Integrand size = 90, antiderivative size = 21 \[ \int \frac {\left (-8 x+12 x^2+72 x^3+32 x^4\right ) \log (x)+\left (64-216 x-148 x^2-44 x^3-16 x^4+\left (224 x+284 x^2+60 x^3+32 x^4\right ) \log (x)\right ) \log \left (8+x+x^2\right )}{\left (8 x+x^2+x^3\right ) \log ^2(x)} \, dx=\frac {4 (2+x) (-1+4 x) \log \left (8+x+x^2\right )}{\log (x)} \] Output:
4*(2+x)/ln(x)*ln(x^2+x+8)*(-1+4*x)
Time = 0.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {\left (-8 x+12 x^2+72 x^3+32 x^4\right ) \log (x)+\left (64-216 x-148 x^2-44 x^3-16 x^4+\left (224 x+284 x^2+60 x^3+32 x^4\right ) \log (x)\right ) \log \left (8+x+x^2\right )}{\left (8 x+x^2+x^3\right ) \log ^2(x)} \, dx=\frac {4 \left (-2+7 x+4 x^2\right ) \log \left (8+x+x^2\right )}{\log (x)} \] Input:
Integrate[((-8*x + 12*x^2 + 72*x^3 + 32*x^4)*Log[x] + (64 - 216*x - 148*x^ 2 - 44*x^3 - 16*x^4 + (224*x + 284*x^2 + 60*x^3 + 32*x^4)*Log[x])*Log[8 + x + x^2])/((8*x + x^2 + x^3)*Log[x]^2),x]
Output:
(4*(-2 + 7*x + 4*x^2)*Log[8 + x + x^2])/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.
\(\displaystyle \int \frac {\left (32 x^4+72 x^3+12 x^2-8 x\right ) \log (x)+\left (-16 x^4-44 x^3-148 x^2+\left (32 x^4+60 x^3+284 x^2+224 x\right ) \log (x)-216 x+64\right ) \log \left (x^2+x+8\right )}{\left (x^3+x^2+8 x\right ) \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {\left (32 x^4+72 x^3+12 x^2-8 x\right ) \log (x)+\left (-16 x^4-44 x^3-148 x^2+\left (32 x^4+60 x^3+284 x^2+224 x\right ) \log (x)-216 x+64\right ) \log \left (x^2+x+8\right )}{x \left (x^2+x+8\right ) \log ^2(x)}dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {4 \left (-4 x^2+8 x^2 \log (x)-7 x+7 x \log (x)+2\right ) \log \left (x^2+x+8\right )}{x \log ^2(x)}+\frac {4 (x+2) (2 x+1) (4 x-1)}{\left (x^2+x+8\right ) \log (x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {56 i \int \frac {\operatorname {LogIntegral}(x)}{-2 x+i \sqrt {31}-1}dx}{\sqrt {31}}-\frac {56}{31} \left (31+i \sqrt {31}\right ) \int \frac {\operatorname {LogIntegral}(x)}{2 x-i \sqrt {31}+1}dx-\frac {56}{31} \left (31-i \sqrt {31}\right ) \int \frac {\operatorname {LogIntegral}(x)}{2 x+i \sqrt {31}+1}dx-\frac {56 i \int \frac {\operatorname {LogIntegral}(x)}{2 x+i \sqrt {31}+1}dx}{\sqrt {31}}-28 \int \frac {\log \left (x^2+x+8\right )}{\log ^2(x)}dx+8 \int \frac {\log \left (x^2+x+8\right )}{x \log ^2(x)}dx-16 \int \frac {x \log \left (x^2+x+8\right )}{\log ^2(x)}dx+4 \int \frac {(x+2) (2 x+1) (4 x-1)}{\left (x^2+x+8\right ) \log (x)}dx+32 \int \frac {x \log \left (x^2+x+8\right )}{\log (x)}dx+28 \operatorname {LogIntegral}(x) \log \left (x^2+x+8\right )\) |
Input:
Int[((-8*x + 12*x^2 + 72*x^3 + 32*x^4)*Log[x] + (64 - 216*x - 148*x^2 - 44 *x^3 - 16*x^4 + (224*x + 284*x^2 + 60*x^3 + 32*x^4)*Log[x])*Log[8 + x + x^ 2])/((8*x + x^2 + x^3)*Log[x]^2),x]
Output:
$Aborted
Time = 0.55 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14
\[\frac {4 \left (4 x^{2}+7 x -2\right ) \ln \left (x^{2}+x +8\right )}{\ln \left (x \right )}\]
Input:
int((((32*x^4+60*x^3+284*x^2+224*x)*ln(x)-16*x^4-44*x^3-148*x^2-216*x+64)* ln(x^2+x+8)+(32*x^4+72*x^3+12*x^2-8*x)*ln(x))/(x^3+x^2+8*x)/ln(x)^2,x)
Output:
4*(4*x^2+7*x-2)/ln(x)*ln(x^2+x+8)
Time = 0.09 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {\left (-8 x+12 x^2+72 x^3+32 x^4\right ) \log (x)+\left (64-216 x-148 x^2-44 x^3-16 x^4+\left (224 x+284 x^2+60 x^3+32 x^4\right ) \log (x)\right ) \log \left (8+x+x^2\right )}{\left (8 x+x^2+x^3\right ) \log ^2(x)} \, dx=\frac {4 \, {\left (4 \, x^{2} + 7 \, x - 2\right )} \log \left (x^{2} + x + 8\right )}{\log \left (x\right )} \] Input:
