Integrand size = 107, antiderivative size = 19 \[ \int \frac {-3300 x+(26400 x-6600 x \log (x)) \log (4-\log (x)) \log (\log (4-\log (x)))}{(-576+144 \log (x)) \log (4-\log (x))+\left (-26400 x^2+6600 x^2 \log (x)\right ) \log (4-\log (x)) \log (\log (4-\log (x)))+\left (-302500 x^4+75625 x^4 \log (x)\right ) \log (4-\log (x)) \log ^2(\log (4-\log (x)))} \, dx=\frac {1}{1+\frac {275}{12} x^2 \log (\log (4-\log (x)))} \] Output:
1/(275/12*x^2*ln(ln(-ln(x)+4))+1)
Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-3300 x+(26400 x-6600 x \log (x)) \log (4-\log (x)) \log (\log (4-\log (x)))}{(-576+144 \log (x)) \log (4-\log (x))+\left (-26400 x^2+6600 x^2 \log (x)\right ) \log (4-\log (x)) \log (\log (4-\log (x)))+\left (-302500 x^4+75625 x^4 \log (x)\right ) \log (4-\log (x)) \log ^2(\log (4-\log (x)))} \, dx=\frac {12}{12+275 x^2 \log (\log (4-\log (x)))} \] Input:
Integrate[(-3300*x + (26400*x - 6600*x*Log[x])*Log[4 - Log[x]]*Log[Log[4 - Log[x]]])/((-576 + 144*Log[x])*Log[4 - Log[x]] + (-26400*x^2 + 6600*x^2*L og[x])*Log[4 - Log[x]]*Log[Log[4 - Log[x]]] + (-302500*x^4 + 75625*x^4*Log [x])*Log[4 - Log[x]]*Log[Log[4 - Log[x]]]^2),x]
Output:
12/(12 + 275*x^2*Log[Log[4 - Log[x]]])
Time = 0.55 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {7239, 27, 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.
\(\displaystyle \int \frac {(26400 x-6600 x \log (x)) \log (4-\log (x)) \log (\log (4-\log (x)))-3300 x}{\left (75625 x^4 \log (x)-302500 x^4\right ) \log (4-\log (x)) \log ^2(\log (4-\log (x)))+\left (6600 x^2 \log (x)-26400 x^2\right ) \log (4-\log (x)) \log (\log (4-\log (x)))+(144 \log (x)-576) \log (4-\log (x))} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {3300 (x+2 x (\log (x)-4) \log (4-\log (x)) \log (\log (4-\log (x))))}{(4-\log (x)) \log (4-\log (x)) \left (275 x^2 \log (\log (4-\log (x)))+12\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 3300 \int \frac {x-2 x (4-\log (x)) \log (4-\log (x)) \log (\log (4-\log (x)))}{(4-\log (x)) \log (4-\log (x)) \left (275 \log (\log (4-\log (x))) x^2+12\right )^2}dx\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle \frac {12}{275 x^2 \log (\log (4-\log (x)))+12}\) |
Input:
Int[(-3300*x + (26400*x - 6600*x*Log[x])*Log[4 - Log[x]]*Log[Log[4 - Log[x ]]])/((-576 + 144*Log[x])*Log[4 - Log[x]] + (-26400*x^2 + 6600*x^2*Log[x]) *Log[4 - Log[x]]*Log[Log[4 - Log[x]]] + (-302500*x^4 + 75625*x^4*Log[x])*L og[4 - Log[x]]*Log[Log[4 - Log[x]]]^2),x]
Output:
12/(12 + 275*x^2*Log[Log[4 - Log[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_)*(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]]
Time = 14.81 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05
method | result | size |
risch | \(\frac {12}{275 x^{2} \ln \left (\ln \left (-\ln \left (x \right )+4\right )\right )+12}\) | \(20\) |
parallelrisch | \(\frac {12}{275 x^{2} \ln \left (\ln \left (-\ln \left (x \right )+4\right )\right )+12}\) | \(20\) |
