Integrand size = 63, antiderivative size = 22 \[ \int \frac {10 x+160 x^2+3 x^3+\left (150 x+3 x^2\right ) \log (x)+\left (10 x^2+10 x \log (x)\right ) \log \left (x^2+2 x \log (x)+\log ^2(x)\right )}{10 x+10 \log (x)} \, dx=x^2 \left (7+\frac {1}{10} \left (5+x+5 \log \left ((x+\log (x))^2\right )\right )\right ) \] Output:
x^2*(15/2+1/10*x+1/2*ln((x+ln(x))^2))
Time = 0.10 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {10 x+160 x^2+3 x^3+\left (150 x+3 x^2\right ) \log (x)+\left (10 x^2+10 x \log (x)\right ) \log \left (x^2+2 x \log (x)+\log ^2(x)\right )}{10 x+10 \log (x)} \, dx=\frac {1}{10} \left (75 x^2+x^3+5 x^2 \log \left ((x+\log (x))^2\right )\right ) \] Input:
Integrate[(10*x + 160*x^2 + 3*x^3 + (150*x + 3*x^2)*Log[x] + (10*x^2 + 10* x*Log[x])*Log[x^2 + 2*x*Log[x] + Log[x]^2])/(10*x + 10*Log[x]),x]
Output:
(75*x^2 + x^3 + 5*x^2*Log[(x + Log[x])^2])/10
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.
\(\displaystyle \int \frac {3 x^3+160 x^2+\left (10 x^2+10 x \log (x)\right ) \log \left (x^2+\log ^2(x)+2 x \log (x)\right )+\left (3 x^2+150 x\right ) \log (x)+10 x}{10 x+10 \log (x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {3 x^3+160 x^2+\left (10 x^2+10 x \log (x)\right ) \log \left (x^2+\log ^2(x)+2 x \log (x)\right )+\left (3 x^2+150 x\right ) \log (x)+10 x}{10 (x+\log (x))}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{10} \int \frac {3 x^3+160 x^2+10 x+3 \left (x^2+50 x\right ) \log (x)+10 \left (x^2+\log (x) x\right ) \log \left (x^2+2 \log (x) x+\log ^2(x)\right )}{x+\log (x)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{10} \int \left (\frac {x \left (3 x^2+3 \log (x) x+160 x+150 \log (x)+10\right )}{x+\log (x)}+10 x \log \left ((x+\log (x))^2\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{10} \left (10 \int \frac {x^2}{x+\log (x)}dx+10 \int \frac {x}{x+\log (x)}dx+10 \int x \log \left ((x+\log (x))^2\right )dx+x^3+75 x^2\right )\) |
Input:
Int[(10*x + 160*x^2 + 3*x^3 + (150*x + 3*x^2)*Log[x] + (10*x^2 + 10*x*Log[ x])*Log[x^2 + 2*x*Log[x] + Log[x]^2])/(10*x + 10*Log[x]),x]
Output:
$Aborted
Time = 0.70 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.41
method | result | size |
parallelrisch | \(\frac {x^{3}}{10}+\frac {\ln \left (\ln \left (x \right )^{2}+2 x \ln \left (x \right )+x^{2}\right ) x^{2}}{2}+\frac {15 x^{2}}{2}\) | \(31\) |
risch | \(x^{2} \ln \left (x +\ln \left (x \right )\right )+\frac {x^{3}}{10}-\frac {i \pi \,x^{2} \operatorname {csgn}\left (i \left (x +\ln \left (x \right )\right )\right )^{2} \operatorname {csgn}\left (i \left (x +\ln \left (x \right )\right )^{2}\right )}{4}+\frac {i \pi \,x^{2} \operatorname {csgn}\left (i \left (x +\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i \left (x +\ln \left (x \right )\right )^{2}\right )^{2}}{2}-\frac {i \pi \,x^{2} \operatorname {csgn}\left (i \left (x +\ln \left (x \right )\right )^{2}\right )^{3}}{4}+\frac {15 x^{2}}{2}\) | \(94\) |
Input:
int(((10*x*ln(x)+10*x^2)*ln(ln(x)^2+2*x*ln(x)+x^2)+(3*x^2+150*x)*ln(x)+3*x ^3+160*x^2+10*x)/(10*ln(x)+10*x),x,method=_RETURNVERBOSE)
Output:
1/10*x^3+1/2*ln(ln(x)^2+2*x*ln(x)+x^2)*x^2+15/2*x^2
