Integrand size = 45, antiderivative size = 22 \[ \int \frac {1-2 e^7+\left (7-6 e^7\right ) \log (x)+6 \log ^2(x)}{\left (x-2 e^7 x\right ) \log (x)+2 x \log ^2(x)} \, dx=2+\log \left (3 x^3 \log (x) \left (1+2 \left (-e^7+\log (x)\right )\right )\right ) \] Output:
ln(3*ln(x)*x^3*(1+2*ln(x)-2*exp(7)))+2
Time = 0.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {1-2 e^7+\left (7-6 e^7\right ) \log (x)+6 \log ^2(x)}{\left (x-2 e^7 x\right ) \log (x)+2 x \log ^2(x)} \, dx=3 \log (x)+\log (\log (x))+\log \left (1-2 e^7+2 \log (x)\right ) \] Input:
Integrate[(1 - 2*E^7 + (7 - 6*E^7)*Log[x] + 6*Log[x]^2)/((x - 2*E^7*x)*Log [x] + 2*x*Log[x]^2),x]
Output:
3*Log[x] + Log[Log[x]] + Log[1 - 2*E^7 + 2*Log[x]]
Time = 0.22 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {3039, 6, 1195, 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.
\(\displaystyle \int \frac {6 \log ^2(x)+\left (7-6 e^7\right ) \log (x)-2 e^7+1}{2 x \log ^2(x)+\left (x-2 e^7 x\right ) \log (x)} \, dx\) |
\(\Big \downarrow \) 3039 |
\(\displaystyle \int \frac {6 \log ^2(x)-6 e^7 \log (x)+7 \log (x)-2 e^7+1}{\log (x) \left (2 \log (x)-2 e^7+1\right )}d\log (x)\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {6 \log ^2(x)+\left (7-6 e^7\right ) \log (x)-2 e^7+1}{\log (x) \left (2 \log (x)-2 e^7+1\right )}d\log (x)\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle \int \left (\frac {1}{\log (x)}-\frac {2}{-2 \log (x)+2 e^7-1}+3\right )d\log (x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \log (x)+\log (\log (x))+\log \left (2 \log (x)-2 e^7+1\right )\) |
Input:
Int[(1 - 2*E^7 + (7 - 6*E^7)*Log[x] + 6*Log[x]^2)/((x - 2*E^7*x)*Log[x] + 2*x*Log[x]^2),x]
Output:
3*Log[x] + Log[Log[x]] + Log[1 - 2*E^7 + 2*Log[x]]
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[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
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]
Time = 0.11 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82
method | result | size |
parallelrisch | \(\ln \left (\ln \left (x \right )\right )+\ln \left (\frac {1}{2}+\ln \left (x \right )-{\mathrm e}^{7}\right )+3 \ln \left (x \right )\) | \(18\) |
default | \(3 \ln \left (x \right )+\ln \left (1+2 \ln \left (x \right )-2 \,{\mathrm e}^{7}\right )+\ln \left (\ln \left (x \right )\right )\) | \(20\) |
norman | \(3 \ln \left (x \right )+\ln \left (\ln \left (x \right )\right )+\ln \left (2 \,{\mathrm e}^{7}-2 \ln \left (x \right )-1\right )\) | \(20\) |
risch | \(3 \ln \left (x \right )+\ln \left (\ln \left (x \right )^{2}+\left (-{\mathrm e}^{7}+\frac {1}{2}\right ) \ln \left (x \right )\right )\) | \(21\) |
Input:
int((6*ln(x)^2+(-6*exp(7)+7)*ln(x)-2*exp(7)+1)/(2*x*ln(x)^2+(-2*x*exp(7)+x )*ln(x)),x,method=_RETURNVERBOSE)
Output:
ln(ln(x))+ln(1/2+ln(x)-exp(7))+3*ln(x)
Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {1-2 e^7+\left (7-6 e^7\right ) \log (x)+6 \log ^2(x)}{\left (x-2 e^7 x\right ) \log (x)+2 x \log ^2(x)} \, dx=3 \, \log \left (x\right ) + \log \left (-2 \, e^{7} + 2 \, \log \left (x\right ) + 1\right ) + \log \left (\log \left (x\right )\right ) \] Input:
integrate((6*log(x)^2+(-6*exp(7)+7)*log(x)-2*exp(7)+1)/(2*x*log(x)^2+(-2*x *exp(7)+x)*log(x)),x, algorithm="fricas")
Output:
3*log(x) + log(-2*e^7 + 2*log(x) + 1) + log(log(x))
Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {1-2 e^7+\left (7-6 e^7\right ) \log (x)+6 \log ^2(x)}{\left (x-2 e^7 x\right ) \log (x)+2 x \log ^2(x)} \, dx=3 \log {\left (x \right )} + \log {\left (\log {\left (x \right )}^{2} + \left (\frac {1}{2} - e^{7}\right ) \log {\left (x \right )} \right )} \] Input:
integrate((6*ln(x)**2+(-6*exp(7)+7)*ln(x)-2*exp(7)+1)/(2*x*ln(x)**2+(-2*x* exp(7)+x)*ln(x)),x)
Output:
3*log(x) + log(log(x)**2 + (1/2 - exp(7))*log(x))
Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (20) = 40\).