default | \(-\frac {3300 x^{2}}{\left (24 \ln \left (-\ln \left (x \right )+4\right ) \ln \left (x \right )-275 x^{2}-96 \ln \left (-\ln \left (x \right )+4\right )\right ) \left (275 x^{2} \ln \left (\ln \left (-\ln \left (x \right )+4\right )\right )+12\right )}+\frac {288 \left (\ln \left (x \right )-4\right ) \ln \left (-\ln \left (x \right )+4\right )}{\left (24 \ln \left (-\ln \left (x \right )+4\right ) \ln \left (x \right )-275 x^{2}-96 \ln \left (-\ln \left (x \right )+4\right )\right ) \left (275 x^{2} \ln \left (\ln \left (-\ln \left (x \right )+4\right )\right )+12\right )}\) | \(110\) |
Input:
int(((-6600*x*ln(x)+26400*x)*ln(-ln(x)+4)*ln(ln(-ln(x)+4))-3300*x)/((75625 *x^4*ln(x)-302500*x^4)*ln(-ln(x)+4)*ln(ln(-ln(x)+4))^2+(6600*x^2*ln(x)-264 00*x^2)*ln(-ln(x)+4)*ln(ln(-ln(x)+4))+(144*ln(x)-576)*ln(-ln(x)+4)),x,meth od=_RETURNVERBOSE)
Output:
12/(275*x^2*ln(ln(-ln(x)+4))+12)
Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-3300 x+(26400 x-6600 x \log (x)) \log (4-\log (x)) \log (\log (4-\log (x)))}{(-576+144 \log (x)) \log (4-\log (x))+\left (-26400 x^2+6600 x^2 \log (x)\right ) \log (4-\log (x)) \log (\log (4-\log (x)))+\left (-302500 x^4+75625 x^4 \log (x)\right ) \log (4-\log (x)) \log ^2(\log (4-\log (x)))} \, dx=\frac {12}{275 \, x^{2} \log \left (\log \left (-\log \left (x\right ) + 4\right )\right ) + 12} \] Input:
integrate(((-6600*x*log(x)+26400*x)*log(-log(x)+4)*log(log(-log(x)+4))-330 0*x)/((75625*x^4*log(x)-302500*x^4)*log(-log(x)+4)*log(log(-log(x)+4))^2+( 6600*x^2*log(x)-26400*x^2)*log(-log(x)+4)*log(log(-log(x)+4))+(144*log(x)- 576)*log(-log(x)+4)),x, algorithm="fricas")
Output:
12/(275*x^2*log(log(-log(x) + 4)) + 12)
Time = 0.15 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {-3300 x+(26400 x-6600 x \log (x)) \log (4-\log (x)) \log (\log (4-\log (x)))}{(-576+144 \log (x)) \log (4-\log (x))+\left (-26400 x^2+6600 x^2 \log (x)\right ) \log (4-\log (x)) \log (\log (4-\log (x)))+\left (-302500 x^4+75625 x^4 \log (x)\right ) \log (4-\log (x)) \log ^2(\log (4-\log (x)))} \, dx=\frac {12}{275 x^{2} \log {\left (\log {\left (4 - \log {\left (x \right )} \right )} \right )} + 12} \] Input:
integrate(((-6600*x*ln(x)+26400*x)*ln(-ln(x)+4)*ln(ln(-ln(x)+4))-3300*x)/( (75625*x**4*ln(x)-302500*x**4)*ln(-ln(x)+4)*ln(ln(-ln(x)+4))**2+(6600*x**2 *ln(x)-26400*x**2)*ln(-ln(x)+4)*ln(ln(-ln(x)+4))+(144*ln(x)-576)*ln(-ln(x) +4)),x)
Output:
12/(275*x**2*log(log(4 - log(x))) + 12)
Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-3300 x+(26400 x-6600 x \log (x)) \log (4-\log (x)) \log (\log (4-\log (x)))}{(-576+144 \log (x)) \log (4-\log (x))+\left (-26400 x^2+6600 x^2 \log (x)\right ) \log (4-\log (x)) \log (\log (4-\log (x)))+\left (-302500 x^4+75625 x^4 \log (x)\right ) \log (4-\log (x)) \log ^2(\log (4-\log (x)))} \, dx=\frac {12}{275 \, x^{2} \log \left (\log \left (-\log \left (x\right ) + 4\right )\right ) + 12} \] Input:
integrate(((-6600*x*log(x)+26400*x)*log(-log(x)+4)*log(log(-log(x)+4))-330 0*x)/((75625*x^4*log(x)-302500*x^4)*log(-log(x)+4)*log(log(-log(x)+4))^2+( 6600*x^2*log(x)-26400*x^2)*log(-log(x)+4)*log(log(-log(x)+4))+(144*log(x)- 576)*log(-log(x)+4)),x, algorithm="maxima")
Output:
12/(275*x^2*log(log(-log(x) + 4)) + 12)
Time = 0.22 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-3300 x+(26400 x-6600 x \log (x)) \log (4-\log (x)) \log (\log (4-\log (x)))}{(-576+144 \log (x)) \log (4-\log (x))+\left (-26400 x^2+6600 x^2 \log (x)\right ) \log (4-\log (x)) \log (\log (4-\log (x)))+\left (-302500 x^4+75625 x^4 \log (x)\right ) \log (4-\log (x)) \log ^2(\log (4-\log (x)))} \, dx=\frac {12}{275 \, x^{2} \log \left (\log \left (-\log \left (x\right ) + 4\right )\right ) + 12} \] Input:
integrate(((-6600*x*log(x)+26400*x)*log(-log(x)+4)*log(log(-log(x)+4))-330 0*x)/((75625*x^4*log(x)-302500*x^4)*log(-log(x)+4)*log(log(-log(x)+4))^2+( 6600*x^2*log(x)-26400*x^2)*log(-log(x)+4)*log(log(-log(x)+4))+(144*log(x)- 576)*log(-log(x)+4)),x, algorithm="giac")
Output:
12/(275*x^2*log(log(-log(x) + 4)) + 12)
Time = 2.22 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-3300 x+(26400 x-6600 x \log (x)) \log (4-\log (x)) \log (\log (4-\log (x)))}{(-576+144 \log (x)) \log (4-\log (x))+\left (-26400 x^2+6600 x^2 \log (x)\right ) \log (4-\log (x)) \log (\log (4-\log (x)))+\left (-302500 x^4+75625 x^4 \log (x)\right ) \log (4-\log (x)) \log ^2(\log (4-\log (x)))} \, dx=\frac {12}{275\,x^2\,\ln \left (\ln \left (4-\ln \left (x\right )\right )\right )+12} \] Input:
int(-(3300*x - log(4 - log(x))*log(log(4 - log(x)))*(26400*x - 6600*x*log( x)))/(log(4 - log(x))*(144*log(x) - 576) + log(4 - log(x))*log(log(4 - log (x)))*(6600*x^2*log(x) - 26400*x^2) + log(4 - log(x))*log(log(4 - log(x))) ^2*(75625*x^4*log(x) - 302500*x^4)),x)
Output:
12/(275*x^2*log(log(4 - log(x))) + 12)
Time = 0.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.58 \[ \int \frac {-3300 x+(26400 x-6600 x \log (x)) \log (4-\log (x)) \log (\log (4-\log (x)))}{(-576+144 \log (x)) \log (4-\log (x))+\left (-26400 x^2+6600 x^2 \log (x)\right ) \log (4-\log (x)) \log (\log (4-\log (x)))+\left (-302500 x^4+75625 x^4 \log (x)\right ) \log (4-\log (x)) \log ^2(\log (4-\log (x)))} \, dx=-\frac {275 \,\mathrm {log}\left (\mathrm {log}\left (-\mathrm {log}\left (x \right )+4\right )\right ) x^{2}}{275 \,\mathrm {log}\left (\mathrm {log}\left (-\mathrm {log}\left (x \right )+4\right )\right ) x^{2}+12} \] Input:
int(((-6600*x*log(x)+26400*x)*log(-log(x)+4)*log(log(-log(x)+4))-3300*x)/( (75625*x^4*log(x)-302500*x^4)*log(-log(x)+4)*log(log(-log(x)+4))^2+(6600*x ^2*log(x)-26400*x^2)*log(-log(x)+4)*log(log(-log(x)+4))+(144*log(x)-576)*l og(-log(x)+4)),x)
Output:
( - 275*log(log( - log(x) + 4))*x**2)/(275*log(log( - log(x) + 4))*x**2 + 12)