Time = 0.07 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {10 x+160 x^2+3 x^3+\left (150 x+3 x^2\right ) \log (x)+\left (10 x^2+10 x \log (x)\right ) \log \left (x^2+2 x \log (x)+\log ^2(x)\right )}{10 x+10 \log (x)} \, dx=\frac {1}{10} \, x^{3} + \frac {1}{2} \, x^{2} \log \left (x^{2} + 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}\right ) + \frac {15}{2} \, x^{2} \] Input:
integrate(((10*x*log(x)+10*x^2)*log(log(x)^2+2*x*log(x)+x^2)+(3*x^2+150*x) *log(x)+3*x^3+160*x^2+10*x)/(10*log(x)+10*x),x, algorithm="fricas")
Output:
1/10*x^3 + 1/2*x^2*log(x^2 + 2*x*log(x) + log(x)^2) + 15/2*x^2
Time = 0.19 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int \frac {10 x+160 x^2+3 x^3+\left (150 x+3 x^2\right ) \log (x)+\left (10 x^2+10 x \log (x)\right ) \log \left (x^2+2 x \log (x)+\log ^2(x)\right )}{10 x+10 \log (x)} \, dx=\frac {x^{3}}{10} + \frac {x^{2} \log {\left (x^{2} + 2 x \log {\left (x \right )} + \log {\left (x \right )}^{2} \right )}}{2} + \frac {15 x^{2}}{2} \] Input:
integrate(((10*x*ln(x)+10*x**2)*ln(ln(x)**2+2*x*ln(x)+x**2)+(3*x**2+150*x) *ln(x)+3*x**3+160*x**2+10*x)/(10*ln(x)+10*x),x)
Output:
x**3/10 + x**2*log(x**2 + 2*x*log(x) + log(x)**2)/2 + 15*x**2/2
Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {10 x+160 x^2+3 x^3+\left (150 x+3 x^2\right ) \log (x)+\left (10 x^2+10 x \log (x)\right ) \log \left (x^2+2 x \log (x)+\log ^2(x)\right )}{10 x+10 \log (x)} \, dx=\frac {1}{10} \, x^{3} + x^{2} \log \left (x + \log \left (x\right )\right ) + \frac {15}{2} \, x^{2} \] Input:
integrate(((10*x*log(x)+10*x^2)*log(log(x)^2+2*x*log(x)+x^2)+(3*x^2+150*x) *log(x)+3*x^3+160*x^2+10*x)/(10*log(x)+10*x),x, algorithm="maxima")
Output:
1/10*x^3 + x^2*log(x + log(x)) + 15/2*x^2
Time = 0.16 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {10 x+160 x^2+3 x^3+\left (150 x+3 x^2\right ) \log (x)+\left (10 x^2+10 x \log (x)\right ) \log \left (x^2+2 x \log (x)+\log ^2(x)\right )}{10 x+10 \log (x)} \, dx=\frac {1}{10} \, x^{3} + \frac {1}{2} \, x^{2} \log \left (x^{2} + 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}\right ) + \frac {15}{2} \, x^{2} \] Input:
integrate(((10*x*log(x)+10*x^2)*log(log(x)^2+2*x*log(x)+x^2)+(3*x^2+150*x) *log(x)+3*x^3+160*x^2+10*x)/(10*log(x)+10*x),x, algorithm="giac")
Output:
1/10*x^3 + 1/2*x^2*log(x^2 + 2*x*log(x) + log(x)^2) + 15/2*x^2
Time = 2.37 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {10 x+160 x^2+3 x^3+\left (150 x+3 x^2\right ) \log (x)+\left (10 x^2+10 x \log (x)\right ) \log \left (x^2+2 x \log (x)+\log ^2(x)\right )}{10 x+10 \log (x)} \, dx=\frac {15\,x^2}{2}+\frac {x^3}{10}+\frac {x^2\,\ln \left (x^2+2\,x\,\ln \left (x\right )+{\ln \left (x\right )}^2\right )}{2} \] Input:
int((10*x + log(log(x)^2 + 2*x*log(x) + x^2)*(10*x*log(x) + 10*x^2) + log( x)*(150*x + 3*x^2) + 160*x^2 + 3*x^3)/(10*x + 10*log(x)),x)
Output:
(15*x^2)/2 + x^3/10 + (x^2*log(log(x)^2 + 2*x*log(x) + x^2))/2
Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {10 x+160 x^2+3 x^3+\left (150 x+3 x^2\right ) \log (x)+\left (10 x^2+10 x \log (x)\right ) \log \left (x^2+2 x \log (x)+\log ^2(x)\right )}{10 x+10 \log (x)} \, dx=\frac {x^{2} \left (5 \,\mathrm {log}\left (\mathrm {log}\left (x \right )^{2}+2 \,\mathrm {log}\left (x \right ) x +x^{2}\right )+x +75\right )}{10} \] Input:
int(((10*x*log(x)+10*x^2)*log(log(x)^2+2*x*log(x)+x^2)+(3*x^2+150*x)*log(x )+3*x^3+160*x^2+10*x)/(10*log(x)+10*x),x)
Output:
(x**2*(5*log(log(x)**2 + 2*log(x)*x + x**2) + x + 75))/10