Integrand size = 75, antiderivative size = 16 \[ \int \frac {-50 x^4-50 x \log (x)+\left (-100-25 x-300 x^3-75 x^4\right ) \log (4+x)}{\left (4 x^7+x^8+\left (8 x^4+2 x^5\right ) \log (x)+\left (4 x+x^2\right ) \log ^2(x)\right ) \log ^3(4+x)} \, dx=\frac {25}{\left (x^3+\log (x)\right ) \log ^2(4+x)} \]
Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {-50 x^4-50 x \log (x)+\left (-100-25 x-300 x^3-75 x^4\right ) \log (4+x)}{\left (4 x^7+x^8+\left (8 x^4+2 x^5\right ) \log (x)+\left (4 x+x^2\right ) \log ^2(x)\right ) \log ^3(4+x)} \, dx=\frac {25}{\left (x^3+\log (x)\right ) \log ^2(4+x)} \]
Integrate[(-50*x^4 - 50*x*Log[x] + (-100 - 25*x - 300*x^3 - 75*x^4)*Log[4 + x])/((4*x^7 + x^8 + (8*x^4 + 2*x^5)*Log[x] + (4*x + x^2)*Log[x]^2)*Log[4 + 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.
\(\displaystyle \int \frac {-50 x^4+\left (-75 x^4-300 x^3-25 x-100\right ) \log (x+4)-50 x \log (x)}{\left (x^8+4 x^7+\left (x^2+4 x\right ) \log ^2(x)+\left (2 x^5+8 x^4\right ) \log (x)\right ) \log ^3(x+4)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-50 x^4+\left (-75 x^4-300 x^3-25 x-100\right ) \log (x+4)-50 x \log (x)}{x (x+4) \left (x^3+\log (x)\right )^2 \log ^3(x+4)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {50}{(x+4) \left (x^3+\log (x)\right ) \log ^3(x+4)}-\frac {25 \left (3 x^3+1\right )}{x \left (x^3+\log (x)\right )^2 \log ^2(x+4)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -50 \int \frac {1}{(x+4) \left (x^3+\log (x)\right ) \log ^3(x+4)}dx-25 \int \frac {1}{x \left (x^3+\log (x)\right )^2 \log ^2(x+4)}dx-75 \int \frac {x^2}{\left (x^3+\log (x)\right )^2 \log ^2(x+4)}dx\) |
Int[(-50*x^4 - 50*x*Log[x] + (-100 - 25*x - 300*x^3 - 75*x^4)*Log[4 + x])/ ((4*x^7 + x^8 + (8*x^4 + 2*x^5)*Log[x] + (4*x + x^2)*Log[x]^2)*Log[4 + x]^ 3),x]
3.10.93.3.1 Defintions of rubi rules used
Time = 6.31 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06
method | result | size |
risch | \(\frac {25}{\ln \left (4+x \right )^{2} \left (\ln \left (x \right )+x^{3}\right )}\) | \(17\) |
parallelrisch | \(\frac {25}{\ln \left (4+x \right )^{2} \left (\ln \left (x \right )+x^{3}\right )}\) | \(17\) |
int(((-75*x^4-300*x^3-25*x-100)*ln(4+x)-50*x*ln(x)-50*x^4)/((x^2+4*x)*ln(x )^2+(2*x^5+8*x^4)*ln(x)+x^8+4*x^7)/ln(4+x)^3,x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {-50 x^4-50 x \log (x)+\left (-100-25 x-300 x^3-75 x^4\right ) \log (4+x)}{\left (4 x^7+x^8+\left (8 x^4+2 x^5\right ) \log (x)+\left (4 x+x^2\right ) \log ^2(x)\right ) \log ^3(4+x)} \, dx=\frac {25}{{\left (x^{3} + \log \left (x\right )\right )} \log \left (x + 4\right )^{2}} \]
integrate(((-75*x^4-300*x^3-25*x-100)*log(4+x)-50*x*log(x)-50*x^4)/((x^2+4 *x)*log(x)^2+(2*x^5+8*x^4)*log(x)+x^8+4*x^7)/log(4+x)^3,x, algorithm=\
Time = 0.07 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {-50 x^4-50 x \log (x)+\left (-100-25 x-300 x^3-75 x^4\right ) \log (4+x)}{\left (4 x^7+x^8+\left (8 x^4+2 x^5\right ) \log (x)+\left (4 x+x^2\right ) \log ^2(x)\right ) \log ^3(4+x)} \, dx=\frac {25}{\left (x^{3} + \log {\left (x \right )}\right ) \log {\left (x + 4 \right )}^{2}} \]
