Integrand size = 116, antiderivative size = 26 \[ \int \frac {-2-8 x-2 x^2-10 x^3-8 x^4+\left (-4 x^2+22 x^3+24 x^4\right ) \log (x)+\left (-2 x^2+11 x^3+12 x^4\right ) \log ^2(x)+\left (2 x+8 x^2+\left (-2 x-8 x^2\right ) \log (x)+\left (-x-4 x^2\right ) \log ^2(x)\right ) \log (1+4 x)}{\left (x+4 x^2\right ) \log ^2(x)} \, dx=\left (1+\frac {2}{\log (x)}\right ) \left (1+x \left (x+x^2-\log (1+4 x)\right )\right ) \] Output:
((x^2+x-ln(1+4*x))*x+1)*(2/ln(x)+1)
Time = 0.85 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.46 \[ \int \frac {-2-8 x-2 x^2-10 x^3-8 x^4+\left (-4 x^2+22 x^3+24 x^4\right ) \log (x)+\left (-2 x^2+11 x^3+12 x^4\right ) \log ^2(x)+\left (2 x+8 x^2+\left (-2 x-8 x^2\right ) \log (x)+\left (-x-4 x^2\right ) \log ^2(x)\right ) \log (1+4 x)}{\left (x+4 x^2\right ) \log ^2(x)} \, dx=x^2+x^3+\frac {2 \left (1+x^2+x^3\right )}{\log (x)}-\frac {x (2+\log (x)) \log (1+4 x)}{\log (x)} \] Input:
Integrate[(-2 - 8*x - 2*x^2 - 10*x^3 - 8*x^4 + (-4*x^2 + 22*x^3 + 24*x^4)* Log[x] + (-2*x^2 + 11*x^3 + 12*x^4)*Log[x]^2 + (2*x + 8*x^2 + (-2*x - 8*x^ 2)*Log[x] + (-x - 4*x^2)*Log[x]^2)*Log[1 + 4*x])/((x + 4*x^2)*Log[x]^2),x]
Output:
x^2 + x^3 + (2*(1 + x^2 + x^3))/Log[x] - (x*(2 + Log[x])*Log[1 + 4*x])/Log [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.
\(\displaystyle \int \frac {-8 x^4-10 x^3-2 x^2+\left (8 x^2+\left (-4 x^2-x\right ) \log ^2(x)+\left (-8 x^2-2 x\right ) \log (x)+2 x\right ) \log (4 x+1)+\left (12 x^4+11 x^3-2 x^2\right ) \log ^2(x)+\left (24 x^4+22 x^3-4 x^2\right ) \log (x)-8 x-2}{\left (4 x^2+x\right ) \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {-8 x^4-10 x^3-2 x^2+\left (8 x^2+\left (-4 x^2-x\right ) \log ^2(x)+\left (-8 x^2-2 x\right ) \log (x)+2 x\right ) \log (4 x+1)+\left (12 x^4+11 x^3-2 x^2\right ) \log ^2(x)+\left (24 x^4+22 x^3-4 x^2\right ) \log (x)-8 x-2}{x (4 x+1) \log ^2(x)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {8 x^3}{(4 x+1) \log ^2(x)}+\frac {\left (12 x^2+11 x-2\right ) x}{4 x+1}-\frac {10 x^2}{(4 x+1) \log ^2(x)}+\frac {2 \left (12 x^2+11 x-2\right ) x}{(4 x+1) \log (x)}-\frac {2 x}{(4 x+1) \log ^2(x)}-\frac {\left (\log ^2(x)+2 \log (x)-2\right ) \log (4 x+1)}{\log ^2(x)}-\frac {8}{(4 x+1) \log ^2(x)}-\frac {2}{(4 x+1) x \log ^2(x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -8 \int \frac {x^3}{(4 x+1) \log ^2(x)}dx-10 \int \frac {x^2}{(4 x+1) \log ^2(x)}dx+2 \int \frac {x \left (12 x^2+11 x-2\right )}{(4 x+1) \log (x)}dx-8 \int \frac {1}{(4 x+1) \log ^2(x)}dx-2 \int \frac {1}{x (4 x+1) \log ^2(x)}dx-2 \int \frac {x}{(4 x+1) \log ^2(x)}dx+2 \int \frac {\log (4 x+1)}{\log ^2(x)}dx-2 \int \frac {\log (4 x+1)}{\log (x)}dx+x^3+x^2-\frac {1}{4} (4 x+1) \log (4 x+1)+\frac {1}{4} \log (4 x+1)\) |
Input:
Int[(-2 - 8*x - 2*x^2 - 10*x^3 - 8*x^4 + (-4*x^2 + 22*x^3 + 24*x^4)*Log[x] + (-2*x^2 + 11*x^3 + 12*x^4)*Log[x]^2 + (2*x + 8*x^2 + (-2*x - 8*x^2)*Log [x] + (-x - 4*x^2)*Log[x]^2)*Log[1 + 4*x])/((x + 4*x^2)*Log[x]^2),x]
Output:
$Aborted
Time = 9.94 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.85
