Integrand size = 81, antiderivative size = 28 \[ \int \frac {\left (12+2 x+22 x^2-8 x^3\right ) \log \left (\frac {4 e^x x^2}{1+8 x+16 x^2}\right )+\left (-x-4 x^2\right ) \log ^2\left (\frac {4 e^x x^2}{1+8 x+16 x^2}\right )}{x+4 x^2} \, dx=(3-x) \log ^2\left (\frac {e^x}{\left (1+\frac {1}{4} \left (4+\frac {2}{x}\right )\right )^2}\right ) \] Output:
(3-x)*ln(exp(x)/(1/2/x+2)^2)^2
Leaf count is larger than twice the leaf count of optimal. \(151\) vs. \(2(28)=56\).
Time = 0.20 (sec) , antiderivative size = 151, normalized size of antiderivative = 5.39 \[ \int \frac {\left (12+2 x+22 x^2-8 x^3\right ) \log \left (\frac {4 e^x x^2}{1+8 x+16 x^2}\right )+\left (-x-4 x^2\right ) \log ^2\left (\frac {4 e^x x^2}{1+8 x+16 x^2}\right )}{x+4 x^2} \, dx=-3 x^2+3 \log ^2\left (\frac {x^2}{(1+4 x)^2}\right )-12 \log (x) \left (x+\log \left (\frac {x^2}{(1+4 x)^2}\right )-\log \left (\frac {4 e^x x^2}{(1+4 x)^2}\right )\right )+6 x \log \left (\frac {4 e^x x^2}{(1+4 x)^2}\right )-x \log ^2\left (\frac {4 e^x x^2}{(1+4 x)^2}\right )+12 x \log (1+4 x)+12 \log \left (\frac {x^2}{(1+4 x)^2}\right ) \log (1+4 x)-12 \log \left (\frac {4 e^x x^2}{(1+4 x)^2}\right ) \log (1+4 x) \] Input:
Integrate[((12 + 2*x + 22*x^2 - 8*x^3)*Log[(4*E^x*x^2)/(1 + 8*x + 16*x^2)] + (-x - 4*x^2)*Log[(4*E^x*x^2)/(1 + 8*x + 16*x^2)]^2)/(x + 4*x^2),x]
Output:
-3*x^2 + 3*Log[x^2/(1 + 4*x)^2]^2 - 12*Log[x]*(x + Log[x^2/(1 + 4*x)^2] - Log[(4*E^x*x^2)/(1 + 4*x)^2]) + 6*x*Log[(4*E^x*x^2)/(1 + 4*x)^2] - x*Log[( 4*E^x*x^2)/(1 + 4*x)^2]^2 + 12*x*Log[1 + 4*x] + 12*Log[x^2/(1 + 4*x)^2]*Lo g[1 + 4*x] - 12*Log[(4*E^x*x^2)/(1 + 4*x)^2]*Log[1 + 4*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.
| \(\displaystyle \int \frac {\left (-4 x^2-x\right ) \log ^2\left (\frac {4 e^x x^2}{16 x^2+8 x+1}\right )+\left (-8 x^3+22 x^2+2 x+12\right ) \log \left (\frac {4 e^x x^2}{16 x^2+8 x+1}\right )}{4 x^2+x} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {\left (-4 x^2-x\right ) \log ^2\left (\frac {4 e^x x^2}{16 x^2+8 x+1}\right )+\left (-8 x^3+22 x^2+2 x+12\right ) \log \left (\frac {4 e^x x^2}{16 x^2+8 x+1}\right )}{x (4 x+1)}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\log \left (\frac {4 e^x x^2}{(4 x+1)^2}\right ) \left (-8 x^3+22 x^2-4 x^2 \log \left (\frac {4 e^x x^2}{(4 x+1)^2}\right )-x \log \left (\frac {4 e^x x^2}{(4 x+1)^2}\right )+2 x+12\right )}{x (4 x+1)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\log ^2\left (\frac {4 e^x x^2}{(4 x+1)^2}\right )-\frac {2 \left (4 x^3-11 x^2-x-6\right ) \log \left (\frac {4 e^x x^2}{(4 x+1)^2}\right )}{x (4 x+1)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\int \log ^2\left (\frac {4 e^x x^2}{(4 x+1)^2}\right )dx+12 \int \frac {\log \left (\frac {4 e^x x^2}{(4 x+1)^2}\right )}{x}dx-52 \int \frac {\log \left (\frac {4 e^x x^2}{(4 x+1)^2}\right )}{4 x+1}dx+\frac {x^3}{3}-3 x^2-x^2 \log \left (\frac {4 e^x x^2}{(4 x+1)^2}\right )+6 x \log \left (\frac {4 e^x x^2}{(4 x+1)^2}\right )+\frac {x}{2}-\frac {25}{8} \log (4 x+1)\) |
Input:
Int[((12 + 2*x + 22*x^2 - 8*x^3)*Log[(4*E^x*x^2)/(1 + 8*x + 16*x^2)] + (-x - 4*x^2)*Log[(4*E^x*x^2)/(1 + 8*x + 16*x^2)]^2)/(x + 4*x^2),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(50\) vs. \(2(21)=42\).
