Integrand size = 114, antiderivative size = 27 \[ \int \frac {78-48 x+390 x^2+\left (-48 x+5004 x^2-3072 x^3+384 x^4\right ) \log (x)+\left (16224 x^2-19968 x^3+8640 x^4-1536 x^5+96 x^6\right ) \log ^2(x)}{4 x+\left (52 x-32 x^2+4 x^3\right ) \log (x)+\left (169 x-208 x^2+90 x^3-16 x^4+x^5\right ) \log ^2(x)} \, dx=3 \left (e^4+16 x^2-\frac {2}{2+\left (-3+(-4+x)^2\right ) \log (x)}\right ) \] Output:
3*exp(4)-6/(ln(x)*((-4+x)^2-3)+2)+48*x^2
Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {78-48 x+390 x^2+\left (-48 x+5004 x^2-3072 x^3+384 x^4\right ) \log (x)+\left (16224 x^2-19968 x^3+8640 x^4-1536 x^5+96 x^6\right ) \log ^2(x)}{4 x+\left (52 x-32 x^2+4 x^3\right ) \log (x)+\left (169 x-208 x^2+90 x^3-16 x^4+x^5\right ) \log ^2(x)} \, dx=6 \left (8 x^2-\frac {1}{2+\left (13-8 x+x^2\right ) \log (x)}\right ) \] Input:
Integrate[(78 - 48*x + 390*x^2 + (-48*x + 5004*x^2 - 3072*x^3 + 384*x^4)*L og[x] + (16224*x^2 - 19968*x^3 + 8640*x^4 - 1536*x^5 + 96*x^6)*Log[x]^2)/( 4*x + (52*x - 32*x^2 + 4*x^3)*Log[x] + (169*x - 208*x^2 + 90*x^3 - 16*x^4 + x^5)*Log[x]^2),x]
Output:
6*(8*x^2 - (2 + (13 - 8*x + x^2)*Log[x])^(-1))
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.
| \(\displaystyle \int \frac {390 x^2+\left (384 x^4-3072 x^3+5004 x^2-48 x\right ) \log (x)+\left (96 x^6-1536 x^5+8640 x^4-19968 x^3+16224 x^2\right ) \log ^2(x)-48 x+78}{\left (4 x^3-32 x^2+52 x\right ) \log (x)+\left (x^5-16 x^4+90 x^3-208 x^2+169 x\right ) \log ^2(x)+4 x} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {390 x^2+\left (384 x^4-3072 x^3+5004 x^2-48 x\right ) \log (x)+\left (96 x^6-1536 x^5+8640 x^4-19968 x^3+16224 x^2\right ) \log ^2(x)-48 x+78}{x \left (x^2 \log (x)-8 x \log (x)+13 \log (x)+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {12 (x-4)}{\left (x^2-8 x+13\right ) \left (x^2 \log (x)-8 x \log (x)+13 \log (x)+2\right )}+\frac {6 \left (x^4-16 x^3+86 x^2-192 x+169\right )}{x \left (x^2-8 x+13\right ) \left (x^2 \log (x)-8 x \log (x)+13 \log (x)+2\right )^2}+96 x\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -48 \int \frac {1}{\left (\log (x) x^2-8 \log (x) x+13 \log (x)+2\right )^2}dx-32 \sqrt {3} \int \frac {1}{\left (-2 x+2 \sqrt {3}+8\right ) \left (\log (x) x^2-8 \log (x) x+13 \log (x)+2\right )^2}dx+78 \int \frac {1}{x \left (\log (x) x^2-8 \log (x) x+13 \log (x)+2\right )^2}dx+6 \int \frac {x}{\left (\log (x) x^2-8 \log (x) x+13 \log (x)+2\right )^2}dx-8 \left (3+4 \sqrt {3}\right ) \int \frac {1}{\left (2 x-2 \sqrt {3}-8\right ) \left (\log (x) x^2-8 \log (x) x+13 \log (x)+2\right )^2}dx-8 \left (3-4 \sqrt {3}\right ) \int \frac {1}{\left (2 x+2 \sqrt {3}-8\right ) \left (\log (x) x^2-8 \log (x) x+13 \log (x)+2\right )^2}dx-32 \sqrt {3} \int \frac {1}{\left (2 x+2 \sqrt {3}-8\right ) \left (\log (x) x^2-8 \log (x) x+13 \log (x)+2\right )^2}dx+16 \sqrt {3} \int \frac {1}{\left (-2 x+2 \sqrt {3}+8\right ) \left (\log (x) x^2-8 \log (x) x+13 \log (x)+2\right )}dx+4 \left (3+4 \sqrt {3}\right ) \int \frac {1}{\left (2 x-2 \sqrt {3}-8\right ) \left (\log (x) x^2-8 \log (x) x+13 \log (x)+2\right )}dx+4 \left (3-4 \sqrt {3}\right ) \int \frac {1}{\left (2 x+2 \sqrt {3}-8\right ) \left (\log (x) x^2-8 \log (x) x+13 \log (x)+2\right )}dx+16 \sqrt {3} \int \frac {1}{\left (2 x+2 \sqrt {3}-8\right ) \left (\log (x) x^2-8 \log (x) x+13 \log (x)+2\right )}dx+48 x^2\) |
Input:
Int[(78 - 48*x + 390*x^2 + (-48*x + 5004*x^2 - 3072*x^3 + 384*x^4)*Log[x] + (16224*x^2 - 19968*x^3 + 8640*x^4 - 1536*x^5 + 96*x^6)*Log[x]^2)/(4*x + (52*x - 32*x^2 + 4*x^3)*Log[x] + (169*x - 208*x^2 + 90*x^3 - 16*x^4 + x^5) *Log[x]^2),x]
Output:
$Aborted
Time = 0.90 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04
| method | result | size |
| risch | \(48 x^{2}-\frac {6}{x^{2} \ln \left (x \right )-8 x \ln \left (x \right )+13 \ln \left (x \right )+2}\) | \(28\) |
| norman | \(\frac {-8112 \ln \left (x \right )+4992 x \ln \left (x \right )+96 x^{2}-384 x^{3} \ln \left (x \right )+48 x^{4} \ln \left (x \right )-1254}{x^{2} \ln \left (x \right )-8 x \ln \left (x \right )+13 \ln \left (x \right )+2}\) | \(51\) |
| parallelrisch | \(\frac {-8112 \ln \left (x \right )+4992 x \ln \left (x \right )+96 x^{2}-384 x^{3} \ln \left (x \right )+48 x^{4} \ln \left (x \right )-1254}{x^{2} \ln \left (x \right )-8 x \ln \left (x \right )+13 \ln \left (x \right )+2}\) | \(51\) |
| default | \(\frac {-8112 \ln \left (x \right )+4992 x \ln \left (x \right )+96 x^{2}-384 x^{3} \ln \left (x \right )+48 x^{4} \ln \left (x \right )-1254}{x^{2} \ln \left (x \right )-8 x \ln \left (x \right )+13 \ln \left (x \right )+2}\) | \(52\) |
Input:
int(((96*x^6-1536*x^5+8640*x^4-19968*x^3+16224*x^2)*ln(x)^2+(384*x^4-3072* x^3+5004*x^2-48*x)*ln(x)+390*x^2-48*x+78)/((x^5-16*x^4+90*x^3-208*x^2+169* x)*ln(x)^2+(4*x^3-32*x^2+52*x)*ln(x)+4*x),x,method=_RETURNVERBOSE)
Output:
48*x^2-6/(x^2*ln(x)-8*x*ln(x)+13*ln(x)+2)
Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {78-48 x+390 x^2+\left (-48 x+5004 x^2-3072 x^3+384 x^4\right ) \log (x)+\left (16224 x^2-19968 x^3+8640 x^4-1536 x^5+96 x^6\right ) \log ^2(x)}{4 x+\left (52 x-32 x^2+4 x^3\right ) \log (x)+\left (169 x-208 x^2+90 x^3-16 x^4+x^5\right ) \log ^2(x)} \, dx=\frac {6 \, {\left (16 \, x^{2} + 8 \, {\left (x^{4} - 8 \, x^{3} + 13 \, x^{2}\right )} \log \left (x\right ) - 1\right )}}{{\left (x^{2} - 8 \, x + 13\right )} \log \left (x\right ) + 2} \] Input:
integrate(((96*x^6-1536*x^5+8640*x^4-19968*x^3+16224*x^2)*log(x)^2+(384*x^ 4-3072*x^3+5004*x^2-48*x)*log(x)+390*x^2-48*x+78)/((x^5-16*x^4+90*x^3-208* x^2+169*x)*log(x)^2+(4*x^3-32*x^2+52*x)*log(x)+4*x),x, algorithm="fricas")
Output:
6*(16*x^2 + 8*(x^4 - 8*x^3 + 13*x^2)*log(x) - 1)/((x^2 - 8*x + 13)*log(x) + 2)
Time = 0.11 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {78-48 x+390 x^2+\left (-48 x+5004 x^2-3072 x^3+384 x^4\right ) \log (x)+\left (16224 x^2-19968 x^3+8640 x^4-1536 x^5+96 x^6\right ) \log ^2(x)}{4 x+\left (52 x-32 x^2+4 x^3\right ) \log (x)+\left (169 x-208 x^2+90 x^3-16 x^4+x^5\right ) \log ^2(x)} \, dx=48 x^{2} - \frac {6}{\left (x^{2} - 8 x + 13\right ) \log {\left (x \right )} + 2} \] Input:
integrate(((96*x**6-1536*x**5+8640*x**4-19968*x**3+16224*x**2)*ln(x)**2+(3 84*x**4-3072*x**3+5004*x**2-48*x)*ln(x)+390*x**2-48*x+78)/((x**5-16*x**4+9 0*x**3-208*x**2+169*x)*ln(x)**2+(4*x**3-32*x**2+52*x)*ln(x)+4*x),x)
Output:
48*x**2 - 6/((x**2 - 8*x + 13)*log(x) + 2)
Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {78-48 x+390 x^2+\left (-48 x+5004 x^2-3072 x^3+384 x^4\right ) \log (x)+\left (16224 x^2-19968 x^3+8640 x^4-1536 x^5+96 x^6\right ) \log ^2(x)}{4 x+\left (52 x-32 x^2+4 x^3\right ) \log (x)+\left (169 x-208 x^2+90 x^3-16 x^4+x^5\right ) \log ^2(x)} \, dx=\frac {6 \, {\left (16 \, x^{2} + 8 \, {\left (x^{4} - 8 \, x^{3} + 13 \, x^{2}\right )} \log \left (x\right ) - 1\right )}}{{\left (x^{2} - 8 \, x + 13\right )} \log \left (x\right ) + 2} \] Input:
integrate(((96*x^6-1536*x^5+8640*x^4-19968*x^3+16224*x^2)*log(x)^2+(384*x^ 4-3072*x^3+5004*x^2-48*x)*log(x)+390*x^2-48*x+78)/((x^5-16*x^4+90*x^3-208* x^2+169*x)*log(x)^2+(4*x^3-32*x^2+52*x)*log(x)+4*x),x, algorithm="maxima")
Output:
6*(16*x^2 + 8*(x^4 - 8*x^3 + 13*x^2)*log(x) - 1)/((x^2 - 8*x + 13)*log(x) + 2)
Time = 0.13 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {78-48 x+390 x^2+\left (-48 x+5004 x^2-3072 x^3+384 x^4\right ) \log (x)+\left (16224 x^2-19968 x^3+8640 x^4-1536 x^5+96 x^6\right ) \log ^2(x)}{4 x+\left (52 x-32 x^2+4 x^3\right ) \log (x)+\left (169 x-208 x^2+90 x^3-16 x^4+x^5\right ) \log ^2(x)} \, dx=48 \, x^{2} - \frac {6}{x^{2} \log \left (x\right ) - 8 \, x \log \left (x\right ) + 13 \, \log \left (x\right ) + 2} \] Input:
integrate(((96*x^6-1536*x^5+8640*x^4-19968*x^3+16224*x^2)*log(x)^2+(384*x^ 4-3072*x^3+5004*x^2-48*x)*log(x)+390*x^2-48*x+78)/((x^5-16*x^4+90*x^3-208* x^2+169*x)*log(x)^2+(4*x^3-32*x^2+52*x)*log(x)+4*x),x, algorithm="giac")
Output:
48*x^2 - 6/(x^2*log(x) - 8*x*log(x) + 13*log(x) + 2)
Timed out. \[ \int \frac {78-48 x+390 x^2+\left (-48 x+5004 x^2-3072 x^3+384 x^4\right ) \log (x)+\left (16224 x^2-19968 x^3+8640 x^4-1536 x^5+96 x^6\right ) \log ^2(x)}{4 x+\left (52 x-32 x^2+4 x^3\right ) \log (x)+\left (169 x-208 x^2+90 x^3-16 x^4+x^5\right ) \log ^2(x)} \, dx=\int \frac {{\ln \left (x\right )}^2\,\left (96\,x^6-1536\,x^5+8640\,x^4-19968\,x^3+16224\,x^2\right )-48\,x-\ln \left (x\right )\,\left (-384\,x^4+3072\,x^3-5004\,x^2+48\,x\right )+390\,x^2+78}{\left (x^5-16\,x^4+90\,x^3-208\,x^2+169\,x\right )\,{\ln \left (x\right )}^2+\left (4\,x^3-32\,x^2+52\,x\right )\,\ln \left (x\right )+4\,x} \,d x \] Input:
int((log(x)^2*(16224*x^2 - 19968*x^3 + 8640*x^4 - 1536*x^5 + 96*x^6) - 48* x - log(x)*(48*x - 5004*x^2 + 3072*x^3 - 384*x^4) + 390*x^2 + 78)/(4*x + l og(x)^2*(169*x - 208*x^2 + 90*x^3 - 16*x^4 + x^5) + log(x)*(52*x - 32*x^2 + 4*x^3)),x)
Output:
int((log(x)^2*(16224*x^2 - 19968*x^3 + 8640*x^4 - 1536*x^5 + 96*x^6) - 48* x - log(x)*(48*x - 5004*x^2 + 3072*x^3 - 384*x^4) + 390*x^2 + 78)/(4*x + l og(x)^2*(169*x - 208*x^2 + 90*x^3 - 16*x^4 + x^5) + log(x)*(52*x - 32*x^2 + 4*x^3)), x)
Time = 0.18 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.81 \[ \int \frac {78-48 x+390 x^2+\left (-48 x+5004 x^2-3072 x^3+384 x^4\right ) \log (x)+\left (16224 x^2-19968 x^3+8640 x^4-1536 x^5+96 x^6\right ) \log ^2(x)}{4 x+\left (52 x-32 x^2+4 x^3\right ) \log (x)+\left (169 x-208 x^2+90 x^3-16 x^4+x^5\right ) \log ^2(x)} \, dx=\frac {48 \,\mathrm {log}\left (x \right ) x^{4}-384 \,\mathrm {log}\left (x \right ) x^{3}+624 \,\mathrm {log}\left (x \right ) x^{2}+96 x^{2}-6}{\mathrm {log}\left (x \right ) x^{2}-8 \,\mathrm {log}\left (x \right ) x +13 \,\mathrm {log}\left (x \right )+2} \] Input:
int(((96*x^6-1536*x^5+8640*x^4-19968*x^3+16224*x^2)*log(x)^2+(384*x^4-3072 *x^3+5004*x^2-48*x)*log(x)+390*x^2-48*x+78)/((x^5-16*x^4+90*x^3-208*x^2+16 9*x)*log(x)^2+(4*x^3-32*x^2+52*x)*log(x)+4*x),x)
Output:
(6*(8*log(x)*x**4 - 64*log(x)*x**3 + 104*log(x)*x**2 + 16*x**2 - 1))/(log( x)*x**2 - 8*log(x)*x + 13*log(x) + 2)