Integrand size = 96, antiderivative size = 32 \begin {dmath*} \int \frac {-36 x+36 x^2+\left (3 x-3 x^2\right ) \log (x)+\left (33 x-108 x^2+\left (-3 x+9 x^2\right ) \log (x)\right ) \log (2 x)+(-36 x+3 x \log (x)+(72 x-6 x \log (x)) \log (2 x)) \log \left (\frac {-12+\log (x)}{8 x}\right )}{(-12+\log (x)) \log ^2(2 x)} \, dx=\frac {3 x^2 \left (-1+x-\log \left (\frac {-3+\frac {\log (x)}{4}}{2 x}\right )\right )}{\log (2 x)} \end {dmath*}
Time = 0.42 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \begin {dmath*} \int \frac {-36 x+36 x^2+\left (3 x-3 x^2\right ) \log (x)+\left (33 x-108 x^2+\left (-3 x+9 x^2\right ) \log (x)\right ) \log (2 x)+(-36 x+3 x \log (x)+(72 x-6 x \log (x)) \log (2 x)) \log \left (\frac {-12+\log (x)}{8 x}\right )}{(-12+\log (x)) \log ^2(2 x)} \, dx=\frac {3 x^2 \left (-1+x-\log \left (\frac {-12+\log (x)}{8 x}\right )\right )}{\log (2 x)} \end {dmath*}
Integrate[(-36*x + 36*x^2 + (3*x - 3*x^2)*Log[x] + (33*x - 108*x^2 + (-3*x + 9*x^2)*Log[x])*Log[2*x] + (-36*x + 3*x*Log[x] + (72*x - 6*x*Log[x])*Log [2*x])*Log[(-12 + Log[x])/(8*x)])/((-12 + Log[x])*Log[2*x]^2),x]
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.
\(\displaystyle \int \frac {36 x^2+\left (3 x-3 x^2\right ) \log (x)+\left (-108 x^2+\left (9 x^2-3 x\right ) \log (x)+33 x\right ) \log (2 x)-36 x+(-36 x+3 x \log (x)+(72 x-6 x \log (x)) \log (2 x)) \log \left (\frac {\log (x)-12}{8 x}\right )}{(\log (x)-12) \log ^2(2 x)} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {3 x (12 x+x (-\log (x))+3 x \log (x) \log (2 x)-36 x \log (2 x)+\log (x)-\log (x) \log (2 x)+11 \log (2 x)-12)}{(\log (x)-12) \log ^2(2 x)}-\frac {3 x (2 \log (2 x)-1) \log \left (\frac {\log (x)-12}{8 x}\right )}{\log ^2(2 x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -108 \int \frac {x^2}{(\log (x)-12) \log (2 x)}dx+9 \int \frac {x^2 \log (x)}{(\log (x)-12) \log (2 x)}dx+3 \int \frac {x \log \left (\frac {\log (x)-12}{8 x}\right )}{\log ^2(2 x)}dx+33 \int \frac {x}{(\log (x)-12) \log (2 x)}dx-3 \int \frac {x \log (x)}{(\log (x)-12) \log (2 x)}dx-6 \int \frac {x \log \left (\frac {\log (x)-12}{8 x}\right )}{\log (2 x)}dx+\frac {3}{2} \operatorname {ExpIntegralEi}(2 \log (2 x))-\frac {9}{8} \operatorname {ExpIntegralEi}(3 \log (2 x))+\frac {3 x^3}{\log (2 x)}-\frac {3 x^2}{\log (2 x)}\) |
Int[(-36*x + 36*x^2 + (3*x - 3*x^2)*Log[x] + (33*x - 108*x^2 + (-3*x + 9*x ^2)*Log[x])*Log[2*x] + (-36*x + 3*x*Log[x] + (72*x - 6*x*Log[x])*Log[2*x]) *Log[(-12 + Log[x])/(8*x)])/((-12 + Log[x])*Log[2*x]^2),x]
3.2.11.3.1 Defintions of rubi rules used
Time = 2.88 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09
method | result | size |
parallelrisch | \(-\frac {-72 x^{3}+72 x^{2}+72 \ln \left (\frac {\ln \left (x \right )-12}{8 x}\right ) x^{2}}{24 \ln \left (2 x \right )}\) | \(35\) |
