Integrand size = 107, antiderivative size = 22 \[ \int \frac {-240 x-48 x^2+\left (240 x+48 x^2\right ) \log (x)+\left (\left (240 x+96 x^2\right ) \log (x)+(45+18 x) \log ^2(x)\right ) \log \left (\frac {16 x+3 \log (x)}{\log (x)}\right ) \log \left (2 \log \left (\frac {16 x+3 \log (x)}{\log (x)}\right )\right )}{\left (16 x \log (x)+3 \log ^2(x)\right ) \log \left (\frac {16 x+3 \log (x)}{\log (x)}\right )} \, dx=\log (5)+3 x (5+x) \log \left (2 \log \left (3+\frac {16 x}{\log (x)}\right )\right ) \] Output:
3*ln(2*ln(3+16*x/ln(x)))*(5+x)*x+ln(5)
Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {-240 x-48 x^2+\left (240 x+48 x^2\right ) \log (x)+\left (\left (240 x+96 x^2\right ) \log (x)+(45+18 x) \log ^2(x)\right ) \log \left (\frac {16 x+3 \log (x)}{\log (x)}\right ) \log \left (2 \log \left (\frac {16 x+3 \log (x)}{\log (x)}\right )\right )}{\left (16 x \log (x)+3 \log ^2(x)\right ) \log \left (\frac {16 x+3 \log (x)}{\log (x)}\right )} \, dx=3 x (5+x) \log \left (2 \log \left (3+\frac {16 x}{\log (x)}\right )\right ) \] Input:
Integrate[(-240*x - 48*x^2 + (240*x + 48*x^2)*Log[x] + ((240*x + 96*x^2)*L og[x] + (45 + 18*x)*Log[x]^2)*Log[(16*x + 3*Log[x])/Log[x]]*Log[2*Log[(16* x + 3*Log[x])/Log[x]]])/((16*x*Log[x] + 3*Log[x]^2)*Log[(16*x + 3*Log[x])/ Log[x]]),x]
Output:
3*x*(5 + x)*Log[2*Log[3 + (16*x)/Log[x]]]
Time = 4.10 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.82, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {7292, 7293, 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.
\(\displaystyle \int \frac {-48 x^2+\left (\left (96 x^2+240 x\right ) \log (x)+(18 x+45) \log ^2(x)\right ) \log \left (\frac {16 x+3 \log (x)}{\log (x)}\right ) \log \left (2 \log \left (\frac {16 x+3 \log (x)}{\log (x)}\right )\right )+\left (48 x^2+240 x\right ) \log (x)-240 x}{\left (3 \log ^2(x)+16 x \log (x)\right ) \log \left (\frac {16 x+3 \log (x)}{\log (x)}\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-48 x^2+\left (\left (96 x^2+240 x\right ) \log (x)+(18 x+45) \log ^2(x)\right ) \log \left (\frac {16 x+3 \log (x)}{\log (x)}\right ) \log \left (2 \log \left (\frac {16 x+3 \log (x)}{\log (x)}\right )\right )+\left (48 x^2+240 x\right ) \log (x)-240 x}{\log (x) (16 x+3 \log (x)) \log \left (\frac {16 x+3 \log (x)}{\log (x)}\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {48 x (x+5) (\log (x)-1)}{\log (x) (16 x+3 \log (x)) \log \left (\frac {16 x}{\log (x)}+3\right )}+3 (2 x+5) \log \left (2 \log \left (\frac {16 x}{\log (x)}+3\right )\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3}{4} (2 x+5)^2 \log \left (2 \log \left (\frac {16 x}{\log (x)}+3\right )\right )-\frac {75}{4} \log \left (\log \left (\frac {16 x}{\log (x)}+3\right )\right )\) |
Input:
Int[(-240*x - 48*x^2 + (240*x + 48*x^2)*Log[x] + ((240*x + 96*x^2)*Log[x] + (45 + 18*x)*Log[x]^2)*Log[(16*x + 3*Log[x])/Log[x]]*Log[2*Log[(16*x + 3* Log[x])/Log[x]]])/((16*x*Log[x] + 3*Log[x]^2)*Log[(16*x + 3*Log[x])/Log[x] ]),x]
Output:
(-75*Log[Log[3 + (16*x)/Log[x]]])/4 + (3*(5 + 2*x)^2*Log[2*Log[3 + (16*x)/ Log[x]]])/4
Time = 2.57 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.00
method | result | size |