Time = 1.71 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.82
| method | result | size |
| parallelrisch | \(-\ln \left (\frac {4 x^{2} {\mathrm e}^{x}}{16 x^{2}+8 x +1}\right )^{2} x +3 \ln \left (\frac {4 x^{2} {\mathrm e}^{x}}{16 x^{2}+8 x +1}\right )^{2}\) | \(51\) |
| default | \(-\frac {13}{4}-4 x \ln \left (x \right )^{2}-12 x \ln \left (x \right )-2 x^{2} \ln \left (x \right )-12 \ln \left (x \right )^{2}-3 x^{2}-x {\left (\ln \left (\frac {4 x^{2} {\mathrm e}^{x}}{16 x^{2}+8 x +1}\right )-2 \ln \left (x \right )-x +\ln \left (16 x^{2}+8 x +1\right )\right )}^{2}-x \ln \left (16 x^{2}+8 x +1\right )^{2}+4 x \ln \left (16 x^{2}+8 x +1\right )-\left (\ln \left (\frac {4 x^{2} {\mathrm e}^{x}}{16 x^{2}+8 x +1}\right )-2 \ln \left (x \right )-x +\ln \left (16 x^{2}+8 x +1\right )\right ) x^{2}+x^{2} \ln \left (16 x^{2}+8 x +1\right )+26 \ln \left (x \right ) \ln \left (1+4 x \right )-\ln \left (\frac {4 x^{2} {\mathrm e}^{x}}{16 x^{2}+8 x +1}\right ) x^{2}+6 x \ln \left (\frac {4 x^{2} {\mathrm e}^{x}}{16 x^{2}+8 x +1}\right )-13 \ln \left (1+4 x \right )^{2}+12 \ln \left (\frac {4 x^{2} {\mathrm e}^{x}}{16 x^{2}+8 x +1}\right ) \ln \left (x \right )-13 \ln \left (\frac {4 x^{2} {\mathrm e}^{x}}{16 x^{2}+8 x +1}\right ) \ln \left (1+4 x \right )+\frac {13 \ln \left (1+4 x \right ) \left (1+4 x \right )}{4}-\frac {\ln \left (16 x^{2}+8 x +1\right )^{2}}{4}-\frac {21 \ln \left (1+4 x \right )}{4}+\ln \left (16 x^{2}+8 x +1\right )+\left (\ln \left (\frac {4 x^{2} {\mathrm e}^{x}}{16 x^{2}+8 x +1}\right )-2 \ln \left (x \right )-x +\ln \left (16 x^{2}+8 x +1\right )\right ) \ln \left (1+4 x \right )+2 \left (\ln \left (\frac {4 x^{2} {\mathrm e}^{x}}{16 x^{2}+8 x +1}\right )-2 \ln \left (x \right )-x +\ln \left (16 x^{2}+8 x +1\right )\right ) x \ln \left (16 x^{2}+8 x +1\right )-4 \ln \left (x \right ) x \left (\ln \left (\frac {4 x^{2} {\mathrm e}^{x}}{16 x^{2}+8 x +1}\right )-2 \ln \left (x \right )-x +\ln \left (16 x^{2}+8 x +1\right )\right )+4 \left (-2+2 \ln \left (x \right )\right ) x \ln \left (x +\frac {1}{4}\right )-2 i \pi \,\operatorname {csgn}\left (i \left (x +\frac {1}{4}\right )^{2}\right ) \left (\operatorname {csgn}\left (i \left (x +\frac {1}{4}\right )\right )^{2}-2 \,\operatorname {csgn}\left (i \left (x +\frac {1}{4}\right )^{2}\right ) \operatorname {csgn}\left (i \left (x +\frac {1}{4}\right )\right )+\operatorname {csgn}\left (i \left (x +\frac {1}{4}\right )^{2}\right )^{2}\right ) \left (x \ln \left (x \right )-x \right )\) | \(549\) |
