Integrand size = 114, antiderivative size = 27 \[ \int \frac {144-2244 x-1728 x^2-1425 x^3-360 x^4-144 x^5+e^5 \left (48-60 x-96 x^2\right )+\left (1104 x+360 x^2+288 x^3\right ) \log (x)-144 x \log ^2(x)}{256 x+160 x^2+153 x^3+40 x^4+16 x^5+\left (-128 x-40 x^2-32 x^3\right ) \log (x)+16 x \log ^2(x)} \, dx=\left (3+e^5+x\right ) \left (-9+\frac {3}{4+\frac {5 x}{4}+x^2-\log (x)}\right ) \]
Time = 0.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {144-2244 x-1728 x^2-1425 x^3-360 x^4-144 x^5+e^5 \left (48-60 x-96 x^2\right )+\left (1104 x+360 x^2+288 x^3\right ) \log (x)-144 x \log ^2(x)}{256 x+160 x^2+153 x^3+40 x^4+16 x^5+\left (-128 x-40 x^2-32 x^3\right ) \log (x)+16 x \log ^2(x)} \, dx=-3 \left (3 x+\frac {4 \left (3+e^5+x\right )}{-16-5 x-4 x^2+4 \log (x)}\right ) \]
Integrate[(144 - 2244*x - 1728*x^2 - 1425*x^3 - 360*x^4 - 144*x^5 + E^5*(4 8 - 60*x - 96*x^2) + (1104*x + 360*x^2 + 288*x^3)*Log[x] - 144*x*Log[x]^2) /(256*x + 160*x^2 + 153*x^3 + 40*x^4 + 16*x^5 + (-128*x - 40*x^2 - 32*x^3) *Log[x] + 16*x*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.
\(\displaystyle \int \frac {-144 x^5-360 x^4-1425 x^3-1728 x^2+e^5 \left (-96 x^2-60 x+48\right )+\left (288 x^3+360 x^2+1104 x\right ) \log (x)-2244 x-144 x \log ^2(x)+144}{16 x^5+40 x^4+153 x^3+160 x^2+\left (-32 x^3-40 x^2-128 x\right ) \log (x)+256 x+16 x \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-144 x^5-360 x^4-1425 x^3-1728 x^2+e^5 \left (-96 x^2-60 x+48\right )+\left (288 x^3+360 x^2+1104 x\right ) \log (x)-2244 x-144 x \log ^2(x)+144}{x \left (4 x^2+5 x-4 \log (x)+16\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {12 \left (x+e^5+3\right ) \left (8 x^2+5 x-4\right )}{x \left (4 x^2+5 x-4 \log (x)+16\right )^2}+\frac {12}{4 x^2+5 x-4 \log (x)+16}-9\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -12 \left (11+5 e^5\right ) \int \frac {1}{\left (4 x^2+5 x-4 \log (x)+16\right )^2}dx+48 \left (3+e^5\right ) \int \frac {1}{x \left (4 x^2+5 x-4 \log (x)+16\right )^2}dx-12 \left (29+8 e^5\right ) \int \frac {x}{\left (4 x^2+5 x-4 \log (x)+16\right )^2}dx-96 \int \frac {x^2}{\left (4 x^2+5 x-4 \log (x)+16\right )^2}dx+12 \int \frac {1}{4 x^2+5 x-4 \log (x)+16}dx-9 x\) |
Int[(144 - 2244*x - 1728*x^2 - 1425*x^3 - 360*x^4 - 144*x^5 + E^5*(48 - 60 *x - 96*x^2) + (1104*x + 360*x^2 + 288*x^3)*Log[x] - 144*x*Log[x]^2)/(256* x + 160*x^2 + 153*x^3 + 40*x^4 + 16*x^5 + (-128*x - 40*x^2 - 32*x^3)*Log[x ] + 16*x*Log[x]^2),x]
3.28.10.3.1 Defintions of rubi rules used
Time = 0.82 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04
method | result | size |
risch | \(-9 x +\frac {12 \,{\mathrm e}^{5}+36+12 x}{4 x^{2}+5 x -4 \ln \left (x \right )+16}\) | \(28\) |
norman | \(\frac {-\frac {303 x}{4}-45 \ln \left (x \right )+36 x \ln \left (x \right )-36 x^{3}+216+12 \,{\mathrm e}^{5}}{4 x^{2}+5 x -4 \ln \left (x \right )+16}\) | \(41\) |
default | \(-\frac {3 \left (-44 x -15 x^{2}-12 x^{3}+12 x \ln \left (x \right )+12+4 \,{\mathrm e}^{5}\right )}{-4 x^{2}+4 \ln \left (x \right )-5 x -16}\) | \(43\) |
parallelrisch | \(\frac {-144 x^{3}+144-180 x^{2}+144 x \ln \left (x \right )+48 \,{\mathrm e}^{5}-528 x}{16 x^{2}+20 x -16 \ln \left (x \right )+64}\) | \(43\) |
int((-144*x*ln(x)^2+(288*x^3+360*x^2+1104*x)*ln(x)+(-96*x^2-60*x+48)*exp(5 )-144*x^5-360*x^4-1425*x^3-1728*x^2-2244*x+144)/(16*x*ln(x)^2+(-32*x^3-40* x^2-128*x)*ln(x)+16*x^5+40*x^4+153*x^3+160*x^2+256*x),x,method=_RETURNVERB OSE)
Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {144-2244 x-1728 x^2-1425 x^3-360 x^4-144 x^5+e^5 \left (48-60 x-96 x^2\right )+\left (1104 x+360 x^2+288 x^3\right ) \log (x)-144 x \log ^2(x)}{256 x+160 x^2+153 x^3+40 x^4+16 x^5+\left (-128 x-40 x^2-32 x^3\right ) \log (x)+16 x \log ^2(x)} \, dx=-\frac {3 \, {\left (12 \, x^{3} + 15 \, x^{2} - 12 \, x \log \left (x\right ) + 44 \, x - 4 \, e^{5} - 12\right )}}{4 \, x^{2} + 5 \, x - 4 \, \log \left (x\right ) + 16} \]
integrate((-144*x*log(x)^2+(288*x^3+360*x^2+1104*x)*log(x)+(-96*x^2-60*x+4 8)*exp(5)-144*x^5-360*x^4-1425*x^3-1728*x^2-2244*x+144)/(16*x*log(x)^2+(-3 2*x^3-40*x^2-128*x)*log(x)+16*x^5+40*x^4+153*x^3+160*x^2+256*x),x, algorit hm=\
Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {144-2244 x-1728 x^2-1425 x^3-360 x^4-144 x^5+e^5 \left (48-60 x-96 x^2\right )+\left (1104 x+360 x^2+288 x^3\right ) \log (x)-144 x \log ^2(x)}{256 x+160 x^2+153 x^3+40 x^4+16 x^5+\left (-128 x-40 x^2-32 x^3\right ) \log (x)+16 x \log ^2(x)} \, dx=- 9 x + \frac {- 12 x - 12 e^{5} - 36}{- 4 x^{2} - 5 x + 4 \log {\left (x \right )} - 16} \]
integrate((-144*x*ln(x)**2+(288*x**3+360*x**2+1104*x)*ln(x)+(-96*x**2-60*x +48)*exp(5)-144*x**5-360*x**4-1425*x**3-1728*x**2-2244*x+144)/(16*x*ln(x)* *2+(-32*x**3-40*x**2-128*x)*ln(x)+16*x**5+40*x**4+153*x**3+160*x**2+256*x) ,x)
Time = 0.22 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {144-2244 x-1728 x^2-1425 x^3-360 x^4-144 x^5+e^5 \left (48-60 x-96 x^2\right )+\left (1104 x+360 x^2+288 x^3\right ) \log (x)-144 x \log ^2(x)}{256 x+160 x^2+153 x^3+40 x^4+16 x^5+\left (-128 x-40 x^2-32 x^3\right ) \log (x)+16 x \log ^2(x)} \, dx=-\frac {3 \, {\left (12 \, x^{3} + 15 \, x^{2} - 12 \, x \log \left (x\right ) + 44 \, x - 4 \, e^{5} - 12\right )}}{4 \, x^{2} + 5 \, x - 4 \, \log \left (x\right ) + 16} \]
integrate((-144*x*log(x)^2+(288*x^3+360*x^2+1104*x)*log(x)+(-96*x^2-60*x+4 8)*exp(5)-144*x^5-360*x^4-1425*x^3-1728*x^2-2244*x+144)/(16*x*log(x)^2+(-3 2*x^3-40*x^2-128*x)*log(x)+16*x^5+40*x^4+153*x^3+160*x^2+256*x),x, algorit hm=\
Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {144-2244 x-1728 x^2-1425 x^3-360 x^4-144 x^5+e^5 \left (48-60 x-96 x^2\right )+\left (1104 x+360 x^2+288 x^3\right ) \log (x)-144 x \log ^2(x)}{256 x+160 x^2+153 x^3+40 x^4+16 x^5+\left (-128 x-40 x^2-32 x^3\right ) \log (x)+16 x \log ^2(x)} \, dx=-\frac {3 \, {\left (12 \, x^{3} + 15 \, x^{2} - 12 \, x \log \left (x\right ) + 44 \, x - 4 \, e^{5} - 12\right )}}{4 \, x^{2} + 5 \, x - 4 \, \log \left (x\right ) + 16} \]
integrate((-144*x*log(x)^2+(288*x^3+360*x^2+1104*x)*log(x)+(-96*x^2-60*x+4 8)*exp(5)-144*x^5-360*x^4-1425*x^3-1728*x^2-2244*x+144)/(16*x*log(x)^2+(-3 2*x^3-40*x^2-128*x)*log(x)+16*x^5+40*x^4+153*x^3+160*x^2+256*x),x, algorit hm=\
Time = 15.50 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {144-2244 x-1728 x^2-1425 x^3-360 x^4-144 x^5+e^5 \left (48-60 x-96 x^2\right )+\left (1104 x+360 x^2+288 x^3\right ) \log (x)-144 x \log ^2(x)}{256 x+160 x^2+153 x^3+40 x^4+16 x^5+\left (-128 x-40 x^2-32 x^3\right ) \log (x)+16 x \log ^2(x)} \, dx=-\frac {3\,\left (44\,x-4\,{\mathrm {e}}^5-12\,x\,\ln \left (x\right )+15\,x^2+12\,x^3-12\right )}{5\,x-4\,\ln \left (x\right )+4\,x^2+16} \]