Integrand size = 171, antiderivative size = 24 \[ \int \frac {\left (-128 x^3+256 x^4+\left (128 x^2-256 x^3\right ) \log (2)+\left (-128 x^2+256 x^3+\left (128 x-256 x^2\right ) \log (2)\right ) \log (x)\right ) \log \left (-12-x+x^2\right )+\left (768 x+64 x^2-64 x^3+\left (-768-832 x+64 x^3\right ) \log (2)+\left (-768 x-64 x^2+64 x^3\right ) \log (x)\right ) \log ^2\left (-12-x+x^2\right )}{-12 x^3-x^4+x^5+\left (-24 x^2-2 x^3+2 x^4\right ) \log (x)+\left (-12 x-x^2+x^3\right ) \log ^2(x)} \, dx=\frac {64 (x-\log (2)) \log ^2((-4+x) (3+x))}{x+\log (x)} \]
\[ \int \frac {\left (-128 x^3+256 x^4+\left (128 x^2-256 x^3\right ) \log (2)+\left (-128 x^2+256 x^3+\left (128 x-256 x^2\right ) \log (2)\right ) \log (x)\right ) \log \left (-12-x+x^2\right )+\left (768 x+64 x^2-64 x^3+\left (-768-832 x+64 x^3\right ) \log (2)+\left (-768 x-64 x^2+64 x^3\right ) \log (x)\right ) \log ^2\left (-12-x+x^2\right )}{-12 x^3-x^4+x^5+\left (-24 x^2-2 x^3+2 x^4\right ) \log (x)+\left (-12 x-x^2+x^3\right ) \log ^2(x)} \, dx=\int \frac {\left (-128 x^3+256 x^4+\left (128 x^2-256 x^3\right ) \log (2)+\left (-128 x^2+256 x^3+\left (128 x-256 x^2\right ) \log (2)\right ) \log (x)\right ) \log \left (-12-x+x^2\right )+\left (768 x+64 x^2-64 x^3+\left (-768-832 x+64 x^3\right ) \log (2)+\left (-768 x-64 x^2+64 x^3\right ) \log (x)\right ) \log ^2\left (-12-x+x^2\right )}{-12 x^3-x^4+x^5+\left (-24 x^2-2 x^3+2 x^4\right ) \log (x)+\left (-12 x-x^2+x^3\right ) \log ^2(x)} \, dx \]
Integrate[((-128*x^3 + 256*x^4 + (128*x^2 - 256*x^3)*Log[2] + (-128*x^2 + 256*x^3 + (128*x - 256*x^2)*Log[2])*Log[x])*Log[-12 - x + x^2] + (768*x + 64*x^2 - 64*x^3 + (-768 - 832*x + 64*x^3)*Log[2] + (-768*x - 64*x^2 + 64*x ^3)*Log[x])*Log[-12 - x + x^2]^2)/(-12*x^3 - x^4 + x^5 + (-24*x^2 - 2*x^3 + 2*x^4)*Log[x] + (-12*x - x^2 + x^3)*Log[x]^2),x]
Integrate[((-128*x^3 + 256*x^4 + (128*x^2 - 256*x^3)*Log[2] + (-128*x^2 + 256*x^3 + (128*x - 256*x^2)*Log[2])*Log[x])*Log[-12 - x + x^2] + (768*x + 64*x^2 - 64*x^3 + (-768 - 832*x + 64*x^3)*Log[2] + (-768*x - 64*x^2 + 64*x ^3)*Log[x])*Log[-12 - x + x^2]^2)/(-12*x^3 - x^4 + x^5 + (-24*x^2 - 2*x^3 + 2*x^4)*Log[x] + (-12*x - x^2 + x^3)*Log[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.
