Integrand size = 103, antiderivative size = 23 \[ \int \frac {94 x+4 x^3+28 x \log (x)+2 x \log ^2(x)+\left (126 x-4 x^3+32 x \log (x)+2 x \log ^2(x)\right ) \log \left (\frac {63-2 x^2+16 \log (x)+\log ^2(x)}{x}\right )}{\left (63-2 x^2+16 \log (x)+\log ^2(x)\right ) \log ^3\left (\frac {63-2 x^2+16 \log (x)+\log ^2(x)}{x}\right )} \, dx=\frac {x^2}{\log ^2\left (-2 x+\frac {-1+(8+\log (x))^2}{x}\right )} \]
Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {94 x+4 x^3+28 x \log (x)+2 x \log ^2(x)+\left (126 x-4 x^3+32 x \log (x)+2 x \log ^2(x)\right ) \log \left (\frac {63-2 x^2+16 \log (x)+\log ^2(x)}{x}\right )}{\left (63-2 x^2+16 \log (x)+\log ^2(x)\right ) \log ^3\left (\frac {63-2 x^2+16 \log (x)+\log ^2(x)}{x}\right )} \, dx=\frac {x^2}{\log ^2\left (\frac {63-2 x^2+16 \log (x)+\log ^2(x)}{x}\right )} \]
Integrate[(94*x + 4*x^3 + 28*x*Log[x] + 2*x*Log[x]^2 + (126*x - 4*x^3 + 32 *x*Log[x] + 2*x*Log[x]^2)*Log[(63 - 2*x^2 + 16*Log[x] + Log[x]^2)/x])/((63 - 2*x^2 + 16*Log[x] + Log[x]^2)*Log[(63 - 2*x^2 + 16*Log[x] + Log[x]^2)/x ]^3),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.
| \(\displaystyle \int \frac {4 x^3+\left (-4 x^3+126 x+2 x \log ^2(x)+32 x \log (x)\right ) \log \left (\frac {-2 x^2+\log ^2(x)+16 \log (x)+63}{x}\right )+94 x+2 x \log ^2(x)+28 x \log (x)}{\left (-2 x^2+\log ^2(x)+16 \log (x)+63\right ) \log ^3\left (\frac {-2 x^2+\log ^2(x)+16 \log (x)+63}{x}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {4 x^3+\left (-4 x^3+126 x+2 x \log ^2(x)+32 x \log (x)\right ) \log \left (\frac {-2 x^2+\log ^2(x)+16 \log (x)+63}{x}\right )+94 x+2 x \log ^2(x)+28 x \log (x)}{\left (-2 x^2+\log ^2(x)+16 \log (x)+63\right ) \log ^3\left (-2 x+\frac {63}{x}+\frac {\log ^2(x)}{x}+\frac {16 \log (x)}{x}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 x \log ^2(x)}{\left (-2 x^2+\log ^2(x)+16 \log (x)+63\right ) \log ^2\left (-2 x+\frac {63}{x}+\frac {\log ^2(x)}{x}+\frac {16 \log (x)}{x}\right )}+\frac {32 x \log (x)}{\left (-2 x^2+\log ^2(x)+16 \log (x)+63\right ) \log ^2\left (-2 x+\frac {63}{x}+\frac {\log ^2(x)}{x}+\frac {16 \log (x)}{x}\right )}+\frac {126 x}{\left (-2 x^2+\log ^2(x)+16 \log (x)+63\right ) \log ^2\left (-2 x+\frac {63}{x}+\frac {\log ^2(x)}{x}+\frac {16 \log (x)}{x}\right )}+\frac {2 x \log ^2(x)}{\left (-2 x^2+\log ^2(x)+16 \log (x)+63\right ) \log ^3\left (-2 x+\frac {63}{x}+\frac {\log ^2(x)}{x}+\frac {16 \log (x)}{x}\right )}+\frac {28 x \log (x)}{\left (-2 x^2+\log ^2(x)+16 \log (x)+63\right ) \log ^3\left (-2 x+\frac {63}{x}+\frac {\log ^2(x)}{x}+\frac {16 \log (x)}{x}\right )}+\frac {94 x}{\left (-2 x^2+\log ^2(x)+16 \log (x)+63\right ) \log ^3\left (-2 x+\frac {63}{x}+\frac {\log ^2(x)}{x}+\frac {16 \log (x)}{x}\right )}+\frac {4 x^3}{\left (2 x^2-\log ^2(x)-16 \log (x)-63\right ) \log ^2\left (-2 x+\frac {63}{x}+\frac {\log ^2(x)}{x}+\frac {16 \log (x)}{x}\right )}+\frac {4 x^3}{\left (-2 x^2+\log ^2(x)+16 \log (x)+63\right ) \log ^3\left (-2 x+\frac {63}{x}+\frac {\log ^2(x)}{x}+\frac {16 \log (x)}{x}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 126 \int \frac {x}{\left (-2 x^2+\log ^2(x)+16 \log (x)+63\right ) \log ^2\left (\frac {\log ^2(x)}{x}+\frac {16 \log (x)}{x}-2 x+\frac {63}{x}\right )}dx+32 \int \frac {x \log (x)}{\left (-2 x^2+\log ^2(x)+16 \log (x)+63\right ) \log ^2\left (\frac {\log ^2(x)}{x}+\frac {16 \log (x)}{x}-2 x+\frac {63}{x}\right )}dx+2 \int \frac {x \log ^2(x)}{\left (-2 x^2+\log ^2(x)+16 \log (x)+63\right ) \log ^2\left (\frac {\log ^2(x)}{x}+\frac {16 \log (x)}{x}-2 x+\frac {63}{x}\right )}dx+94 \int \frac {x}{\left (-2 x^2+\log ^2(x)+16 \log (x)+63\right ) \log ^3\left (\frac {\log ^2(x)}{x}+\frac {16 \log (x)}{x}-2 x+\frac {63}{x}\right )}dx+28 \int \frac {x \log (x)}{\left (-2 x^2+\log ^2(x)+16 \log (x)+63\right ) \log ^3\left (\frac {\log ^2(x)}{x}+\frac {16 \log (x)}{x}-2 x+\frac {63}{x}\right )}dx+2 \int \frac {x \log ^2(x)}{\left (-2 x^2+\log ^2(x)+16 \log (x)+63\right ) \log ^3\left (\frac {\log ^2(x)}{x}+\frac {16 \log (x)}{x}-2 x+\frac {63}{x}\right )}dx+4 \int \frac {x^3}{\left (2 x^2-\log ^2(x)-16 \log (x)-63\right ) \log ^2\left (\frac {\log ^2(x)}{x}+\frac {16 \log (x)}{x}-2 x+\frac {63}{x}\right )}dx+4 \int \frac {x^3}{\left (-2 x^2+\log ^2(x)+16 \log (x)+63\right ) \log ^3\left (\frac {\log ^2(x)}{x}+\frac {16 \log (x)}{x}-2 x+\frac {63}{x}\right )}dx\) |
Int[(94*x + 4*x^3 + 28*x*Log[x] + 2*x*Log[x]^2 + (126*x - 4*x^3 + 32*x*Log [x] + 2*x*Log[x]^2)*Log[(63 - 2*x^2 + 16*Log[x] + Log[x]^2)/x])/((63 - 2*x ^2 + 16*Log[x] + Log[x]^2)*Log[(63 - 2*x^2 + 16*Log[x] + Log[x]^2)/x]^3),x ]
3.12.18.3.1 Defintions of rubi rules used
Time = 22.58 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17
| method | result | size |
| parallelrisch | \(\frac {x^{2}}{\ln \left (\frac {\ln \left (x \right )^{2}+16 \ln \left (x \right )-2 x^{2}+63}{x}\right )^{2}}\) | \(27\) |
| risch | \(\frac {4 x^{2}}{{\left (2 \ln \left (2\right )+2 i \pi -2 \ln \left (x \right )+2 \ln \left (x^{2}-\frac {\ln \left (x \right )^{2}}{2}-8 \ln \left (x \right )-\frac {63}{2}\right )-i \pi \,\operatorname {csgn}\left (i \left (-x^{2}+\frac {\ln \left (x \right )^{2}}{2}+8 \ln \left (x \right )+\frac {63}{2}\right )\right ) {\operatorname {csgn}\left (\frac {i \left (-x^{2}+\frac {\ln \left (x \right )^{2}}{2}+8 \ln \left (x \right )+\frac {63}{2}\right )}{x}\right )}^{2}-i \pi \,\operatorname {csgn}\left (i \left (-x^{2}+\frac {\ln \left (x \right )^{2}}{2}+8 \ln \left (x \right )+\frac {63}{2}\right )\right ) \operatorname {csgn}\left (\frac {i \left (-x^{2}+\frac {\ln \left (x \right )^{2}}{2}+8 \ln \left (x \right )+\frac {63}{2}\right )}{x}\right ) \operatorname {csgn}\left (\frac {i}{x}\right )+i \pi {\operatorname {csgn}\left (\frac {i \left (-x^{2}+\frac {\ln \left (x \right )^{2}}{2}+8 \ln \left (x \right )+\frac {63}{2}\right )}{x}\right )}^{2} \operatorname {csgn}\left (\frac {i}{x}\right )-2 i \pi {\operatorname {csgn}\left (\frac {i \left (-x^{2}+\frac {\ln \left (x \right )^{2}}{2}+8 \ln \left (x \right )+\frac {63}{2}\right )}{x}\right )}^{2}-i \pi {\operatorname {csgn}\left (\frac {i \left (-x^{2}+\frac {\ln \left (x \right )^{2}}{2}+8 \ln \left (x \right )+\frac {63}{2}\right )}{x}\right )}^{3}\right )}^{2}}\) | \(243\) |
| default | \(\text {Expression too large to display}\) | \(1273\) |
int(((2*x*ln(x)^2+32*x*ln(x)-4*x^3+126*x)*ln((ln(x)^2+16*ln(x)-2*x^2+63)/x )+2*x*ln(x)^2+28*x*ln(x)+4*x^3+94*x)/(ln(x)^2+16*ln(x)-2*x^2+63)/ln((ln(x) ^2+16*ln(x)-2*x^2+63)/x)^3,x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {94 x+4 x^3+28 x \log (x)+2 x \log ^2(x)+\left (126 x-4 x^3+32 x \log (x)+2 x \log ^2(x)\right ) \log \left (\frac {63-2 x^2+16 \log (x)+\log ^2(x)}{x}\right )}{\left (63-2 x^2+16 \log (x)+\log ^2(x)\right ) \log ^3\left (\frac {63-2 x^2+16 \log (x)+\log ^2(x)}{x}\right )} \, dx=\frac {x^{2}}{\log \left (-\frac {2 \, x^{2} - \log \left (x\right )^{2} - 16 \, \log \left (x\right ) - 63}{x}\right )^{2}} \]
integrate(((2*x*log(x)^2+32*x*log(x)-4*x^3+126*x)*log((log(x)^2+16*log(x)- 2*x^2+63)/x)+2*x*log(x)^2+28*x*log(x)+4*x^3+94*x)/(log(x)^2+16*log(x)-2*x^ 2+63)/log((log(x)^2+16*log(x)-2*x^2+63)/x)^3,x, algorithm=\
Time = 0.14 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {94 x+4 x^3+28 x \log (x)+2 x \log ^2(x)+\left (126 x-4 x^3+32 x \log (x)+2 x \log ^2(x)\right ) \log \left (\frac {63-2 x^2+16 \log (x)+\log ^2(x)}{x}\right )}{\left (63-2 x^2+16 \log (x)+\log ^2(x)\right ) \log ^3\left (\frac {63-2 x^2+16 \log (x)+\log ^2(x)}{x}\right )} \, dx=\frac {x^{2}}{\log {\left (\frac {- 2 x^{2} + \log {\left (x \right )}^{2} + 16 \log {\left (x \right )} + 63}{x} \right )}^{2}} \]
integrate(((2*x*ln(x)**2+32*x*ln(x)-4*x**3+126*x)*ln((ln(x)**2+16*ln(x)-2* x**2+63)/x)+2*x*ln(x)**2+28*x*ln(x)+4*x**3+94*x)/(ln(x)**2+16*ln(x)-2*x**2 +63)/ln((ln(x)**2+16*ln(x)-2*x**2+63)/x)**3,x)
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (23) = 46\).
Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.13 \[ \int \frac {94 x+4 x^3+28 x \log (x)+2 x \log ^2(x)+\left (126 x-4 x^3+32 x \log (x)+2 x \log ^2(x)\right ) \log \left (\frac {63-2 x^2+16 \log (x)+\log ^2(x)}{x}\right )}{\left (63-2 x^2+16 \log (x)+\log ^2(x)\right ) \log ^3\left (\frac {63-2 x^2+16 \log (x)+\log ^2(x)}{x}\right )} \, dx=\frac {x^{2}}{\log \left (-2 \, x^{2} + \log \left (x\right )^{2} + 16 \, \log \left (x\right ) + 63\right )^{2} - 2 \, \log \left (-2 \, x^{2} + \log \left (x\right )^{2} + 16 \, \log \left (x\right ) + 63\right ) \log \left (x\right ) + \log \left (x\right )^{2}} \]
integrate(((2*x*log(x)^2+32*x*log(x)-4*x^3+126*x)*log((log(x)^2+16*log(x)- 2*x^2+63)/x)+2*x*log(x)^2+28*x*log(x)+4*x^3+94*x)/(log(x)^2+16*log(x)-2*x^ 2+63)/log((log(x)^2+16*log(x)-2*x^2+63)/x)^3,x, algorithm=\
x^2/(log(-2*x^2 + log(x)^2 + 16*log(x) + 63)^2 - 2*log(-2*x^2 + log(x)^2 + 16*log(x) + 63)*log(x) + log(x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (23) = 46\).
Time = 0.36 (sec) , antiderivative size = 230, normalized size of antiderivative = 10.00 \[ \int \frac {94 x+4 x^3+28 x \log (x)+2 x \log ^2(x)+\left (126 x-4 x^3+32 x \log (x)+2 x \log ^2(x)\right ) \log \left (\frac {63-2 x^2+16 \log (x)+\log ^2(x)}{x}\right )}{\left (63-2 x^2+16 \log (x)+\log ^2(x)\right ) \log ^3\left (\frac {63-2 x^2+16 \log (x)+\log ^2(x)}{x}\right )} \, dx=\frac {2 \, x^{4} + x^{2} \log \left (x\right )^{2} + 14 \, x^{2} \log \left (x\right ) + 47 \, x^{2}}{2 \, x^{2} \log \left (-2 \, x^{2} + \log \left (x\right )^{2} + 16 \, \log \left (x\right ) + 63\right )^{2} - 4 \, x^{2} \log \left (-2 \, x^{2} + \log \left (x\right )^{2} + 16 \, \log \left (x\right ) + 63\right ) \log \left (x\right ) + 2 \, x^{2} \log \left (x\right )^{2} + \log \left (-2 \, x^{2} + \log \left (x\right )^{2} + 16 \, \log \left (x\right ) + 63\right )^{2} \log \left (x\right )^{2} - 2 \, \log \left (-2 \, x^{2} + \log \left (x\right )^{2} + 16 \, \log \left (x\right ) + 63\right ) \log \left (x\right )^{3} + \log \left (x\right )^{4} + 14 \, \log \left (-2 \, x^{2} + \log \left (x\right )^{2} + 16 \, \log \left (x\right ) + 63\right )^{2} \log \left (x\right ) - 28 \, \log \left (-2 \, x^{2} + \log \left (x\right )^{2} + 16 \, \log \left (x\right ) + 63\right ) \log \left (x\right )^{2} + 14 \, \log \left (x\right )^{3} + 47 \, \log \left (-2 \, x^{2} + \log \left (x\right )^{2} + 16 \, \log \left (x\right ) + 63\right )^{2} - 94 \, \log \left (-2 \, x^{2} + \log \left (x\right )^{2} + 16 \, \log \left (x\right ) + 63\right ) \log \left (x\right ) + 47 \, \log \left (x\right )^{2}} \]
integrate(((2*x*log(x)^2+32*x*log(x)-4*x^3+126*x)*log((log(x)^2+16*log(x)- 2*x^2+63)/x)+2*x*log(x)^2+28*x*log(x)+4*x^3+94*x)/(log(x)^2+16*log(x)-2*x^ 2+63)/log((log(x)^2+16*log(x)-2*x^2+63)/x)^3,x, algorithm=\
(2*x^4 + x^2*log(x)^2 + 14*x^2*log(x) + 47*x^2)/(2*x^2*log(-2*x^2 + log(x) ^2 + 16*log(x) + 63)^2 - 4*x^2*log(-2*x^2 + log(x)^2 + 16*log(x) + 63)*log (x) + 2*x^2*log(x)^2 + log(-2*x^2 + log(x)^2 + 16*log(x) + 63)^2*log(x)^2 - 2*log(-2*x^2 + log(x)^2 + 16*log(x) + 63)*log(x)^3 + log(x)^4 + 14*log(- 2*x^2 + log(x)^2 + 16*log(x) + 63)^2*log(x) - 28*log(-2*x^2 + log(x)^2 + 1 6*log(x) + 63)*log(x)^2 + 14*log(x)^3 + 47*log(-2*x^2 + log(x)^2 + 16*log( x) + 63)^2 - 94*log(-2*x^2 + log(x)^2 + 16*log(x) + 63)*log(x) + 47*log(x) ^2)
Timed out. \[ \int \frac {94 x+4 x^3+28 x \log (x)+2 x \log ^2(x)+\left (126 x-4 x^3+32 x \log (x)+2 x \log ^2(x)\right ) \log \left (\frac {63-2 x^2+16 \log (x)+\log ^2(x)}{x}\right )}{\left (63-2 x^2+16 \log (x)+\log ^2(x)\right ) \log ^3\left (\frac {63-2 x^2+16 \log (x)+\log ^2(x)}{x}\right )} \, dx=\int \frac {94\,x+2\,x\,{\ln \left (x\right )}^2+\ln \left (\frac {-2\,x^2+{\ln \left (x\right )}^2+16\,\ln \left (x\right )+63}{x}\right )\,\left (-4\,x^3+2\,x\,{\ln \left (x\right )}^2+32\,x\,\ln \left (x\right )+126\,x\right )+28\,x\,\ln \left (x\right )+4\,x^3}{{\ln \left (\frac {-2\,x^2+{\ln \left (x\right )}^2+16\,\ln \left (x\right )+63}{x}\right )}^3\,\left (-2\,x^2+{\ln \left (x\right )}^2+16\,\ln \left (x\right )+63\right )} \,d x \]
int((94*x + 2*x*log(x)^2 + log((16*log(x) + log(x)^2 - 2*x^2 + 63)/x)*(126 *x + 2*x*log(x)^2 + 32*x*log(x) - 4*x^3) + 28*x*log(x) + 4*x^3)/(log((16*l og(x) + log(x)^2 - 2*x^2 + 63)/x)^3*(16*log(x) + log(x)^2 - 2*x^2 + 63)),x )