risch | \(\frac {6 i x^{2} \ln \left (\ln \left (x \right )-12\right )}{-2 i \ln \left (2\right )-2 i \ln \left (x \right )}+\frac {18 x^{2} \ln \left (2\right )+6 x^{2} \ln \left (x \right )-3 i \pi \,x^{2} \operatorname {csgn}\left (i \left (\ln \left (x \right )-12\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-12\right )}{x}\right )^{2}+3 i \pi \,x^{2} \operatorname {csgn}\left (i \left (\ln \left (x \right )-12\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-12\right )}{x}\right ) \operatorname {csgn}\left (\frac {i}{x}\right )-3 i \pi \,x^{2} \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-12\right )}{x}\right )^{2} \operatorname {csgn}\left (\frac {i}{x}\right )+3 i \pi \,x^{2} \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-12\right )}{x}\right )^{3}+6 x^{3}-6 x^{2}}{2 \ln \left (x \right )+2 \ln \left (2\right )}\) | \(172\) |
int((((-6*x*ln(x)+72*x)*ln(2*x)+3*x*ln(x)-36*x)*ln(1/8*(ln(x)-12)/x)+((9*x ^2-3*x)*ln(x)-108*x^2+33*x)*ln(2*x)+(-3*x^2+3*x)*ln(x)+36*x^2-36*x)/(ln(x) -12)/ln(2*x)^2,x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \begin {dmath*} \int \frac {-36 x+36 x^2+\left (3 x-3 x^2\right ) \log (x)+\left (33 x-108 x^2+\left (-3 x+9 x^2\right ) \log (x)\right ) \log (2 x)+(-36 x+3 x \log (x)+(72 x-6 x \log (x)) \log (2 x)) \log \left (\frac {-12+\log (x)}{8 x}\right )}{(-12+\log (x)) \log ^2(2 x)} \, dx=\frac {3 \, {\left (x^{3} - x^{2} \log \left (\frac {\log \left (x\right ) - 12}{8 \, x}\right ) - x^{2}\right )}}{\log \left (2\right ) + \log \left (x\right )} \end {dmath*}
integrate((((-6*x*log(x)+72*x)*log(2*x)+3*x*log(x)-36*x)*log(1/8*(log(x)-1 2)/x)+((9*x^2-3*x)*log(x)-108*x^2+33*x)*log(2*x)+(-3*x^2+3*x)*log(x)+36*x^ 2-36*x)/(log(x)-12)/log(2*x)^2,x, algorithm=\
Exception generated. \begin {dmath*} \int \frac {-36 x+36 x^2+\left (3 x-3 x^2\right ) \log (x)+\left (33 x-108 x^2+\left (-3 x+9 x^2\right ) \log (x)\right ) \log (2 x)+(-36 x+3 x \log (x)+(72 x-6 x \log (x)) \log (2 x)) \log \left (\frac {-12+\log (x)}{8 x}\right )}{(-12+\log (x)) \log ^2(2 x)} \, dx=\text {Exception raised: TypeError} \end {dmath*}
integrate((((-6*x*ln(x)+72*x)*ln(2*x)+3*x*ln(x)-36*x)*ln(1/8*(ln(x)-12)/x) +((9*x**2-3*x)*ln(x)-108*x**2+33*x)*ln(2*x)+(-3*x**2+3*x)*ln(x)+36*x**2-36 *x)/(ln(x)-12)/ln(2*x)**2,x)
Time = 0.32 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.22 \begin {dmath*} \int \frac {-36 x+36 x^2+\left (3 x-3 x^2\right ) \log (x)+\left (33 x-108 x^2+\left (-3 x+9 x^2\right ) \log (x)\right ) \log (2 x)+(-36 x+3 x \log (x)+(72 x-6 x \log (x)) \log (2 x)) \log \left (\frac {-12+\log (x)}{8 x}\right )}{(-12+\log (x)) \log ^2(2 x)} \, dx=\frac {3 \, {\left (x^{3} + x^{2} {\left (3 \, \log \left (2\right ) - 1\right )} + x^{2} \log \left (x\right ) - x^{2} \log \left (\log \left (x\right ) - 12\right )\right )}}{\log \left (2\right ) + \log \left (x\right )} \end {dmath*}
integrate((((-6*x*log(x)+72*x)*log(2*x)+3*x*log(x)-36*x)*log(1/8*(log(x)-1 2)/x)+((9*x^2-3*x)*log(x)-108*x^2+33*x)*log(2*x)+(-3*x^2+3*x)*log(x)+36*x^ 2-36*x)/(log(x)-12)/log(2*x)^2,x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (26) = 52\).