integrate(((-75*x**4-300*x**3-25*x-100)*ln(4+x)-50*x*ln(x)-50*x**4)/((x**2 +4*x)*ln(x)**2+(2*x**5+8*x**4)*ln(x)+x**8+4*x**7)/ln(4+x)**3,x)
Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {-50 x^4-50 x \log (x)+\left (-100-25 x-300 x^3-75 x^4\right ) \log (4+x)}{\left (4 x^7+x^8+\left (8 x^4+2 x^5\right ) \log (x)+\left (4 x+x^2\right ) \log ^2(x)\right ) \log ^3(4+x)} \, dx=\frac {25}{{\left (x^{3} + \log \left (x\right )\right )} \log \left (x + 4\right )^{2}} \]
integrate(((-75*x^4-300*x^3-25*x-100)*log(4+x)-50*x*log(x)-50*x^4)/((x^2+4 *x)*log(x)^2+(2*x^5+8*x^4)*log(x)+x^8+4*x^7)/log(4+x)^3,x, algorithm=\
Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.50 \[ \int \frac {-50 x^4-50 x \log (x)+\left (-100-25 x-300 x^3-75 x^4\right ) \log (4+x)}{\left (4 x^7+x^8+\left (8 x^4+2 x^5\right ) \log (x)+\left (4 x+x^2\right ) \log ^2(x)\right ) \log ^3(4+x)} \, dx=\frac {25}{x^{3} \log \left (x + 4\right )^{2} + \log \left (x + 4\right )^{2} \log \left (x\right )} \]
integrate(((-75*x^4-300*x^3-25*x-100)*log(4+x)-50*x*log(x)-50*x^4)/((x^2+4 *x)*log(x)^2+(2*x^5+8*x^4)*log(x)+x^8+4*x^7)/log(4+x)^3,x, algorithm=\
Time = 9.68 (sec) , antiderivative size = 956, normalized size of antiderivative = 59.75 \[ \int \frac {-50 x^4-50 x \log (x)+\left (-100-25 x-300 x^3-75 x^4\right ) \log (4+x)}{\left (4 x^7+x^8+\left (8 x^4+2 x^5\right ) \log (x)+\left (4 x+x^2\right ) \log ^2(x)\right ) \log ^3(4+x)} \, dx =\text {Too large to display} \]
int(-(log(x + 4)*(25*x + 300*x^3 + 75*x^4 + 100) + 50*x*log(x) + 50*x^4)/( log(x + 4)^3*(log(x)*(8*x^4 + 2*x^5) + log(x)^2*(4*x + x^2) + 4*x^7 + x^8) ),x)
((25*(48*x + 6*x^2 + 960*x^3 + 456*x^4 + 54*x^5 + 2336*x^6 + 1012*x^7 + 10 8*x^8 + 960*x^9 + 336*x^10 + 27*x^11 + 96))/(6*x^2*(3*x^3 + 1)) + (25*log( x)^2*(4*x + 96*x^3 + 120*x^4 + 27*x^5 + 32))/(6*x^2*(3*x^3 + 1)) - (25*log (x)*(112*x^3 - 12*x + 140*x^4 + 27*x^5 + 768*x^6 + 420*x^7 + 54*x^8 - 48)) /(3*x^2*(3*x^3 + 1)))/(3*x^6*log(x) + log(x)^3 + 3*x^3*log(x)^2 + x^9) - ( (25*log(x)*(8*x + 640*x^3 + 292*x^4 + 54*x^5 + 2112*x^6 + 1320*x^7 + 243*x ^8 + 2304*x^9 + 2520*x^10 + 486*x^11 + 64))/(6*x^2*(3*x^3 + 1)^3) - (25*(1 28*x^4 - 416*x^3 - 24*x + 54*x^5 + 1376*x^6 + 2012*x^7 + 378*x^8 + 9120*x^ 9 + 6240*x^10 + 891*x^11 + 11520*x^12 + 5040*x^13 + 486*x^14 - 96))/(12*x^ 2*(3*x^3 + 1)^3) + (25*log(x)^2*(4*x + 384*x^3 - 192*x^4 - 81*x^5 + 576*x^ 6 + 360*x^7 + 64))/(12*x^2*(3*x^3 + 1)^3))/(2*x^3*log(x) + log(x)^2 + x^6) - ((25*x)/729 + (5600*x^3)/729 - (1025*x^4)/729 - (25*x^5)/36 + (400*x^6) /9 - (175*x^7)/81 - (25*x^8)/12 + (3200*x^9)/27 - (1250*x^10)/27 - (75*x^1 1)/4 + (3200*x^12)/27 + (1750*x^13)/27 + 400/729)/(x^2/243 + (5*x^5)/81 + (10*x^8)/27 + (10*x^11)/9 + (5*x^14)/3 + x^17) + (25/(log(x) + x^3) + (25* log(x + 4)*(3*x^3 + 1)*(x + 4))/(2*x*(log(x) + x^3)^2))/log(x + 4)^2 + ((2 5*(8*x + 1088*x^3 + 344*x^4 + 54*x^5 + 7424*x^6 + 2980*x^7 + 486*x^8 + 259 20*x^9 + 12816*x^10 + 2187*x^11 + 46656*x^12 + 21600*x^13 + 2916*x^14 + 34 560*x^15 + 30240*x^16 + 4374*x^17 + 64))/(12*x^2*(3*x^3 + 1)^5) + (25*log( x)^2*(4*x + 1728*x^3 + 504*x^4 + 243*x^5 + 6912*x^6 - 5832*x^7 - 1458*x...