \left (x \right )-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\ln \left (x \right )^{2}-2 \,{\mathrm e}^{3} \ln \left (x \right )+{\mathrm e}^{6}-\ln \left (x \right )+4\right )-\frac {\ln \left (\ln \left (x \right )-\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{2 \left (2 \underline {\hspace {1.25 ex}}\alpha -2 \,{\mathrm e}^{3}-1\right )}-\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha -2 \,{\mathrm e}^{3}-1\right ) \ln \left (\ln \left (x \right )-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\underline {\hspace {1.25 ex}}\alpha -2 \,{\mathrm e}^{3}+\ln \left (x \right )-1}{2 \underline {\hspace {1.25 ex}}\alpha -2 \,{\mathrm e}^{3}-1}\right )}{-4 \,{\mathrm e}^{3}+15}-\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha -2 \,{\mathrm e}^{3}-1\right ) \operatorname {dilog}\left (\frac {\underline {\hspace {1.25 ex}}\alpha -2 \,{\mathrm e}^{3}+\ln \left (x \right )-1}{2 \underline {\hspace {1.25 ex}}\alpha -2 \,{\mathrm e}^{3}-1}\right )}{-4 \,{\mathrm e}^{3}+15}\right )\right )\) | \(414\) |
Input:
int(((2*ln(x)^2+(-4*exp(3)+2)*ln(x)+2*exp(3)^2-4*exp(3)+6)*ln(ln(x)^2+(-2* exp(3)-1)*ln(x)+exp(3)^2+4)+2*ln(x)^3-4*exp(3)*ln(x)^2+(2*exp(3)^2+4)*ln(x )-2*exp(3)^2+4*exp(3)-6)/(x*ln(x)^2+(-2*x*exp(3)-x)*ln(x)+x*exp(3)^2+4*x), x,method=_RETURNVERBOSE)
Output:
ln(ln(x)^2+(-2*exp(3)-1)*ln(x)+exp(6)+4)^2+2*ln(x)*ln(ln(x)^2+(-2*exp(3)-1 )*ln(x)+exp(6)+4)+ln(x)^2-2*ln(x)-2*ln(ln(x)^2+(-2*exp(3)-1)*ln(x)+exp(6)+ 4)
Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (22) = 44\).
Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.39 \[ \int \frac {-6+4 e^3-2 e^6+\left (4+2 e^6\right ) \log (x)-4 e^3 \log ^2(x)+2 \log ^3(x)+\left (6-4 e^3+2 e^6+\left (2-4 e^3\right ) \log (x)+2 \log ^2(x)\right ) \log \left (4+e^6+\left (-1-2 e^3\right ) \log (x)+\log ^2(x)\right )}{4 x+e^6 x+\left (-x-2 e^3 x\right ) \log (x)+x \log ^2(x)} \, dx=2 \, {\left (\log \left (x\right ) - 1\right )} \log \left (-{\left (2 \, e^{3} + 1\right )} \log \left (x\right ) + \log \left (x\right )^{2} + e^{6} + 4\right ) + \log \left (-{\left (2 \, e^{3} + 1\right )} \log \left (x\right ) + \log \left (x\right )^{2} + e^{6} + 4\right )^{2} + \log \left (x\right )^{2} - 2 \, \log \left (x\right ) \] Input:
integrate(((2*log(x)^2+(-4*exp(3)+2)*log(x)+2*exp(3)^2-4*exp(3)+6)*log(log (x)^2+(-2*exp(3)-1)*log(x)+exp(3)^2+4)+2*log(x)^3-4*exp(3)*log(x)^2+(2*exp (3)^2+4)*log(x)-2*exp(3)^2+4*exp(3)-6)/(x*log(x)^2+(-2*x*exp(3)-x)*log(x)+ x*exp(3)^2+4*x),x, algorithm="fricas")
Output:
2*(log(x) - 1)*log(-(2*e^3 + 1)*log(x) + log(x)^2 + e^6 + 4) + log(-(2*e^3 + 1)*log(x) + log(x)^2 + e^6 + 4)^2 + log(x)^2 - 2*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (20) = 40\).