Time = 0.12 (sec) , antiderivative size = 373, normalized size of antiderivative = 16.95 \[ \int \frac {1-2 e^7+\left (7-6 e^7\right ) \log (x)+6 \log ^2(x)}{\left (x-2 e^7 x\right ) \log (x)+2 x \log ^2(x)} \, dx=-6 \, {\left (\frac {\log \left (-e^{7} + \log \left (x\right ) + \frac {1}{2}\right )}{2 \, e^{7} - 1} - \frac {\log \left (\log \left (x\right )\right )}{2 \, e^{7} - 1}\right )} e^{7} \log \left (x\right ) + 6 \, {\left (\frac {\log \left (-e^{7} + \log \left (x\right ) + \frac {1}{2}\right )}{2 \, e^{7} - 1} - \frac {\log \left (\log \left (x\right )\right )}{2 \, e^{7} - 1}\right )} \log \left (x\right )^{2} - 3 \, {\left (\frac {{\left (2 \, e^{7} - 2 \, \log \left (x\right ) - 1\right )} \log \left (-e^{7} + \log \left (x\right ) + \frac {1}{2}\right ) - 2 \, e^{7} + 2 \, \log \left (x\right ) + 1}{2 \, e^{7} - 1} + \frac {2 \, {\left (\log \left (x\right ) \log \left (\log \left (x\right )\right ) - \log \left (x\right )\right )}}{2 \, e^{7} - 1}\right )} e^{7} - 2 \, {\left (\frac {\log \left (-e^{7} + \log \left (x\right ) + \frac {1}{2}\right )}{2 \, e^{7} - 1} - \frac {\log \left (\log \left (x\right )\right )}{2 \, e^{7} - 1}\right )} e^{7} + 7 \, {\left (\frac {\log \left (-e^{7} + \log \left (x\right ) + \frac {1}{2}\right )}{2 \, e^{7} - 1} - \frac {\log \left (\log \left (x\right )\right )}{2 \, e^{7} - 1}\right )} \log \left (x\right ) + \frac {3 \, {\left (4 \, \log \left (2\right ) \log \left (x\right )^{2} + 4 \, \log \left (x\right )^{2} \log \left (\log \left (x\right )\right ) + 2 \, {\left (2 \, e^{7} - 1\right )} \log \left (x\right ) - {\left (4 \, \log \left (x\right )^{2} - 4 \, e^{14} + 4 \, e^{7} - 1\right )} \log \left (-2 \, e^{7} + 2 \, \log \left (x\right ) + 1\right )\right )}}{2 \, {\left (2 \, e^{7} - 1\right )}} + \frac {7 \, {\left ({\left (2 \, e^{7} - 2 \, \log \left (x\right ) - 1\right )} \log \left (-e^{7} + \log \left (x\right ) + \frac {1}{2}\right ) - 2 \, e^{7} + 2 \, \log \left (x\right ) + 1\right )}}{2 \, {\left (2 \, e^{7} - 1\right )}} + \frac {7 \, {\left (\log \left (x\right ) \log \left (\log \left (x\right )\right ) - \log \left (x\right )\right )}}{2 \, e^{7} - 1} + \frac {\log \left (-e^{7} + \log \left (x\right ) + \frac {1}{2}\right )}{2 \, e^{7} - 1} - \frac {\log \left (\log \left (x\right )\right )}{2 \, e^{7} - 1} \] Input:
integrate((6*log(x)^2+(-6*exp(7)+7)*log(x)-2*exp(7)+1)/(2*x*log(x)^2+(-2*x *exp(7)+x)*log(x)),x, algorithm="maxima")
Output:
-6*(log(-e^7 + log(x) + 1/2)/(2*e^7 - 1) - log(log(x))/(2*e^7 - 1))*e^7*lo g(x) + 6*(log(-e^7 + log(x) + 1/2)/(2*e^7 - 1) - log(log(x))/(2*e^7 - 1))* log(x)^2 - 3*(((2*e^7 - 2*log(x) - 1)*log(-e^7 + log(x) + 1/2) - 2*e^7 + 2 *log(x) + 1)/(2*e^7 - 1) + 2*(log(x)*log(log(x)) - log(x))/(2*e^7 - 1))*e^ 7 - 2*(log(-e^7 + log(x) + 1/2)/(2*e^7 - 1) - log(log(x))/(2*e^7 - 1))*e^7 + 7*(log(-e^7 + log(x) + 1/2)/(2*e^7 - 1) - log(log(x))/(2*e^7 - 1))*log( x) + 3/2*(4*log(2)*log(x)^2 + 4*log(x)^2*log(log(x)) + 2*(2*e^7 - 1)*log(x ) - (4*log(x)^2 - 4*e^14 + 4*e^7 - 1)*log(-2*e^7 + 2*log(x) + 1))/(2*e^7 - 1) + 7/2*((2*e^7 - 2*log(x) - 1)*log(-e^7 + log(x) + 1/2) - 2*e^7 + 2*log (x) + 1)/(2*e^7 - 1) + 7*(log(x)*log(log(x)) - log(x))/(2*e^7 - 1) + log(- e^7 + log(x) + 1/2)/(2*e^7 - 1) - log(log(x))/(2*e^7 - 1)
Time = 0.12 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.64 \[ \int \frac {1-2 e^7+\left (7-6 e^7\right ) \log (x)+6 \log ^2(x)}{\left (x-2 e^7 x\right ) \log (x)+2 x \log ^2(x)} \, dx=\frac {1}{2} \, \log \left (\pi ^{2} {\left (\mathrm {sgn}\left (x\right ) - 1\right )}^{2} + {\left (2 \, e^{7} - 2 \, \log \left ({\left | x \right |}\right ) - 1\right )}^{2}\right ) + 3 \, \log \left (x\right ) + \log \left ({\left | \log \left (x\right ) \right |}\right ) \] Input:
integrate((6*log(x)^2+(-6*exp(7)+7)*log(x)-2*exp(7)+1)/(2*x*log(x)^2+(-2*x *exp(7)+x)*log(x)),x, algorithm="giac")
Output:
1/2*log(pi^2*(sgn(x) - 1)^2 + (2*e^7 - 2*log(abs(x)) - 1)^2) + 3*log(x) + log(abs(log(x)))
Time = 3.66 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {1-2 e^7+\left (7-6 e^7\right ) \log (x)+6 \log ^2(x)}{\left (x-2 e^7 x\right ) \log (x)+2 x \log ^2(x)} \, dx=\ln \left (\ln \left (x\right )\right )+\ln \left (2\,\ln \left (x\right )-2\,{\mathrm {e}}^7+1\right )+3\,\ln \left (x\right ) \] Input:
int(-(2*exp(7) - 6*log(x)^2 + log(x)*(6*exp(7) - 7) - 1)/(2*x*log(x)^2 + l og(x)*(x - 2*x*exp(7))),x)
Output:
log(log(x)) + log(2*log(x) - 2*exp(7) + 1) + 3*log(x)
Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {1-2 e^7+\left (7-6 e^7\right ) \log (x)+6 \log ^2(x)}{\left (x-2 e^7 x\right ) \log (x)+2 x \log ^2(x)} \, dx=\mathrm {log}\left (\mathrm {log}\left (x \right )\right )+\mathrm {log}\left (2 \,\mathrm {log}\left (x \right )-2 e^{7}+1\right )+3 \,\mathrm {log}\left (x \right ) \] Input:
int((6*log(x)^2+(-6*exp(7)+7)*log(x)-2*exp(7)+1)/(2*x*log(x)^2+(-2*x*exp(7 )+x)*log(x)),x)
Output:
log(log(x)) + log(2*log(x) - 2*e**7 + 1) + 3*log(x)