| parts | \(-\frac {13}{4}-4 x \ln \left (x \right )^{2}-12 x \ln \left (x \right )-2 x^{2} \ln \left (x \right )-12 \ln \left (x \right )^{2}-3 x^{2}-x {\left (\ln \left (\frac {4 x^{2} {\mathrm e}^{x}}{16 x^{2}+8 x +1}\right )-2 \ln \left (x \right )-x +\ln \left (16 x^{2}+8 x +1\right )\right )}^{2}-x \ln \left (16 x^{2}+8 x +1\right )^{2}+4 x \ln \left (16 x^{2}+8 x +1\right )-\left (\ln \left (\frac {4 x^{2} {\mathrm e}^{x}}{16 x^{2}+8 x +1}\right )-2 \ln \left (x \right )-x +\ln \left (16 x^{2}+8 x +1\right )\right ) x^{2}+x^{2} \ln \left (16 x^{2}+8 x +1\right )+26 \ln \left (x \right ) \ln \left (1+4 x \right )-\ln \left (\frac {4 x^{2} {\mathrm e}^{x}}{16 x^{2}+8 x +1}\right ) x^{2}+6 x \ln \left (\frac {4 x^{2} {\mathrm e}^{x}}{16 x^{2}+8 x +1}\right )-13 \ln \left (1+4 x \right )^{2}+12 \ln \left (\frac {4 x^{2} {\mathrm e}^{x}}{16 x^{2}+8 x +1}\right ) \ln \left (x \right )-13 \ln \left (\frac {4 x^{2} {\mathrm e}^{x}}{16 x^{2}+8 x +1}\right ) \ln \left (1+4 x \right )+\frac {13 \ln \left (1+4 x \right ) \left (1+4 x \right )}{4}-\frac {\ln \left (16 x^{2}+8 x +1\right )^{2}}{4}-\frac {21 \ln \left (1+4 x \right )}{4}+\ln \left (16 x^{2}+8 x +1\right )+\left (\ln \left (\frac {4 x^{2} {\mathrm e}^{x}}{16 x^{2}+8 x +1}\right )-2 \ln \left (x \right )-x +\ln \left (16 x^{2}+8 x +1\right )\right ) \ln \left (1+4 x \right )+2 \left (\ln \left (\frac {4 x^{2} {\mathrm e}^{x}}{16 x^{2}+8 x +1}\right )-2 \ln \left (x \right )-x +\ln \left (16 x^{2}+8 x +1\right )\right ) x \ln \left (16 x^{2}+8 x +1\right )-4 \ln \left (x \right ) x \left (\ln \left (\frac {4 x^{2} {\mathrm e}^{x}}{16 x^{2}+8 x +1}\right )-2 \ln \left (x \right )-x +\ln \left (16 x^{2}+8 x +1\right )\right )+4 \left (-2+2 \ln \left (x \right )\right ) x \ln \left (x +\frac {1}{4}\right )-2 i \pi \,\operatorname {csgn}\left (i \left (x +\frac {1}{4}\right )^{2}\right ) \left (\operatorname {csgn}\left (i \left (x +\frac {1}{4}\right )\right )^{2}-2 \,\operatorname {csgn}\left (i \left (x +\frac {1}{4}\right )^{2}\right ) \operatorname {csgn}\left (i \left (x +\frac {1}{4}\right )\right )+\operatorname {csgn}\left (i \left (x +\frac {1}{4}\right )^{2}\right )^{2}\right ) \left (x \ln \left (x \right )-x \right )\) | \(549\) |
| risch | \(\text {Expression too large to display}\) | \(6478\) |
Input:
int(((-4*x^2-x)*ln(4*x^2*exp(x)/(16*x^2+8*x+1))^2+(-8*x^3+22*x^2+2*x+12)*l n(4*x^2*exp(x)/(16*x^2+8*x+1)))/(4*x^2+x),x,method=_RETURNVERBOSE)
Output:
-ln(4*x^2*exp(x)/(16*x^2+8*x+1))^2*x+3*ln(4*x^2*exp(x)/(16*x^2+8*x+1))^2
Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {\left (12+2 x+22 x^2-8 x^3\right ) \log \left (\frac {4 e^x x^2}{1+8 x+16 x^2}\right )+\left (-x-4 x^2\right ) \log ^2\left (\frac {4 e^x x^2}{1+8 x+16 x^2}\right )}{x+4 x^2} \, dx=-{\left (x - 3\right )} \log \left (\frac {4 \, x^{2} e^{x}}{16 \, x^{2} + 8 \, x + 1}\right )^{2} \] Input:
integrate(((-4*x^2-x)*log(4*x^2*exp(x)/(16*x^2+8*x+1))^2+(-8*x^3+22*x^2+2* x+12)*log(4*x^2*exp(x)/(16*x^2+8*x+1)))/(4*x^2+x),x, algorithm="fricas")
Output:
-(x - 3)*log(4*x^2*e^x/(16*x^2 + 8*x + 1))^2
Time = 0.32 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {\left (12+2 x+22 x^2-8 x^3\right ) \log \left (\frac {4 e^x x^2}{1+8 x+16 x^2}\right )+\left (-x-4 x^2\right ) \log ^2\left (\frac {4 e^x x^2}{1+8 x+16 x^2}\right )}{x+4 x^2} \, dx=\left (3 - x\right ) \log {\left (\frac {4 x^{2} e^{x}}{16 x^{2} + 8 x + 1} \right )}^{2} \] Input:
integrate(((-4*x**2-x)*ln(4*x**2*exp(x)/(16*x**2+8*x+1))**2+(-8*x**3+22*x* *2+2*x+12)*ln(4*x**2*exp(x)/(16*x**2+8*x+1)))/(4*x**2+x),x)
Output:
(3 - x)*log(4*x**2*exp(x)/(16*x**2 + 8*x + 1))**2
Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (19) = 38\).
Time = 0.18 (sec) , antiderivative size = 162, normalized size of antiderivative = 5.79 \[ \int \frac {\left (12+2 x+22 x^2-8 x^3\right ) \log \left (\frac {4 e^x x^2}{1+8 x+16 x^2}\right )+\left (-x-4 x^2\right ) \log ^2\left (\frac {4 e^x x^2}{1+8 x+16 x^2}\right )}{x+4 x^2} \, dx=-x^{3} - x^{2} {\left (4 \, \log \left (2\right ) - 3\right )} - 4 \, x \log \left (4 \, x + 1\right )^{2} - 4 \, x \log \left (x\right )^{2} - 4 \, {\left (\log \left (2\right )^{2} - 3 \, \log \left (2\right )\right )} x - 12 \, {\left (\log \left (4 \, x + 1\right ) - \log \left (x\right )\right )} \log \left (\frac {4 \, x^{2} e^{x}}{16 \, x^{2} + 8 \, x + 1}\right ) + {\left (4 \, x^{2} + 4 \, x {\left (2 \, \log \left (2\right ) - 3\right )} + 8 \, x \log \left (x\right ) - 3\right )} \log \left (4 \, x + 1\right ) + 3 \, {\left (4 \, x + 8 \, \log \left (x\right ) + 1\right )} \log \left (4 \, x + 1\right ) - 12 \, \log \left (4 \, x + 1\right )^{2} - 4 \, {\left (x^{2} + x {\left (2 \, \log \left (2\right ) - 3\right )}\right )} \log \left (x\right ) - 12 \, x \log \left (x\right ) - 12 \, \log \left (x\right )^{2} \] Input:
integrate(((-4*x^2-x)*log(4*x^2*exp(x)/(16*x^2+8*x+1))^2+(-8*x^3+22*x^2+2* x+12)*log(4*x^2*exp(x)/(16*x^2+8*x+1)))/(4*x^2+x),x, algorithm="maxima")
Output:
-x^3 - x^2*(4*log(2) - 3) - 4*x*log(4*x + 1)^2 - 4*x*log(x)^2 - 4*(log(2)^ 2 - 3*log(2))*x - 12*(log(4*x + 1) - log(x))*log(4*x^2*e^x/(16*x^2 + 8*x + 1)) + (4*x^2 + 4*x*(2*log(2) - 3) + 8*x*log(x) - 3)*log(4*x + 1) + 3*(4*x + 8*log(x) + 1)*log(4*x + 1) - 12*log(4*x + 1)^2 - 4*(x^2 + x*(2*log(2) - 3))*log(x) - 12*x*log(x) - 12*log(x)^2
\[ \int \frac {\left (12+2 x+22 x^2-8 x^3\right ) \log \left (\frac {4 e^x x^2}{1+8 x+16 x^2}\right )+\left (-x-4 x^2\right ) \log ^2\left (\frac {4 e^x x^2}{1+8 x+16 x^2}\right )}{x+4 x^2} \, dx=\int { -\frac {{\left (4 \, x^{2} + x\right )} \log \left (\frac {4 \, x^{2} e^{x}}{16 \, x^{2} + 8 \, x + 1}\right )^{2} + 2 \, {\left (4 \, x^{3} - 11 \, x^{2} - x - 6\right )} \log \left (\frac {4 \, x^{2} e^{x}}{16 \, x^{2} + 8 \, x + 1}\right )}{4 \, x^{2} + x} \,d x } \] Input:
integrate(((-4*x^2-x)*log(4*x^2*exp(x)/(16*x^2+8*x+1))^2+(-8*x^3+22*x^2+2* x+12)*log(4*x^2*exp(x)/(16*x^2+8*x+1)))/(4*x^2+x),x, algorithm="giac")
Output:
integrate(-((4*x^2 + x)*log(4*x^2*e^x/(16*x^2 + 8*x + 1))^2 + 2*(4*x^3 - 1 1*x^2 - x - 6)*log(4*x^2*e^x/(16*x^2 + 8*x + 1)))/(4*x^2 + x), x)
Time = 4.74 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {\left (12+2 x+22 x^2-8 x^3\right ) \log \left (\frac {4 e^x x^2}{1+8 x+16 x^2}\right )+\left (-x-4 x^2\right ) \log ^2\left (\frac {4 e^x x^2}{1+8 x+16 x^2}\right )}{x+4 x^2} \, dx=-{\ln \left (\frac {4\,x^2\,{\mathrm {e}}^x}{16\,x^2+8\,x+1}\right )}^2\,\left (x-3\right ) \] Input:
int(-(log((4*x^2*exp(x))/(8*x + 16*x^2 + 1))^2*(x + 4*x^2) - log((4*x^2*ex p(x))/(8*x + 16*x^2 + 1))*(2*x + 22*x^2 - 8*x^3 + 12))/(x + 4*x^2),x)
Output:
-log((4*x^2*exp(x))/(8*x + 16*x^2 + 1))^2*(x - 3)
Time = 0.17 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {\left (12+2 x+22 x^2-8 x^3\right ) \log \left (\frac {4 e^x x^2}{1+8 x+16 x^2}\right )+\left (-x-4 x^2\right ) \log ^2\left (\frac {4 e^x x^2}{1+8 x+16 x^2}\right )}{x+4 x^2} \, dx=\mathrm {log}\left (\frac {4 e^{x} x^{2}}{16 x^{2}+8 x +1}\right )^{2} \left (-x +3\right ) \] Input:
int(((-4*x^2-x)*log(4*x^2*exp(x)/(16*x^2+8*x+1))^2+(-8*x^3+22*x^2+2*x+12)* log(4*x^2*exp(x)/(16*x^2+8*x+1)))/(4*x^2+x),x)
Output:
log((4*e**x*x**2)/(16*x**2 + 8*x + 1))**2*( - x + 3)