| \(\displaystyle \int \frac {\left (-64 x^3+\left (64 x^3-832 x-768\right ) \log (2)+64 x^2+\left (64 x^3-64 x^2-768 x\right ) \log (x)+768 x\right ) \log ^2\left (x^2-x-12\right )+\left (256 x^4-128 x^3+\left (256 x^3-128 x^2+\left (128 x-256 x^2\right ) \log (2)\right ) \log (x)+\left (128 x^2-256 x^3\right ) \log (2)\right ) \log \left (x^2-x-12\right )}{x^5-x^4-12 x^3+\left (x^3-x^2-12 x\right ) \log ^2(x)+\left (2 x^4-2 x^3-24 x^2\right ) \log (x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-\left (\left (-64 x^3+\left (64 x^3-832 x-768\right ) \log (2)+64 x^2+\left (64 x^3-64 x^2-768 x\right ) \log (x)+768 x\right ) \log ^2\left (x^2-x-12\right )\right )-\left (256 x^4-128 x^3+\left (256 x^3-128 x^2+\left (128 x-256 x^2\right ) \log (2)\right ) \log (x)+\left (128 x^2-256 x^3\right ) \log (2)\right ) \log \left (x^2-x-12\right )}{x \left (-x^2+x+12\right ) (x+\log (x))^2}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {64 (x \log (x)-x (1-\log (2))+\log (2)) \log ^2\left (x^2-x-12\right )}{x (x+\log (x))^2}+\frac {128 (2 x-1) (x-\log (2)) \log \left (x^2-x-12\right )}{(x-4) (x+3) (x+\log (x))}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -64 (1-\log (2)) \int \frac {\log ^2\left (x^2-x-12\right )}{(x+\log (x))^2}dx+64 \log (2) \int \frac {\log ^2\left (x^2-x-12\right )}{x (x+\log (x))^2}dx+64 \int \frac {\log (x) \log ^2\left (x^2-x-12\right )}{(x+\log (x))^2}dx+256 \int \frac {\log \left (x^2-x-12\right )}{x+\log (x)}dx+128 (4-\log (2)) \int \frac {\log \left (x^2-x-12\right )}{(x-4) (x+\log (x))}dx-128 (3+\log (2)) \int \frac {\log \left (x^2-x-12\right )}{(x+3) (x+\log (x))}dx\) |
Int[((-128*x^3 + 256*x^4 + (128*x^2 - 256*x^3)*Log[2] + (-128*x^2 + 256*x^ 3 + (128*x - 256*x^2)*Log[2])*Log[x])*Log[-12 - x + x^2] + (768*x + 64*x^2 - 64*x^3 + (-768 - 832*x + 64*x^3)*Log[2] + (-768*x - 64*x^2 + 64*x^3)*Lo g[x])*Log[-12 - x + x^2]^2)/(-12*x^3 - x^4 + x^5 + (-24*x^2 - 2*x^3 + 2*x^ 4)*Log[x] + (-12*x - x^2 + x^3)*Log[x]^2),x]
3.13.4.3.1 Defintions of rubi rules used
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 9.44 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08
| method | result | size |
| risch | \(-\frac {64 \left (\ln \left (2\right )-x \right ) \ln \left (x^{2}-x -12\right )^{2}}{x +\ln \left (x \right )}\) | \(26\) |
| parallelrisch | \(-\frac {64 \ln \left (x^{2}-x -12\right )^{2} \ln \left (2\right )-64 \ln \left (x^{2}-x -12\right )^{2} x}{x +\ln \left (x \right )}\) | \(39\) |
int((((64*x^3-64*x^2-768*x)*ln(x)+(64*x^3-832*x-768)*ln(2)-64*x^3+64*x^2+7 68*x)*ln(x^2-x-12)^2+(((-256*x^2+128*x)*ln(2)+256*x^3-128*x^2)*ln(x)+(-256 *x^3+128*x^2)*ln(2)+256*x^4-128*x^3)*ln(x^2-x-12))/((x^3-x^2-12*x)*ln(x)^2 +(2*x^4-2*x^3-24*x^2)*ln(x)+x^5-x^4-12*x^3),x,method=_RETURNVERBOSE)
Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {\left (-128 x^3+256 x^4+\left (128 x^2-256 x^3\right ) \log (2)+\left (-128 x^2+256 x^3+\left (128 x-256 x^2\right ) \log (2)\right ) \log (x)\right ) \log \left (-12-x+x^2\right )+\left (768 x+64 x^2-64 x^3+\left (-768-832 x+64 x^3\right ) \log (2)+\left (-768 x-64 x^2+64 x^3\right ) \log (x)\right ) \log ^2\left (-12-x+x^2\right )}{-12 x^3-x^4+x^5+\left (-24 x^2-2 x^3+2 x^4\right ) \log (x)+\left (-12 x-x^2+x^3\right ) \log ^2(x)} \, dx=\frac {64 \, {\left (x - \log \left (2\right )\right )} \log \left (x^{2} - x - 12\right )^{2}}{x + \log \left (x\right )} \]
integrate((((64*x^3-64*x^2-768*x)*log(x)+(64*x^3-832*x-768)*log(2)-64*x^3+ 64*x^2+768*x)*log(x^2-x-12)^2+(((-256*x^2+128*x)*log(2)+256*x^3-128*x^2)*l og(x)+(-256*x^3+128*x^2)*log(2)+256*x^4-128*x^3)*log(x^2-x-12))/((x^3-x^2- 12*x)*log(x)^2+(2*x^4-2*x^3-24*x^2)*log(x)+x^5-x^4-12*x^3),x, algorithm=\
Exception generated. \[ \int \frac {\left (-128 x^3+256 x^4+\left (128 x^2-256 x^3\right ) \log (2)+\left (-128 x^2+256 x^3+\left (128 x-256 x^2\right ) \log (2)\right ) \log (x)\right ) \log \left (-12-x+x^2\right )+\left (768 x+64 x^2-64 x^3+\left (-768-832 x+64 x^3\right ) \log (2)+\left (-768 x-64 x^2+64 x^3\right ) \log (x)\right ) \log ^2\left (-12-x+x^2\right )}{-12 x^3-x^4+x^5+\left (-24 x^2-2 x^3+2 x^4\right ) \log (x)+\left (-12 x-x^2+x^3\right ) \log ^2(x)} \, dx=\text {Exception raised: TypeError} \]
integrate((((64*x**3-64*x**2-768*x)*ln(x)+(64*x**3-832*x-768)*ln(2)-64*x** 3+64*x**2+768*x)*ln(x**2-x-12)**2+(((-256*x**2+128*x)*ln(2)+256*x**3-128*x **2)*ln(x)+(-256*x**3+128*x**2)*ln(2)+256*x**4-128*x**3)*ln(x**2-x-12))/(( x**3-x**2-12*x)*ln(x)**2+(2*x**4-2*x**3-24*x**2)*ln(x)+x**5-x**4-12*x**3), x)
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (24) = 48\).
Time = 0.33 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.12 \[ \int \frac {\left (-128 x^3+256 x^4+\left (128 x^2-256 x^3\right ) \log (2)+\left (-128 x^2+256 x^3+\left (128 x-256 x^2\right ) \log (2)\right ) \log (x)\right ) \log \left (-12-x+x^2\right )+\left (768 x+64 x^2-64 x^3+\left (-768-832 x+64 x^3\right ) \log (2)+\left (-768 x-64 x^2+64 x^3\right ) \log (x)\right ) \log ^2\left (-12-x+x^2\right )}{-12 x^3-x^4+x^5+\left (-24 x^2-2 x^3+2 x^4\right ) \log (x)+\left (-12 x-x^2+x^3\right ) \log ^2(x)} \, dx=\frac {64 \, {\left ({\left (x - \log \left (2\right )\right )} \log \left (x + 3\right )^{2} + 2 \, {\left (x - \log \left (2\right )\right )} \log \left (x + 3\right ) \log \left (x - 4\right ) + {\left (x - \log \left (2\right )\right )} \log \left (x - 4\right )^{2}\right )}}{x + \log \left (x\right )} \]
