Integrand size = 83, antiderivative size = 25 \[ \int \frac {\left (-7500 x-3750 x^3\right ) \log (x)+\left (3750 x-1875 x^3+\left (-7500 x+3750 x^3\right ) \log (x)\right ) \log \left (\frac {16-16 x^2+4 x^4}{x^2}\right )}{\left (-2+x^2\right ) \log ^2(x) \log ^2\left (\frac {16-16 x^2+4 x^4}{x^2}\right )} \, dx=\frac {1875 x^2}{\log (x) \log \left (\frac {\left (4-2 x^2\right )^2}{x^2}\right )} \] Output:
1875*x^2/ln(x)/ln((-2*x^2+4)^2/x^2)
Time = 0.34 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {\left (-7500 x-3750 x^3\right ) \log (x)+\left (3750 x-1875 x^3+\left (-7500 x+3750 x^3\right ) \log (x)\right ) \log \left (\frac {16-16 x^2+4 x^4}{x^2}\right )}{\left (-2+x^2\right ) \log ^2(x) \log ^2\left (\frac {16-16 x^2+4 x^4}{x^2}\right )} \, dx=\frac {1875 x^2}{\log (x) \log \left (\frac {4 \left (-2+x^2\right )^2}{x^2}\right )} \] Input:
Integrate[((-7500*x - 3750*x^3)*Log[x] + (3750*x - 1875*x^3 + (-7500*x + 3 750*x^3)*Log[x])*Log[(16 - 16*x^2 + 4*x^4)/x^2])/((-2 + x^2)*Log[x]^2*Log[ (16 - 16*x^2 + 4*x^4)/x^2]^2),x]
Output:
(1875*x^2)/(Log[x]*Log[(4*(-2 + x^2)^2)/x^2])
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.
| \(\displaystyle \int \frac {\left (-3750 x^3-7500 x\right ) \log (x)+\left (-1875 x^3+\left (3750 x^3-7500 x\right ) \log (x)+3750 x\right ) \log \left (\frac {4 x^4-16 x^2+16}{x^2}\right )}{\left (x^2-2\right ) \log ^2(x) \log ^2\left (\frac {4 x^4-16 x^2+16}{x^2}\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {1875 x (2 \log (x)-1)}{\log ^2(x) \log \left (\frac {4 \left (x^2-2\right )^2}{x^2}\right )}-\frac {3750 x \left (x^2+2\right )}{\left (x^2-2\right ) \log (x) \log ^2\left (\frac {4 \left (x^2-2\right )^2}{x^2}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 7500 \int \frac {1}{\left (\sqrt {2}-x\right ) \log (x) \log ^2\left (\frac {4 \left (x^2-2\right )^2}{x^2}\right )}dx-3750 \int \frac {x}{\log (x) \log ^2\left (\frac {4 \left (x^2-2\right )^2}{x^2}\right )}dx-7500 \int \frac {1}{\left (x+\sqrt {2}\right ) \log (x) \log ^2\left (\frac {4 \left (x^2-2\right )^2}{x^2}\right )}dx-1875 \int \frac {x}{\log ^2(x) \log \left (\frac {4 \left (x^2-2\right )^2}{x^2}\right )}dx+3750 \int \frac {x}{\log (x) \log \left (\frac {4 \left (x^2-2\right )^2}{x^2}\right )}dx\) |
Input:
Int[((-7500*x - 3750*x^3)*Log[x] + (3750*x - 1875*x^3 + (-7500*x + 3750*x^ 3)*Log[x])*Log[(16 - 16*x^2 + 4*x^4)/x^2])/((-2 + x^2)*Log[x]^2*Log[(16 - 16*x^2 + 4*x^4)/x^2]^2),x]
Output:
$Aborted
Time = 5.09 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12
| method | result | size |
| parallelrisch | \(\frac {1875 x^{2}}{\ln \left (x \right ) \ln \left (\frac {4 x^{4}-16 x^{2}+16}{x^{2}}\right )}\) | \(28\) |
| risch | \(\frac {3750 i x^{2}}{\ln \left (x \right ) \left (\pi \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) \operatorname {csgn}\left (i \left (x^{2}-2\right )^{2}\right ) \operatorname {csgn}\left (\frac {i \left (x^{2}-2\right )^{2}}{x^{2}}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}-2\right )^{2}}{x^{2}}\right )}^{2}+\pi {\operatorname {csgn}\left (i \left (x^{2}-2\right )\right )}^{2} \operatorname {csgn}\left (i \left (x^{2}-2\right )^{2}\right )-2 \pi \,\operatorname {csgn}\left (i \left (x^{2}-2\right )\right ) {\operatorname {csgn}\left (i \left (x^{2}-2\right )^{2}\right )}^{2}+\pi {\operatorname {csgn}\left (i \left (x^{2}-2\right )^{2}\right )}^{3}-\pi \,\operatorname {csgn}\left (i \left (x^{2}-2\right )^{2}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}-2\right )^{2}}{x^{2}}\right )}^{2}+\pi {\operatorname {csgn}\left (\frac {i \left (x^{2}-2\right )^{2}}{x^{2}}\right )}^{3}-\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-\pi \operatorname {csgn}\left (i x^{2}\right )^{3}+4 i \ln \left (2\right )-4 i \ln \left (x \right )+4 i \ln \left (x^{2}-2\right )\right )}\) | \(251\) |
Input:
int((((3750*x^3-7500*x)*ln(x)-1875*x^3+3750*x)*ln((4*x^4-16*x^2+16)/x^2)+( -3750*x^3-7500*x)*ln(x))/(x^2-2)/ln(x)^2/ln((4*x^4-16*x^2+16)/x^2)^2,x,met hod=_RETURNVERBOSE)
Output:
1875*x^2/ln(x)/ln(4*(x^4-4*x^2+4)/x^2)
Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\left (-7500 x-3750 x^3\right ) \log (x)+\left (3750 x-1875 x^3+\left (-7500 x+3750 x^3\right ) \log (x)\right ) \log \left (\frac {16-16 x^2+4 x^4}{x^2}\right )}{\left (-2+x^2\right ) \log ^2(x) \log ^2\left (\frac {16-16 x^2+4 x^4}{x^2}\right )} \, dx=\frac {1875 \, x^{2}}{\log \left (x\right ) \log \left (\frac {4 \, {\left (x^{4} - 4 \, x^{2} + 4\right )}}{x^{2}}\right )} \] Input:
integrate((((3750*x^3-7500*x)*log(x)-1875*x^3+3750*x)*log((4*x^4-16*x^2+16 )/x^2)+(-3750*x^3-7500*x)*log(x))/(x^2-2)/log(x)^2/log((4*x^4-16*x^2+16)/x ^2)^2,x, algorithm="fricas")
Output:
1875*x^2/(log(x)*log(4*(x^4 - 4*x^2 + 4)/x^2))
Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {\left (-7500 x-3750 x^3\right ) \log (x)+\left (3750 x-1875 x^3+\left (-7500 x+3750 x^3\right ) \log (x)\right ) \log \left (\frac {16-16 x^2+4 x^4}{x^2}\right )}{\left (-2+x^2\right ) \log ^2(x) \log ^2\left (\frac {16-16 x^2+4 x^4}{x^2}\right )} \, dx=\frac {1875 x^{2}}{\log {\left (x \right )} \log {\left (\frac {4 x^{4} - 16 x^{2} + 16}{x^{2}} \right )}} \] Input:
integrate((((3750*x**3-7500*x)*ln(x)-1875*x**3+3750*x)*ln((4*x**4-16*x**2+ 16)/x**2)+(-3750*x**3-7500*x)*ln(x))/(x**2-2)/ln(x)**2/ln((4*x**4-16*x**2+ 16)/x**2)**2,x)
Output:
1875*x**2/(log(x)*log((4*x**4 - 16*x**2 + 16)/x**2))
Time = 0.16 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {\left (-7500 x-3750 x^3\right ) \log (x)+\left (3750 x-1875 x^3+\left (-7500 x+3750 x^3\right ) \log (x)\right ) \log \left (\frac {16-16 x^2+4 x^4}{x^2}\right )}{\left (-2+x^2\right ) \log ^2(x) \log ^2\left (\frac {16-16 x^2+4 x^4}{x^2}\right )} \, dx=\frac {1875 \, x^{2}}{2 \, {\left (\log \left (2\right ) \log \left (x\right ) + \log \left (x^{2} - 2\right ) \log \left (x\right ) - \log \left (x\right )^{2}\right )}} \] Input:
integrate((((3750*x^3-7500*x)*log(x)-1875*x^3+3750*x)*log((4*x^4-16*x^2+16 )/x^2)+(-3750*x^3-7500*x)*log(x))/(x^2-2)/log(x)^2/log((4*x^4-16*x^2+16)/x ^2)^2,x, algorithm="maxima")
Output:
1875/2*x^2/(log(2)*log(x) + log(x^2 - 2)*log(x) - log(x)^2)
Time = 0.22 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {\left (-7500 x-3750 x^3\right ) \log (x)+\left (3750 x-1875 x^3+\left (-7500 x+3750 x^3\right ) \log (x)\right ) \log \left (\frac {16-16 x^2+4 x^4}{x^2}\right )}{\left (-2+x^2\right ) \log ^2(x) \log ^2\left (\frac {16-16 x^2+4 x^4}{x^2}\right )} \, dx=\frac {1875 \, x^{2}}{\log \left (4 \, x^{4} - 16 \, x^{2} + 16\right ) \log \left (x\right ) - 2 \, \log \left (x\right )^{2}} \] Input:
integrate((((3750*x^3-7500*x)*log(x)-1875*x^3+3750*x)*log((4*x^4-16*x^2+16 )/x^2)+(-3750*x^3-7500*x)*log(x))/(x^2-2)/log(x)^2/log((4*x^4-16*x^2+16)/x ^2)^2,x, algorithm="giac")
Output:
1875*x^2/(log(4*x^4 - 16*x^2 + 16)*log(x) - 2*log(x)^2)
Time = 3.12 (sec) , antiderivative size = 292, normalized size of antiderivative = 11.68 \[ \int \frac {\left (-7500 x-3750 x^3\right ) \log (x)+\left (3750 x-1875 x^3+\left (-7500 x+3750 x^3\right ) \log (x)\right ) \log \left (\frac {16-16 x^2+4 x^4}{x^2}\right )}{\left (-2+x^2\right ) \log ^2(x) \log ^2\left (\frac {16-16 x^2+4 x^4}{x^2}\right )} \, dx=22500\,\ln \left (x\right )-\frac {45000\,x^4+30000\,x^2}{x^6+6\,x^4+12\,x^2+8}-\frac {\frac {3750\,x^4}{{\left (x^2+2\right )}^2}-\frac {30000\,x^4\,\ln \left (x\right )}{{\left (x^2+2\right )}^3}+\frac {3750\,x^2\,{\ln \left (x\right )}^2\,\left (x^6+6\,x^4+20\,x^2-8\right )}{{\left (x^2+2\right )}^3}}{\ln \left (x\right )}+\frac {\frac {1875\,x^2}{\ln \left (x\right )}-\frac {1875\,x^2\,\ln \left (\frac {4\,x^4-16\,x^2+16}{x^2}\right )\,\left (x^2-2\right )\,\left (2\,\ln \left (x\right )-1\right )}{2\,{\ln \left (x\right )}^2\,\left (x^2+2\right )}}{\ln \left (\frac {4\,x^4-16\,x^2+16}{x^2}\right )}-\frac {\frac {1875\,x^2\,\left (x^2-2\right )}{2\,\left (x^2+2\right )}-\frac {1875\,x^2\,\ln \left (x\right )\,\left (x^4+2\,x^2-4\right )}{{\left (x^2+2\right )}^2}+\frac {1875\,x^2\,{\ln \left (x\right )}^2\,\left (x^4+4\,x^2-4\right )}{{\left (x^2+2\right )}^2}}{{\ln \left (x\right )}^2}+1875\,x^2-\frac {\ln \left (x\right )\,\left (-3750\,x^8+60000\,x^4+300000\,x^2+180000\right )}{x^6+6\,x^4+12\,x^2+8} \] Input:
int(-(log(x)*(7500*x + 3750*x^3) + log((4*x^4 - 16*x^2 + 16)/x^2)*(log(x)* (7500*x - 3750*x^3) - 3750*x + 1875*x^3))/(log((4*x^4 - 16*x^2 + 16)/x^2)^ 2*log(x)^2*(x^2 - 2)),x)
Output:
22500*log(x) - (30000*x^2 + 45000*x^4)/(12*x^2 + 6*x^4 + x^6 + 8) - ((3750 *x^4)/(x^2 + 2)^2 - (30000*x^4*log(x))/(x^2 + 2)^3 + (3750*x^2*log(x)^2*(2 0*x^2 + 6*x^4 + x^6 - 8))/(x^2 + 2)^3)/log(x) + ((1875*x^2)/log(x) - (1875 *x^2*log((4*x^4 - 16*x^2 + 16)/x^2)*(x^2 - 2)*(2*log(x) - 1))/(2*log(x)^2* (x^2 + 2)))/log((4*x^4 - 16*x^2 + 16)/x^2) - ((1875*x^2*(x^2 - 2))/(2*(x^2 + 2)) - (1875*x^2*log(x)*(2*x^2 + x^4 - 4))/(x^2 + 2)^2 + (1875*x^2*log(x )^2*(4*x^2 + x^4 - 4))/(x^2 + 2)^2)/log(x)^2 + 1875*x^2 - (log(x)*(300000* x^2 + 60000*x^4 - 3750*x^8 + 180000))/(12*x^2 + 6*x^4 + x^6 + 8)
Time = 0.18 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {\left (-7500 x-3750 x^3\right ) \log (x)+\left (3750 x-1875 x^3+\left (-7500 x+3750 x^3\right ) \log (x)\right ) \log \left (\frac {16-16 x^2+4 x^4}{x^2}\right )}{\left (-2+x^2\right ) \log ^2(x) \log ^2\left (\frac {16-16 x^2+4 x^4}{x^2}\right )} \, dx=\frac {1875 x^{2}}{\mathrm {log}\left (\frac {4 x^{4}-16 x^{2}+16}{x^{2}}\right ) \mathrm {log}\left (x \right )} \] Input:
int((((3750*x^3-7500*x)*log(x)-1875*x^3+3750*x)*log((4*x^4-16*x^2+16)/x^2) +(-3750*x^3-7500*x)*log(x))/(x^2-2)/log(x)^2/log((4*x^4-16*x^2+16)/x^2)^2, x)
Output:
(1875*x**2)/(log((4*x**4 - 16*x**2 + 16)/x**2)*log(x))