Integrand size = 108, antiderivative size = 28 \[ \int \frac {36 x^3+12 x^4+\left (-36 x^3-9 x^4\right ) \log (x)+\left (-18-9 x-x^2\right ) \log ^2(5) \log ^3(x)+\left (-18 x^3-6 x^4+\left (18 x^3+6 x^4\right ) \log (x)+(9+3 x) \log ^2(5) \log ^3(x)\right ) \log (3+x)}{\left (3 x^2+x^3\right ) \log ^2(5) \log ^3(x)} \, dx=\frac {\left (-3-x+\frac {3 x^3}{\log ^2(5) \log ^2(x)}\right ) (-2+\log (3+x))}{x} \] Output:
(3*x^3/ln(5)^2/ln(x)^2-3-x)*(ln(3+x)-2)/x
Time = 0.17 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {36 x^3+12 x^4+\left (-36 x^3-9 x^4\right ) \log (x)+\left (-18-9 x-x^2\right ) \log ^2(5) \log ^3(x)+\left (-18 x^3-6 x^4+\left (18 x^3+6 x^4\right ) \log (x)+(9+3 x) \log ^2(5) \log ^3(x)\right ) \log (3+x)}{\left (3 x^2+x^3\right ) \log ^2(5) \log ^3(x)} \, dx=\frac {6+\frac {3 x^3 (-2+\log (3+x))}{\log ^2(5) \log ^2(x)}-(3+x) \log (3+x)}{x} \] Input:
Integrate[(36*x^3 + 12*x^4 + (-36*x^3 - 9*x^4)*Log[x] + (-18 - 9*x - x^2)* Log[5]^2*Log[x]^3 + (-18*x^3 - 6*x^4 + (18*x^3 + 6*x^4)*Log[x] + (9 + 3*x) *Log[5]^2*Log[x]^3)*Log[3 + x])/((3*x^2 + x^3)*Log[5]^2*Log[x]^3),x]
Output:
(6 + (3*x^3*(-2 + Log[3 + x]))/(Log[5]^2*Log[x]^2) - (3 + x)*Log[3 + x])/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.
\(\displaystyle \int \frac {12 x^4+36 x^3+\left (-x^2-9 x-18\right ) \log ^2(5) \log ^3(x)+\left (-6 x^4-18 x^3+\left (6 x^4+18 x^3\right ) \log (x)+(3 x+9) \log ^2(5) \log ^3(x)\right ) \log (x+3)+\left (-9 x^4-36 x^3\right ) \log (x)}{\left (x^3+3 x^2\right ) \log ^2(5) \log ^3(x)} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {12 x^4+36 x^3-\left (x^2+9 x+18\right ) \log ^2(5) \log ^3(x)-9 \left (x^4+4 x^3\right ) \log (x)-3 \left (2 x^4+6 x^3-(x+3) \log ^2(5) \log ^3(x)-2 \left (x^4+3 x^3\right ) \log (x)\right ) \log (x+3)}{\left (x^3+3 x^2\right ) \log ^3(x)}dx}{\log ^2(5)}\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \frac {\int \frac {12 x^4+36 x^3-\left (x^2+9 x+18\right ) \log ^2(5) \log ^3(x)-9 \left (x^4+4 x^3\right ) \log (x)-3 \left (2 x^4+6 x^3-(x+3) \log ^2(5) \log ^3(x)-2 \left (x^4+3 x^3\right ) \log (x)\right ) \log (x+3)}{x^2 (x+3) \log ^3(x)}dx}{\log ^2(5)}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {\int \left (\frac {-9 \log (x) x^4+12 x^4-36 \log (x) x^3+36 x^3-\log ^2(5) \log ^3(x) x^2-9 \log ^2(5) \log ^3(x) x-18 \log ^2(5) \log ^3(x)}{x^2 (x+3) \log ^3(x)}+\frac {3 \left (2 \log (x) x^3-2 x^3+\log ^2(5) \log ^3(x)\right ) \log (x+3)}{x^2 \log ^3(x)}\right )dx}{\log ^2(5)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-6 \int \frac {x \log (x+3)}{\log ^3(x)}dx-9 \int \frac {x (x+4)}{(x+3) \log ^2(x)}dx+6 \int \frac {x \log (x+3)}{\log ^2(x)}dx+24 \operatorname {ExpIntegralEi}(2 \log (x))-\frac {6 x^2}{\log ^2(x)}-\frac {12 x^2}{\log (x)}-\log ^2(5) \log (x+3)-\frac {3 \log ^2(5) \log (x+3)}{x}+\frac {6 \log ^2(5)}{x}}{\log ^2(5)}\) |
Input:
Int[(36*x^3 + 12*x^4 + (-36*x^3 - 9*x^4)*Log[x] + (-18 - 9*x - x^2)*Log[5] ^2*Log[x]^3 + (-18*x^3 - 6*x^4 + (18*x^3 + 6*x^4)*Log[x] + (9 + 3*x)*Log[5 ]^2*Log[x]^3)*Log[3 + x])/((3*x^2 + x^3)*Log[5]^2*Log[x]^3),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(76\) vs. \(2(28)=56\).
Time = 0.13 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.75
\[-\frac {3 \left (\ln \left (5\right )^{2} \ln \left (x \right )^{2}-x^{3}\right ) \ln \left (3+x \right )}{\ln \left (5\right )^{2} x \ln \left (x \right )^{2}}-\frac {\ln \left (5\right )^{2} \ln \left (3+x \right ) x \ln \left (x \right )^{2}-6 \ln \left (5\right )^{2} \ln \left (x \right )^{2}+6 x^{3}}{\ln \left (5\right )^{2} x \ln \left (x \right )^{2}}\]
Input:
int((((3*x+9)*ln(5)^2*ln(x)^3+(6*x^4+18*x^3)*ln(x)-6*x^4-18*x^3)*ln(3+x)+( -x^2-9*x-18)*ln(5)^2*ln(x)^3+(-9*x^4-36*x^3)*ln(x)+12*x^4+36*x^3)/(x^3+3*x ^2)/ln(5)^2/ln(x)^3,x)
Output:
-3/ln(5)^2*(ln(5)^2*ln(x)^2-x^3)/x/ln(x)^2*ln(3+x)-1/ln(5)^2*(ln(5)^2*ln(3 +x)*x*ln(x)^2-6*ln(5)^2*ln(x)^2+6*x^3)/x/ln(x)^2
Time = 0.10 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.86 \[ \int \frac {36 x^3+12 x^4+\left (-36 x^3-9 x^4\right ) \log (x)+\left (-18-9 x-x^2\right ) \log ^2(5) \log ^3(x)+\left (-18 x^3-6 x^4+\left (18 x^3+6 x^4\right ) \log (x)+(9+3 x) \log ^2(5) \log ^3(x)\right ) \log (3+x)}{\left (3 x^2+x^3\right ) \log ^2(5) \log ^3(x)} \, dx=\frac {6 \, \log \left (5\right )^{2} \log \left (x\right )^{2} - 6 \, x^{3} - {\left ({\left (x + 3\right )} \log \left (5\right )^{2} \log \left (x\right )^{2} - 3 \, x^{3}\right )} \log \left (x + 3\right )}{x \log \left (5\right )^{2} \log \left (x\right )^{2}} \] Input:
integrate((((3*x+9)*log(5)^2*log(x)^3+(6*x^4+18*x^3)*log(x)-6*x^4-18*x^3)* log(3+x)+(-x^2-9*x-18)*log(5)^2*log(x)^3+(-9*x^4-36*x^3)*log(x)+12*x^4+36* x^3)/(x^3+3*x^2)/log(5)^2/log(x)^3,x, algorithm="fricas")
Output:
(6*log(5)^2*log(x)^2 - 6*x^3 - ((x + 3)*log(5)^2*log(x)^2 - 3*x^3)*log(x + 3))/(x*log(5)^2*log(x)^2)
Exception generated. \[ \int \frac {36 x^3+12 x^4+\left (-36 x^3-9 x^4\right ) \log (x)+\left (-18-9 x-x^2\right ) \log ^2(5) \log ^3(x)+\left (-18 x^3-6 x^4+\left (18 x^3+6 x^4\right ) \log (x)+(9+3 x) \log ^2(5) \log ^3(x)\right ) \log (3+x)}{\left (3 x^2+x^3\right ) \log ^2(5) \log ^3(x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((((3*x+9)*ln(5)**2*ln(x)**3+(6*x**4+18*x**3)*ln(x)-6*x**4-18*x** 3)*ln(3+x)+(-x**2-9*x-18)*ln(5)**2*ln(x)**3+(-9*x**4-36*x**3)*ln(x)+12*x** 4+36*x**3)/(x**3+3*x**2)/ln(5)**2/ln(x)**3,x)
Output:
Exception raised: TypeError >> '>' not supported between instances of 'Pol y' and 'int'
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (27) = 54\).
Time = 0.15 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.07 \[ \int \frac {36 x^3+12 x^4+\left (-36 x^3-9 x^4\right ) \log (x)+\left (-18-9 x-x^2\right ) \log ^2(5) \log ^3(x)+\left (-18 x^3-6 x^4+\left (18 x^3+6 x^4\right ) \log (x)+(9+3 x) \log ^2(5) \log ^3(x)\right ) \log (3+x)}{\left (3 x^2+x^3\right ) \log ^2(5) \log ^3(x)} \, dx=\frac {6 \, \log \left (5\right )^{2} \log \left (x\right )^{2} - 6 \, x^{3} + {\left (3 \, x^{3} - {\left (x \log \left (5\right )^{2} + 3 \, \log \left (5\right )^{2}\right )} \log \left (x\right )^{2}\right )} \log \left (x + 3\right )}{x \log \left (5\right )^{2} \log \left (x\right )^{2}} \] Input:
integrate((((3*x+9)*log(5)^2*log(x)^3+(6*x^4+18*x^3)*log(x)-6*x^4-18*x^3)* log(3+x)+(-x^2-9*x-18)*log(5)^2*log(x)^3+(-9*x^4-36*x^3)*log(x)+12*x^4+36* x^3)/(x^3+3*x^2)/log(5)^2/log(x)^3,x, algorithm="maxima")
Output:
(6*log(5)^2*log(x)^2 - 6*x^3 + (3*x^3 - (x*log(5)^2 + 3*log(5)^2)*log(x)^2 )*log(x + 3))/(x*log(5)^2*log(x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (27) = 54\).