Time = 0.31 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.19 \begin {dmath*} \int \frac {-36 x+36 x^2+\left (3 x-3 x^2\right ) \log (x)+\left (33 x-108 x^2+\left (-3 x+9 x^2\right ) \log (x)\right ) \log (2 x)+(-36 x+3 x \log (x)+(72 x-6 x \log (x)) \log (2 x)) \log \left (\frac {-12+\log (x)}{8 x}\right )}{(-12+\log (x)) \log ^2(2 x)} \, dx=\frac {3 \, x^{3}}{\log \left (2\right ) + \log \left (x\right )} + \frac {9 \, x^{2} \log \left (2\right )}{\log \left (2\right ) + \log \left (x\right )} + \frac {3 \, x^{2} \log \left (x\right )}{\log \left (2\right ) + \log \left (x\right )} - \frac {3 \, x^{2} \log \left (\log \left (x\right ) - 12\right )}{\log \left (2\right ) + \log \left (x\right )} - \frac {3 \, x^{2}}{\log \left (2\right ) + \log \left (x\right )} \end {dmath*}
integrate((((-6*x*log(x)+72*x)*log(2*x)+3*x*log(x)-36*x)*log(1/8*(log(x)-1 2)/x)+((9*x^2-3*x)*log(x)-108*x^2+33*x)*log(2*x)+(-3*x^2+3*x)*log(x)+36*x^ 2-36*x)/(log(x)-12)/log(2*x)^2,x, algorithm=\
3*x^3/(log(2) + log(x)) + 9*x^2*log(2)/(log(2) + log(x)) + 3*x^2*log(x)/(l og(2) + log(x)) - 3*x^2*log(log(x) - 12)/(log(2) + log(x)) - 3*x^2/(log(2) + log(x))
Time = 14.21 (sec) , antiderivative size = 115, normalized size of antiderivative = 3.59 \begin {dmath*} \int \frac {-36 x+36 x^2+\left (3 x-3 x^2\right ) \log (x)+\left (33 x-108 x^2+\left (-3 x+9 x^2\right ) \log (x)\right ) \log (2 x)+(-36 x+3 x \log (x)+(72 x-6 x \log (x)) \log (2 x)) \log \left (\frac {-12+\log (x)}{8 x}\right )}{(-12+\log (x)) \log ^2(2 x)} \, dx=9\,x^3-6\,x^2-\frac {3\,x^2\,\left (2\,\ln \left (x\right )-2\,\ln \left (2\,x\right )-x+3\,x\,\left (\ln \left (2\,x\right )-\ln \left (x\right )\right )+1\right )+3\,x^2\,\ln \left (x\right )\,\left (3\,x-2\right )}{\ln \left (2\,x\right )}-\frac {\ln \left (\frac {\frac {\ln \left (x\right )}{8}-\frac {3}{2}}{x}\right )\,\left (6\,x^2\,\left (\ln \left (2\,x\right )-\ln \left (x\right )\right )+3\,x^2\,\left (2\,\ln \left (x\right )-2\,\ln \left (2\,x\right )+1\right )\right )}{\ln \left (2\,x\right )} \end {dmath*}
int((log((log(x)/8 - 3/2)/x)*(log(2*x)*(72*x - 6*x*log(x)) - 36*x + 3*x*lo g(x)) - 36*x + log(x)*(3*x - 3*x^2) - log(2*x)*(log(x)*(3*x - 9*x^2) - 33* x + 108*x^2) + 36*x^2)/(log(2*x)^2*(log(x) - 12)),x)