integrate((((32*x^4+60*x^3+284*x^2+224*x)*log(x)-16*x^4-44*x^3-148*x^2-216 *x+64)*log(x^2+x+8)+(32*x^4+72*x^3+12*x^2-8*x)*log(x))/(x^3+x^2+8*x)/log(x )^2,x, algorithm="fricas")
Output:
4*(4*x^2 + 7*x - 2)*log(x^2 + x + 8)/log(x)
Exception generated. \[ \int \frac {\left (-8 x+12 x^2+72 x^3+32 x^4\right ) \log (x)+\left (64-216 x-148 x^2-44 x^3-16 x^4+\left (224 x+284 x^2+60 x^3+32 x^4\right ) \log (x)\right ) \log \left (8+x+x^2\right )}{\left (8 x+x^2+x^3\right ) \log ^2(x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((((32*x**4+60*x**3+284*x**2+224*x)*ln(x)-16*x**4-44*x**3-148*x** 2-216*x+64)*ln(x**2+x+8)+(32*x**4+72*x**3+12*x**2-8*x)*ln(x))/(x**3+x**2+8 *x)/ln(x)**2,x)
Output:
Exception raised: TypeError >> '>' not supported between instances of 'Pol y' and 'int'
Time = 0.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {\left (-8 x+12 x^2+72 x^3+32 x^4\right ) \log (x)+\left (64-216 x-148 x^2-44 x^3-16 x^4+\left (224 x+284 x^2+60 x^3+32 x^4\right ) \log (x)\right ) \log \left (8+x+x^2\right )}{\left (8 x+x^2+x^3\right ) \log ^2(x)} \, dx=\frac {4 \, {\left (4 \, x^{2} + 7 \, x - 2\right )} \log \left (x^{2} + x + 8\right )}{\log \left (x\right )} \] Input:
integrate((((32*x^4+60*x^3+284*x^2+224*x)*log(x)-16*x^4-44*x^3-148*x^2-216 *x+64)*log(x^2+x+8)+(32*x^4+72*x^3+12*x^2-8*x)*log(x))/(x^3+x^2+8*x)/log(x )^2,x, algorithm="maxima")
Output:
4*(4*x^2 + 7*x - 2)*log(x^2 + x + 8)/log(x)
Time = 0.13 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {\left (-8 x+12 x^2+72 x^3+32 x^4\right ) \log (x)+\left (64-216 x-148 x^2-44 x^3-16 x^4+\left (224 x+284 x^2+60 x^3+32 x^4\right ) \log (x)\right ) \log \left (8+x+x^2\right )}{\left (8 x+x^2+x^3\right ) \log ^2(x)} \, dx=\frac {4 \, {\left (4 \, x^{2} + 7 \, x - 2\right )} \log \left (x^{2} + x + 8\right )}{\log \left (x\right )} \] Input:
integrate((((32*x^4+60*x^3+284*x^2+224*x)*log(x)-16*x^4-44*x^3-148*x^2-216 *x+64)*log(x^2+x+8)+(32*x^4+72*x^3+12*x^2-8*x)*log(x))/(x^3+x^2+8*x)/log(x )^2,x, algorithm="giac")
Output:
4*(4*x^2 + 7*x - 2)*log(x^2 + x + 8)/log(x)
Time = 3.18 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {\left (-8 x+12 x^2+72 x^3+32 x^4\right ) \log (x)+\left (64-216 x-148 x^2-44 x^3-16 x^4+\left (224 x+284 x^2+60 x^3+32 x^4\right ) \log (x)\right ) \log \left (8+x+x^2\right )}{\left (8 x+x^2+x^3\right ) \log ^2(x)} \, dx=\frac {4\,\ln \left (x^2+x+8\right )\,\left (4\,x^2+7\,x-2\right )}{\ln \left (x\right )} \] Input:
int((log(x)*(12*x^2 - 8*x + 72*x^3 + 32*x^4) - log(x + x^2 + 8)*(216*x - l og(x)*(224*x + 284*x^2 + 60*x^3 + 32*x^4) + 148*x^2 + 44*x^3 + 16*x^4 - 64 ))/(log(x)^2*(8*x + x^2 + x^3)),x)
Output:
(4*log(x + x^2 + 8)*(7*x + 4*x^2 - 2))/log(x)
Time = 0.15 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {\left (-8 x+12 x^2+72 x^3+32 x^4\right ) \log (x)+\left (64-216 x-148 x^2-44 x^3-16 x^4+\left (224 x+284 x^2+60 x^3+32 x^4\right ) \log (x)\right ) \log \left (8+x+x^2\right )}{\left (8 x+x^2+x^3\right ) \log ^2(x)} \, dx=\frac {4 \,\mathrm {log}\left (x^{2}+x +8\right ) \left (4 x^{2}+7 x -2\right )}{\mathrm {log}\left (x \right )} \] Input:
int((((32*x^4+60*x^3+284*x^2+224*x)*log(x)-16*x^4-44*x^3-148*x^2-216*x+64) *log(x^2+x+8)+(32*x^4+72*x^3+12*x^2-8*x)*log(x))/(x^3+x^2+8*x)/log(x)^2,x)
Output:
(4*log(x**2 + x + 8)*(4*x**2 + 7*x - 2))/log(x)