Time = 0.13 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.07 \[ \int \frac {36 x^3+12 x^4+\left (-36 x^3-9 x^4\right ) \log (x)+\left (-18-9 x-x^2\right ) \log ^2(5) \log ^3(x)+\left (-18 x^3-6 x^4+\left (18 x^3+6 x^4\right ) \log (x)+(9+3 x) \log ^2(5) \log ^3(x)\right ) \log (3+x)}{\left (3 x^2+x^3\right ) \log ^2(5) \log ^3(x)} \, dx=-\frac {\log \left (5\right )^{2} \log \left (x + 3\right ) + 3 \, {\left (\frac {\log \left (5\right )^{2}}{x} - \frac {x^{2}}{\log \left (x\right )^{2}}\right )} \log \left (x + 3\right ) - \frac {6 \, \log \left (5\right )^{2}}{x} + \frac {6 \, x^{2}}{\log \left (x\right )^{2}}}{\log \left (5\right )^{2}} \] Input:
integrate((((3*x+9)*log(5)^2*log(x)^3+(6*x^4+18*x^3)*log(x)-6*x^4-18*x^3)* log(3+x)+(-x^2-9*x-18)*log(5)^2*log(x)^3+(-9*x^4-36*x^3)*log(x)+12*x^4+36* x^3)/(x^3+3*x^2)/log(5)^2/log(x)^3,x, algorithm="giac")
Output:
-(log(5)^2*log(x + 3) + 3*(log(5)^2/x - x^2/log(x)^2)*log(x + 3) - 6*log(5 )^2/x + 6*x^2/log(x)^2)/log(5)^2
Time = 3.79 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.82 \[ \int \frac {36 x^3+12 x^4+\left (-36 x^3-9 x^4\right ) \log (x)+\left (-18-9 x-x^2\right ) \log ^2(5) \log ^3(x)+\left (-18 x^3-6 x^4+\left (18 x^3+6 x^4\right ) \log (x)+(9+3 x) \log ^2(5) \log ^3(x)\right ) \log (3+x)}{\left (3 x^2+x^3\right ) \log ^2(5) \log ^3(x)} \, dx=\frac {6}{x}-\frac {3\,\ln \left (x+3\right )}{x}-\ln \left (x+3\right )-\frac {6\,x^2}{{\ln \left (5\right )}^2\,{\ln \left (x\right )}^2}+\frac {3\,x^2\,\ln \left (x+3\right )}{{\ln \left (5\right )}^2\,{\ln \left (x\right )}^2} \] Input:
int((log(x + 3)*(log(x)*(18*x^3 + 6*x^4) - 18*x^3 - 6*x^4 + log(5)^2*log(x )^3*(3*x + 9)) - log(x)*(36*x^3 + 9*x^4) + 36*x^3 + 12*x^4 - log(5)^2*log( x)^3*(9*x + x^2 + 18))/(log(5)^2*log(x)^3*(3*x^2 + x^3)),x)
Output:
6/x - (3*log(x + 3))/x - log(x + 3) - (6*x^2)/(log(5)^2*log(x)^2) + (3*x^2 *log(x + 3))/(log(5)^2*log(x)^2)
Time = 0.17 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.36 \[ \int \frac {36 x^3+12 x^4+\left (-36 x^3-9 x^4\right ) \log (x)+\left (-18-9 x-x^2\right ) \log ^2(5) \log ^3(x)+\left (-18 x^3-6 x^4+\left (18 x^3+6 x^4\right ) \log (x)+(9+3 x) \log ^2(5) \log ^3(x)\right ) \log (3+x)}{\left (3 x^2+x^3\right ) \log ^2(5) \log ^3(x)} \, dx=\frac {-\mathrm {log}\left (x +3\right ) \mathrm {log}\left (x \right )^{2} \mathrm {log}\left (5\right )^{2} x -3 \,\mathrm {log}\left (x +3\right ) \mathrm {log}\left (x \right )^{2} \mathrm {log}\left (5\right )^{2}+3 \,\mathrm {log}\left (x +3\right ) x^{3}+6 \mathrm {log}\left (x \right )^{2} \mathrm {log}\left (5\right )^{2}-6 x^{3}}{\mathrm {log}\left (x \right )^{2} \mathrm {log}\left (5\right )^{2} x} \] Input:
int((((3*x+9)*log(5)^2*log(x)^3+(6*x^4+18*x^3)*log(x)-6*x^4-18*x^3)*log(3+ x)+(-x^2-9*x-18)*log(5)^2*log(x)^3+(-9*x^4-36*x^3)*log(x)+12*x^4+36*x^3)/( x^3+3*x^2)/log(5)^2/log(x)^3,x)
Output:
( - log(x + 3)*log(x)**2*log(5)**2*x - 3*log(x + 3)*log(x)**2*log(5)**2 + 3*log(x + 3)*x**3 + 6*log(x)**2*log(5)**2 - 6*x**3)/(log(x)**2*log(5)**2*x )