Time = 0.24 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.83 \[ \int \frac {-6+4 e^3-2 e^6+\left (4+2 e^6\right ) \log (x)-4 e^3 \log ^2(x)+2 \log ^3(x)+\left (6-4 e^3+2 e^6+\left (2-4 e^3\right ) \log (x)+2 \log ^2(x)\right ) \log \left (4+e^6+\left (-1-2 e^3\right ) \log (x)+\log ^2(x)\right )}{4 x+e^6 x+\left (-x-2 e^3 x\right ) \log (x)+x \log ^2(x)} \, dx=\log {\left (x \right )}^{2} + 2 \log {\left (x \right )} \log {\left (\log {\left (x \right )}^{2} + \left (- 2 e^{3} - 1\right ) \log {\left (x \right )} + 4 + e^{6} \right )} - 2 \log {\left (x \right )} + \log {\left (\log {\left (x \right )}^{2} + \left (- 2 e^{3} - 1\right ) \log {\left (x \right )} + 4 + e^{6} \right )}^{2} - 2 \log {\left (\log {\left (x \right )}^{2} + \left (- 2 e^{3} - 1\right ) \log {\left (x \right )} + 4 + e^{6} \right )} \] Input:
integrate(((2*ln(x)**2+(-4*exp(3)+2)*ln(x)+2*exp(3)**2-4*exp(3)+6)*ln(ln(x )**2+(-2*exp(3)-1)*ln(x)+exp(3)**2+4)+2*ln(x)**3-4*exp(3)*ln(x)**2+(2*exp( 3)**2+4)*ln(x)-2*exp(3)**2+4*exp(3)-6)/(x*ln(x)**2+(-2*x*exp(3)-x)*ln(x)+x *exp(3)**2+4*x),x)
Output:
log(x)**2 + 2*log(x)*log(log(x)**2 + (-2*exp(3) - 1)*log(x) + 4 + exp(6)) - 2*log(x) + log(log(x)**2 + (-2*exp(3) - 1)*log(x) + 4 + exp(6))**2 - 2*l og(log(x)**2 + (-2*exp(3) - 1)*log(x) + 4 + exp(6))
Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (22) = 44\).
Time = 0.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.39 \[ \int \frac {-6+4 e^3-2 e^6+\left (4+2 e^6\right ) \log (x)-4 e^3 \log ^2(x)+2 \log ^3(x)+\left (6-4 e^3+2 e^6+\left (2-4 e^3\right ) \log (x)+2 \log ^2(x)\right ) \log \left (4+e^6+\left (-1-2 e^3\right ) \log (x)+\log ^2(x)\right )}{4 x+e^6 x+\left (-x-2 e^3 x\right ) \log (x)+x \log ^2(x)} \, dx=2 \, {\left (\log \left (x\right ) - 1\right )} \log \left (-{\left (2 \, e^{3} + 1\right )} \log \left (x\right ) + \log \left (x\right )^{2} + e^{6} + 4\right ) + \log \left (-{\left (2 \, e^{3} + 1\right )} \log \left (x\right ) + \log \left (x\right )^{2} + e^{6} + 4\right )^{2} + \log \left (x\right )^{2} - 2 \, \log \left (x\right ) \] Input:
integrate(((2*log(x)^2+(-4*exp(3)+2)*log(x)+2*exp(3)^2-4*exp(3)+6)*log(log (x)^2+(-2*exp(3)-1)*log(x)+exp(3)^2+4)+2*log(x)^3-4*exp(3)*log(x)^2+(2*exp (3)^2+4)*log(x)-2*exp(3)^2+4*exp(3)-6)/(x*log(x)^2+(-2*x*exp(3)-x)*log(x)+ x*exp(3)^2+4*x),x, algorithm="maxima")
Output:
2*(log(x) - 1)*log(-(2*e^3 + 1)*log(x) + log(x)^2 + e^6 + 4) + log(-(2*e^3 + 1)*log(x) + log(x)^2 + e^6 + 4)^2 + log(x)^2 - 2*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (22) = 44\).
Time = 0.22 (sec) , antiderivative size = 74, normalized size of antiderivative = 3.22 \[ \int \frac {-6+4 e^3-2 e^6+\left (4+2 e^6\right ) \log (x)-4 e^3 \log ^2(x)+2 \log ^3(x)+\left (6-4 e^3+2 e^6+\left (2-4 e^3\right ) \log (x)+2 \log ^2(x)\right ) \log \left (4+e^6+\left (-1-2 e^3\right ) \log (x)+\log ^2(x)\right )}{4 x+e^6 x+\left (-x-2 e^3 x\right ) \log (x)+x \log ^2(x)} \, dx=\log \left (-2 \, e^{3} \log \left (x\right ) + \log \left (x\right )^{2} + e^{6} - \log \left (x\right ) + 4\right )^{2} + 2 \, \log \left (-2 \, e^{3} \log \left (x\right ) + \log \left (x\right )^{2} + e^{6} - \log \left (x\right ) + 4\right ) \log \left (x\right ) + \log \left (x\right )^{2} - 2 \, \log \left (-2 \, e^{3} \log \left (x\right ) + \log \left (x\right )^{2} + e^{6} - \log \left (x\right ) + 4\right ) - 2 \, \log \left (x\right ) \] Input:
integrate(((2*log(x)^2+(-4*exp(3)+2)*log(x)+2*exp(3)^2-4*exp(3)+6)*log(log (x)^2+(-2*exp(3)-1)*log(x)+exp(3)^2+4)+2*log(x)^3-4*exp(3)*log(x)^2+(2*exp (3)^2+4)*log(x)-2*exp(3)^2+4*exp(3)-6)/(x*log(x)^2+(-2*x*exp(3)-x)*log(x)+ x*exp(3)^2+4*x),x, algorithm="giac")
Output:
log(-2*e^3*log(x) + log(x)^2 + e^6 - log(x) + 4)^2 + 2*log(-2*e^3*log(x) + log(x)^2 + e^6 - log(x) + 4)*log(x) + log(x)^2 - 2*log(-2*e^3*log(x) + lo g(x)^2 + e^6 - log(x) + 4) - 2*log(x)
Time = 2.80 (sec) , antiderivative size = 74, normalized size of antiderivative = 3.22 \[ \int \frac {-6+4 e^3-2 e^6+\left (4+2 e^6\right ) \log (x)-4 e^3 \log ^2(x)+2 \log ^3(x)+\left (6-4 e^3+2 e^6+\left (2-4 e^3\right ) \log (x)+2 \log ^2(x)\right ) \log \left (4+e^6+\left (-1-2 e^3\right ) \log (x)+\log ^2(x)\right )}{4 x+e^6 x+\left (-x-2 e^3 x\right ) \log (x)+x \log ^2(x)} \, dx={\ln \left ({\mathrm {e}}^6-\ln \left (x\right )+{\ln \left (x\right )}^2-2\,{\mathrm {e}}^3\,\ln \left (x\right )+4\right )}^2+2\,\ln \left ({\mathrm {e}}^6-\ln \left (x\right )+{\ln \left (x\right )}^2-2\,{\mathrm {e}}^3\,\ln \left (x\right )+4\right )\,\ln \left (x\right )-2\,\ln \left ({\mathrm {e}}^6-\ln \left (x\right )+{\ln \left (x\right )}^2-2\,{\mathrm {e}}^3\,\ln \left (x\right )+4\right )+{\ln \left (x\right )}^2-2\,\ln \left (x\right ) \] Input:
int((4*exp(3) - 2*exp(6) - 4*exp(3)*log(x)^2 + log(exp(6) + log(x)^2 - log (x)*(2*exp(3) + 1) + 4)*(2*exp(6) - 4*exp(3) + 2*log(x)^2 - log(x)*(4*exp( 3) - 2) + 6) + 2*log(x)^3 + log(x)*(2*exp(6) + 4) - 6)/(4*x + x*log(x)^2 + x*exp(6) - log(x)*(x + 2*x*exp(3))),x)
Output:
log(x)^2 - 2*log(x) - 2*log(exp(6) - log(x) + log(x)^2 - 2*exp(3)*log(x) + 4) + 2*log(exp(6) - log(x) + log(x)^2 - 2*exp(3)*log(x) + 4)*log(x) + log (exp(6) - log(x) + log(x)^2 - 2*exp(3)*log(x) + 4)^2
Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.48 \[ \int \frac {-6+4 e^3-2 e^6+\left (4+2 e^6\right ) \log (x)-4 e^3 \log ^2(x)+2 \log ^3(x)+\left (6-4 e^3+2 e^6+\left (2-4 e^3\right ) \log (x)+2 \log ^2(x)\right ) \log \left (4+e^6+\left (-1-2 e^3\right ) \log (x)+\log ^2(x)\right )}{4 x+e^6 x+\left (-x-2 e^3 x\right ) \log (x)+x \log ^2(x)} \, dx=\mathrm {log}\left (\mathrm {log}\left (x \right )^{2}-2 \,\mathrm {log}\left (x \right ) e^{3}-\mathrm {log}\left (x \right )+e^{6}+4\right )^{2}+2 \,\mathrm {log}\left (\mathrm {log}\left (x \right )^{2}-2 \,\mathrm {log}\left (x \right ) e^{3}-\mathrm {log}\left (x \right )+e^{6}+4\right ) \mathrm {log}\left (x \right )-2 \,\mathrm {log}\left (\mathrm {log}\left (x \right )^{2}-2 \,\mathrm {log}\left (x \right ) e^{3}-\mathrm {log}\left (x \right )+e^{6}+4\right )+\mathrm {log}\left (x \right )^{2}-2 \,\mathrm {log}\left (x \right ) \] Input:
int(((2*log(x)^2+(-4*exp(3)+2)*log(x)+2*exp(3)^2-4*exp(3)+6)*log(log(x)^2+ (-2*exp(3)-1)*log(x)+exp(3)^2+4)+2*log(x)^3-4*exp(3)*log(x)^2+(2*exp(3)^2+ 4)*log(x)-2*exp(3)^2+4*exp(3)-6)/(x*log(x)^2+(-2*x*exp(3)-x)*log(x)+x*exp( 3)^2+4*x),x)
Output:
log(log(x)**2 - 2*log(x)*e**3 - log(x) + e**6 + 4)**2 + 2*log(log(x)**2 - 2*log(x)*e**3 - log(x) + e**6 + 4)*log(x) - 2*log(log(x)**2 - 2*log(x)*e** 3 - log(x) + e**6 + 4) + log(x)**2 - 2*log(x)