parallelrisch | \(3 \ln \left (2 \ln \left (\frac {3 \ln \left (x \right )+16 x}{\ln \left (x \right )}\right )\right ) x^{2}+15 \ln \left (2 \ln \left (\frac {3 \ln \left (x \right )+16 x}{\ln \left (x \right )}\right )\right ) x\) | \(44\) |
risch | \(\left (3 x^{2}+15 x \right ) \ln \left (8 \ln \left (2\right )-2 \ln \left (\ln \left (x \right )\right )+2 \ln \left (\frac {3 \ln \left (x \right )}{16}+x \right )-i \pi \,\operatorname {csgn}\left (\frac {i \left (\frac {3 \ln \left (x \right )}{16}+x \right )}{\ln \left (x \right )}\right ) \left (-\operatorname {csgn}\left (\frac {i \left (\frac {3 \ln \left (x \right )}{16}+x \right )}{\ln \left (x \right )}\right )+\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left (\frac {3 \ln \left (x \right )}{16}+x \right )}{\ln \left (x \right )}\right )+\operatorname {csgn}\left (i \left (\frac {3 \ln \left (x \right )}{16}+x \right )\right )\right )\right )\) | \(101\) |
Input:
int((((18*x+45)*ln(x)^2+(96*x^2+240*x)*ln(x))*ln((3*ln(x)+16*x)/ln(x))*ln( 2*ln((3*ln(x)+16*x)/ln(x)))+(48*x^2+240*x)*ln(x)-48*x^2-240*x)/(3*ln(x)^2+ 16*x*ln(x))/ln((3*ln(x)+16*x)/ln(x)),x,method=_RETURNVERBOSE)
Output:
3*ln(2*ln((3*ln(x)+16*x)/ln(x)))*x^2+15*ln(2*ln((3*ln(x)+16*x)/ln(x)))*x
Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {-240 x-48 x^2+\left (240 x+48 x^2\right ) \log (x)+\left (\left (240 x+96 x^2\right ) \log (x)+(45+18 x) \log ^2(x)\right ) \log \left (\frac {16 x+3 \log (x)}{\log (x)}\right ) \log \left (2 \log \left (\frac {16 x+3 \log (x)}{\log (x)}\right )\right )}{\left (16 x \log (x)+3 \log ^2(x)\right ) \log \left (\frac {16 x+3 \log (x)}{\log (x)}\right )} \, dx=3 \, {\left (x^{2} + 5 \, x\right )} \log \left (2 \, \log \left (\frac {16 \, x + 3 \, \log \left (x\right )}{\log \left (x\right )}\right )\right ) \] Input:
integrate((((18*x+45)*log(x)^2+(96*x^2+240*x)*log(x))*log((3*log(x)+16*x)/ log(x))*log(2*log((3*log(x)+16*x)/log(x)))+(48*x^2+240*x)*log(x)-48*x^2-24 0*x)/(3*log(x)^2+16*x*log(x))/log((3*log(x)+16*x)/log(x)),x, algorithm="fr icas")
Output:
3*(x^2 + 5*x)*log(2*log((16*x + 3*log(x))/log(x)))
Exception generated. \[ \int \frac {-240 x-48 x^2+\left (240 x+48 x^2\right ) \log (x)+\left (\left (240 x+96 x^2\right ) \log (x)+(45+18 x) \log ^2(x)\right ) \log \left (\frac {16 x+3 \log (x)}{\log (x)}\right ) \log \left (2 \log \left (\frac {16 x+3 \log (x)}{\log (x)}\right )\right )}{\left (16 x \log (x)+3 \log ^2(x)\right ) \log \left (\frac {16 x+3 \log (x)}{\log (x)}\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((((18*x+45)*ln(x)**2+(96*x**2+240*x)*ln(x))*ln((3*ln(x)+16*x)/ln (x))*ln(2*ln((3*ln(x)+16*x)/ln(x)))+(48*x**2+240*x)*ln(x)-48*x**2-240*x)/( 3*ln(x)**2+16*x*ln(x))/ln((3*ln(x)+16*x)/ln(x)),x)
Output:
Exception raised: TypeError >> '>' not supported between instances of 'Pol y' and 'int'
Time = 0.16 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.73 \[ \int \frac {-240 x-48 x^2+\left (240 x+48 x^2\right ) \log (x)+\left (\left (240 x+96 x^2\right ) \log (x)+(45+18 x) \log ^2(x)\right ) \log \left (\frac {16 x+3 \log (x)}{\log (x)}\right ) \log \left (2 \log \left (\frac {16 x+3 \log (x)}{\log (x)}\right )\right )}{\left (16 x \log (x)+3 \log ^2(x)\right ) \log \left (\frac {16 x+3 \log (x)}{\log (x)}\right )} \, dx=3 \, x^{2} \log \left (2\right ) + 15 \, x \log \left (2\right ) + 3 \, {\left (x^{2} + 5 \, x\right )} \log \left (\log \left (16 \, x + 3 \, \log \left (x\right )\right ) - \log \left (\log \left (x\right )\right )\right ) \] Input:
integrate((((18*x+45)*log(x)^2+(96*x^2+240*x)*log(x))*log((3*log(x)+16*x)/ log(x))*log(2*log((3*log(x)+16*x)/log(x)))+(48*x^2+240*x)*log(x)-48*x^2-24 0*x)/(3*log(x)^2+16*x*log(x))/log((3*log(x)+16*x)/log(x)),x, algorithm="ma xima")
Output:
3*x^2*log(2) + 15*x*log(2) + 3*(x^2 + 5*x)*log(log(16*x + 3*log(x)) - log( log(x)))
Time = 0.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23 \[ \int \frac {-240 x-48 x^2+\left (240 x+48 x^2\right ) \log (x)+\left (\left (240 x+96 x^2\right ) \log (x)+(45+18 x) \log ^2(x)\right ) \log \left (\frac {16 x+3 \log (x)}{\log (x)}\right ) \log \left (2 \log \left (\frac {16 x+3 \log (x)}{\log (x)}\right )\right )}{\left (16 x \log (x)+3 \log ^2(x)\right ) \log \left (\frac {16 x+3 \log (x)}{\log (x)}\right )} \, dx=3 \, {\left (x^{2} + 5 \, x\right )} \log \left (2 \, \log \left (16 \, x + 3 \, \log \left (x\right )\right ) - 2 \, \log \left (\log \left (x\right )\right )\right ) \] Input:
integrate((((18*x+45)*log(x)^2+(96*x^2+240*x)*log(x))*log((3*log(x)+16*x)/ log(x))*log(2*log((3*log(x)+16*x)/log(x)))+(48*x^2+240*x)*log(x)-48*x^2-24 0*x)/(3*log(x)^2+16*x*log(x))/log((3*log(x)+16*x)/log(x)),x, algorithm="gi ac")
Output:
3*(x^2 + 5*x)*log(2*log(16*x + 3*log(x)) - 2*log(log(x)))
Time = 2.06 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {-240 x-48 x^2+\left (240 x+48 x^2\right ) \log (x)+\left (\left (240 x+96 x^2\right ) \log (x)+(45+18 x) \log ^2(x)\right ) \log \left (\frac {16 x+3 \log (x)}{\log (x)}\right ) \log \left (2 \log \left (\frac {16 x+3 \log (x)}{\log (x)}\right )\right )}{\left (16 x \log (x)+3 \log ^2(x)\right ) \log \left (\frac {16 x+3 \log (x)}{\log (x)}\right )} \, dx=\left (3\,x^2+15\,x\right )\,\left (\ln \left (2\right )+\ln \left (\ln \left (\frac {16\,x+3\,\ln \left (x\right )}{\ln \left (x\right )}\right )\right )\right ) \] Input:
int(-(240*x - log(x)*(240*x + 48*x^2) + 48*x^2 - log(2*log((16*x + 3*log(x ))/log(x)))*log((16*x + 3*log(x))/log(x))*(log(x)*(240*x + 96*x^2) + log(x )^2*(18*x + 45)))/(log((16*x + 3*log(x))/log(x))*(3*log(x)^2 + 16*x*log(x) )),x)
Output:
(15*x + 3*x^2)*(log(2) + log(log((16*x + 3*log(x))/log(x))))
Time = 0.18 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {-240 x-48 x^2+\left (240 x+48 x^2\right ) \log (x)+\left (\left (240 x+96 x^2\right ) \log (x)+(45+18 x) \log ^2(x)\right ) \log \left (\frac {16 x+3 \log (x)}{\log (x)}\right ) \log \left (2 \log \left (\frac {16 x+3 \log (x)}{\log (x)}\right )\right )}{\left (16 x \log (x)+3 \log ^2(x)\right ) \log \left (\frac {16 x+3 \log (x)}{\log (x)}\right )} \, dx=3 \,\mathrm {log}\left (2 \,\mathrm {log}\left (\frac {3 \,\mathrm {log}\left (x \right )+16 x}{\mathrm {log}\left (x \right )}\right )\right ) x \left (x +5\right ) \] Input:
int((((18*x+45)*log(x)^2+(96*x^2+240*x)*log(x))*log((3*log(x)+16*x)/log(x) )*log(2*log((3*log(x)+16*x)/log(x)))+(48*x^2+240*x)*log(x)-48*x^2-240*x)/( 3*log(x)^2+16*x*log(x))/log((3*log(x)+16*x)/log(x)),x)
Output:
3*log(2*log((3*log(x) + 16*x)/log(x)))*x*(x + 5)