method | result | size |
risch | \(-\frac {x \left (\ln \left (x \right )+2\right ) \ln \left (1+4 x \right )}{\ln \left (x \right )}+\frac {x^{3} \ln \left (x \right )+2 x^{3}+x^{2} \ln \left (x \right )+2 x^{2}+2}{\ln \left (x \right )}\) | \(48\) |
parallelrisch | \(-\frac {-16 x^{3} \ln \left (x \right )-32-32 x^{3}-16 x^{2} \ln \left (x \right )+16 \ln \left (x \right ) \ln \left (1+4 x \right ) x +8 \ln \left (\frac {1}{4}+x \right ) \ln \left (x \right )-32 x^{2}+32 \ln \left (1+4 x \right ) x -8 \ln \left (1+4 x \right ) \ln \left (x \right )}{16 \ln \left (x \right )}\) | \(71\) |
Input:
int((((-4*x^2-x)*ln(x)^2+(-8*x^2-2*x)*ln(x)+8*x^2+2*x)*ln(1+4*x)+(12*x^4+1 1*x^3-2*x^2)*ln(x)^2+(24*x^4+22*x^3-4*x^2)*ln(x)-8*x^4-10*x^3-2*x^2-8*x-2) /(4*x^2+x)/ln(x)^2,x,method=_RETURNVERBOSE)
Output:
-x*(ln(x)+2)/ln(x)*ln(1+4*x)+(x^3*ln(x)+2*x^3+x^2*ln(x)+2*x^2+2)/ln(x)
Time = 0.07 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \frac {-2-8 x-2 x^2-10 x^3-8 x^4+\left (-4 x^2+22 x^3+24 x^4\right ) \log (x)+\left (-2 x^2+11 x^3+12 x^4\right ) \log ^2(x)+\left (2 x+8 x^2+\left (-2 x-8 x^2\right ) \log (x)+\left (-x-4 x^2\right ) \log ^2(x)\right ) \log (1+4 x)}{\left (x+4 x^2\right ) \log ^2(x)} \, dx=\frac {2 \, x^{3} + 2 \, x^{2} - {\left (x \log \left (x\right ) + 2 \, x\right )} \log \left (4 \, x + 1\right ) + {\left (x^{3} + x^{2}\right )} \log \left (x\right ) + 2}{\log \left (x\right )} \] Input:
integrate((((-4*x^2-x)*log(x)^2+(-8*x^2-2*x)*log(x)+8*x^2+2*x)*log(1+4*x)+ (12*x^4+11*x^3-2*x^2)*log(x)^2+(24*x^4+22*x^3-4*x^2)*log(x)-8*x^4-10*x^3-2 *x^2-8*x-2)/(4*x^2+x)/log(x)^2,x, algorithm="fricas")
Output:
(2*x^3 + 2*x^2 - (x*log(x) + 2*x)*log(4*x + 1) + (x^3 + x^2)*log(x) + 2)/l og(x)
Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (20) = 40\).
Time = 0.22 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58 \[ \int \frac {-2-8 x-2 x^2-10 x^3-8 x^4+\left (-4 x^2+22 x^3+24 x^4\right ) \log (x)+\left (-2 x^2+11 x^3+12 x^4\right ) \log ^2(x)+\left (2 x+8 x^2+\left (-2 x-8 x^2\right ) \log (x)+\left (-x-4 x^2\right ) \log ^2(x)\right ) \log (1+4 x)}{\left (x+4 x^2\right ) \log ^2(x)} \, dx=x^{3} + x^{2} + \frac {\left (- x \log {\left (x \right )} - 2 x\right ) \log {\left (4 x + 1 \right )}}{\log {\left (x \right )}} + \frac {2 x^{3} + 2 x^{2} + 2}{\log {\left (x \right )}} \] Input:
integrate((((-4*x**2-x)*ln(x)**2+(-8*x**2-2*x)*ln(x)+8*x**2+2*x)*ln(1+4*x) +(12*x**4+11*x**3-2*x**2)*ln(x)**2+(24*x**4+22*x**3-4*x**2)*ln(x)-8*x**4-1 0*x**3-2*x**2-8*x-2)/(4*x**2+x)/ln(x)**2,x)
Output:
x**3 + x**2 + (-x*log(x) - 2*x)*log(4*x + 1)/log(x) + (2*x**3 + 2*x**2 + 2 )/log(x)
Time = 0.08 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \frac {-2-8 x-2 x^2-10 x^3-8 x^4+\left (-4 x^2+22 x^3+24 x^4\right ) \log (x)+\left (-2 x^2+11 x^3+12 x^4\right ) \log ^2(x)+\left (2 x+8 x^2+\left (-2 x-8 x^2\right ) \log (x)+\left (-x-4 x^2\right ) \log ^2(x)\right ) \log (1+4 x)}{\left (x+4 x^2\right ) \log ^2(x)} \, dx=\frac {2 \, x^{3} + 2 \, x^{2} - {\left (x \log \left (x\right ) + 2 \, x\right )} \log \left (4 \, x + 1\right ) + {\left (x^{3} + x^{2}\right )} \log \left (x\right ) + 2}{\log \left (x\right )} \] Input:
integrate((((-4*x^2-x)*log(x)^2+(-8*x^2-2*x)*log(x)+8*x^2+2*x)*log(1+4*x)+ (12*x^4+11*x^3-2*x^2)*log(x)^2+(24*x^4+22*x^3-4*x^2)*log(x)-8*x^4-10*x^3-2 *x^2-8*x-2)/(4*x^2+x)/log(x)^2,x, algorithm="maxima")
Output:
(2*x^3 + 2*x^2 - (x*log(x) + 2*x)*log(4*x + 1) + (x^3 + x^2)*log(x) + 2)/l og(x)
Time = 0.12 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.46 \[ \int \frac {-2-8 x-2 x^2-10 x^3-8 x^4+\left (-4 x^2+22 x^3+24 x^4\right ) \log (x)+\left (-2 x^2+11 x^3+12 x^4\right ) \log ^2(x)+\left (2 x+8 x^2+\left (-2 x-8 x^2\right ) \log (x)+\left (-x-4 x^2\right ) \log ^2(x)\right ) \log (1+4 x)}{\left (x+4 x^2\right ) \log ^2(x)} \, dx=x^{3} + x^{2} - {\left (x + \frac {2 \, x}{\log \left (x\right )}\right )} \log \left (4 \, x + 1\right ) + \frac {2 \, {\left (x^{3} + x^{2} + 1\right )}}{\log \left (x\right )} \] Input:
integrate((((-4*x^2-x)*log(x)^2+(-8*x^2-2*x)*log(x)+8*x^2+2*x)*log(1+4*x)+ (12*x^4+11*x^3-2*x^2)*log(x)^2+(24*x^4+22*x^3-4*x^2)*log(x)-8*x^4-10*x^3-2 *x^2-8*x-2)/(4*x^2+x)/log(x)^2,x, algorithm="giac")
Output:
x^3 + x^2 - (x + 2*x/log(x))*log(4*x + 1) + 2*(x^3 + x^2 + 1)/log(x)
Time = 2.15 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.31 \[ \int \frac {-2-8 x-2 x^2-10 x^3-8 x^4+\left (-4 x^2+22 x^3+24 x^4\right ) \log (x)+\left (-2 x^2+11 x^3+12 x^4\right ) \log ^2(x)+\left (2 x+8 x^2+\left (-2 x-8 x^2\right ) \log (x)+\left (-x-4 x^2\right ) \log ^2(x)\right ) \log (1+4 x)}{\left (x+4 x^2\right ) \log ^2(x)} \, dx=\frac {2\,x^2+2\,x^3-2\,x^2\,\ln \left (x\right )\,\left (3\,x+2\right )+2}{\ln \left (x\right )}+5\,x^2+7\,x^3-\frac {\ln \left (4\,x+1\right )\,\left (2\,x+x\,\ln \left (x\right )\right )}{\ln \left (x\right )} \] Input:
int(-(8*x - log(x)*(22*x^3 - 4*x^2 + 24*x^4) - log(x)^2*(11*x^3 - 2*x^2 + 12*x^4) - log(4*x + 1)*(2*x - log(x)*(2*x + 8*x^2) - log(x)^2*(x + 4*x^2) + 8*x^2) + 2*x^2 + 10*x^3 + 8*x^4 + 2)/(log(x)^2*(x + 4*x^2)),x)
Output:
(2*x^2 + 2*x^3 - 2*x^2*log(x)*(3*x + 2) + 2)/log(x) + 5*x^2 + 7*x^3 - (log (4*x + 1)*(2*x + x*log(x)))/log(x)
Time = 0.19 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.88 \[ \int \frac {-2-8 x-2 x^2-10 x^3-8 x^4+\left (-4 x^2+22 x^3+24 x^4\right ) \log (x)+\left (-2 x^2+11 x^3+12 x^4\right ) \log ^2(x)+\left (2 x+8 x^2+\left (-2 x-8 x^2\right ) \log (x)+\left (-x-4 x^2\right ) \log ^2(x)\right ) \log (1+4 x)}{\left (x+4 x^2\right ) \log ^2(x)} \, dx=\frac {-\mathrm {log}\left (4 x +1\right ) \mathrm {log}\left (x \right ) x -2 \,\mathrm {log}\left (4 x +1\right ) x +\mathrm {log}\left (x \right ) x^{3}+\mathrm {log}\left (x \right ) x^{2}+2 x^{3}+2 x^{2}+2}{\mathrm {log}\left (x \right )} \] Input:
int((((-4*x^2-x)*log(x)^2+(-8*x^2-2*x)*log(x)+8*x^2+2*x)*log(1+4*x)+(12*x^ 4+11*x^3-2*x^2)*log(x)^2+(24*x^4+22*x^3-4*x^2)*log(x)-8*x^4-10*x^3-2*x^2-8 *x-2)/(4*x^2+x)/log(x)^2,x)
Output:
( - log(4*x + 1)*log(x)*x - 2*log(4*x + 1)*x + log(x)*x**3 + log(x)*x**2 + 2*x**3 + 2*x**2 + 2)/log(x)