integrate((((64*x^3-64*x^2-768*x)*log(x)+(64*x^3-832*x-768)*log(2)-64*x^3+ 64*x^2+768*x)*log(x^2-x-12)^2+(((-256*x^2+128*x)*log(2)+256*x^3-128*x^2)*l og(x)+(-256*x^3+128*x^2)*log(2)+256*x^4-128*x^3)*log(x^2-x-12))/((x^3-x^2- 12*x)*log(x)^2+(2*x^4-2*x^3-24*x^2)*log(x)+x^5-x^4-12*x^3),x, algorithm=\
64*((x - log(2))*log(x + 3)^2 + 2*(x - log(2))*log(x + 3)*log(x - 4) + (x - log(2))*log(x - 4)^2)/(x + log(x))
Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {\left (-128 x^3+256 x^4+\left (128 x^2-256 x^3\right ) \log (2)+\left (-128 x^2+256 x^3+\left (128 x-256 x^2\right ) \log (2)\right ) \log (x)\right ) \log \left (-12-x+x^2\right )+\left (768 x+64 x^2-64 x^3+\left (-768-832 x+64 x^3\right ) \log (2)+\left (-768 x-64 x^2+64 x^3\right ) \log (x)\right ) \log ^2\left (-12-x+x^2\right )}{-12 x^3-x^4+x^5+\left (-24 x^2-2 x^3+2 x^4\right ) \log (x)+\left (-12 x-x^2+x^3\right ) \log ^2(x)} \, dx=\frac {64 \, {\left (x - \log \left (2\right )\right )} \log \left (x^{2} - x - 12\right )^{2}}{x + \log \left (x\right )} \]
integrate((((64*x^3-64*x^2-768*x)*log(x)+(64*x^3-832*x-768)*log(2)-64*x^3+ 64*x^2+768*x)*log(x^2-x-12)^2+(((-256*x^2+128*x)*log(2)+256*x^3-128*x^2)*l og(x)+(-256*x^3+128*x^2)*log(2)+256*x^4-128*x^3)*log(x^2-x-12))/((x^3-x^2- 12*x)*log(x)^2+(2*x^4-2*x^3-24*x^2)*log(x)+x^5-x^4-12*x^3),x, algorithm=\
Time = 10.39 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.79 \[ \int \frac {\left (-128 x^3+256 x^4+\left (128 x^2-256 x^3\right ) \log (2)+\left (-128 x^2+256 x^3+\left (128 x-256 x^2\right ) \log (2)\right ) \log (x)\right ) \log \left (-12-x+x^2\right )+\left (768 x+64 x^2-64 x^3+\left (-768-832 x+64 x^3\right ) \log (2)+\left (-768 x-64 x^2+64 x^3\right ) \log (x)\right ) \log ^2\left (-12-x+x^2\right )}{-12 x^3-x^4+x^5+\left (-24 x^2-2 x^3+2 x^4\right ) \log (x)+\left (-12 x-x^2+x^3\right ) \log ^2(x)} \, dx=-{\ln \left (x^2-x-12\right )}^2\,\left (\frac {\frac {64\,x}{x+1}+\ln \left (x\right )\,\left (\frac {64\,x}{x+1}+\frac {64}{x+1}\right )+\frac {64\,\left (\ln \left (2\right )-x+x\,\ln \left (2\right )\right )}{x+1}}{x+\ln \left (x\right )}-64\right ) \]
int((log(x^2 - x - 12)^2*(log(2)*(832*x - 64*x^3 + 768) - 768*x - 64*x^2 + 64*x^3 + log(x)*(768*x + 64*x^2 - 64*x^3)) - log(x^2 - x - 12)*(log(x)*(l og(2)*(128*x - 256*x^2) - 128*x^2 + 256*x^3) + log(2)*(128*x^2 - 256*x^3) - 128*x^3 + 256*x^4))/(log(x)*(24*x^2 + 2*x^3 - 2*x^4) + 12*x^3 + x^4 - x^ 5 + log(x)^2*(12*x + x^2 - x^3)),x)