Integrand size = 108, antiderivative size = 28 \[ \int \frac {36 x^3+12 x^4+\left (-36 x^3-9 x^4\right ) \log (x)+\left (-18-9 x-x^2\right ) \log ^2(5) \log ^3(x)+\left (-18 x^3-6 x^4+\left (18 x^3+6 x^4\right ) \log (x)+(9+3 x) \log ^2(5) \log ^3(x)\right ) \log (3+x)}{\left (3 x^2+x^3\right ) \log ^2(5) \log ^3(x)} \, dx=\frac {\left (-3-x+\frac {3 x^3}{\log ^2(5) \log ^2(x)}\right ) (-2+\log (3+x))}{x} \]
Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {36 x^3+12 x^4+\left (-36 x^3-9 x^4\right ) \log (x)+\left (-18-9 x-x^2\right ) \log ^2(5) \log ^3(x)+\left (-18 x^3-6 x^4+\left (18 x^3+6 x^4\right ) \log (x)+(9+3 x) \log ^2(5) \log ^3(x)\right ) \log (3+x)}{\left (3 x^2+x^3\right ) \log ^2(5) \log ^3(x)} \, dx=\frac {6+\frac {3 x^3 (-2+\log (3+x))}{\log ^2(5) \log ^2(x)}-(3+x) \log (3+x)}{x} \]
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]
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 |
\(\displaystyle \frac {-6 \int \frac {x \log (x+3)}{\log ^3(x)}dx-9 \int \frac {x (x+4)}{(x+3) \log ^2(x)}dx+6 \int \frac {x \log (x+3)}{\log ^2(x)}dx+24 \operatorname {ExpIntegralEi}(2 \log (x))-\frac {6 x^2}{\log ^2(x)}-\frac {12 x^2}{\log (x)}-\log ^2(5) \log (x+3)-\frac {3 \log ^2(5) \log (x+3)}{x}+\frac {6 \log ^2(5)}{x}}{\log ^2(5)}\) |
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]
3.27.10.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Leaf count of result is larger than twice the leaf count of optimal. \(66\) vs. \(2(28)=56\).
Time = 1.26 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.39
method | result | size |
parallelrisch | \(-\frac {\ln \left (5\right )^{2} \ln \left (3+x \right ) x \ln \left (x \right )^{2}+3 \ln \left (5\right )^{2} \ln \left (3+x \right ) \ln \left (x \right )^{2}-6 \ln \left (5\right )^{2} \ln \left (x \right )^{2}-3 \ln \left (3+x \right ) x^{3}+6 x^{3}}{\ln \left (5\right )^{2} x \ln \left (x \right )^{2}}\) | \(67\) |
risch | \(-\frac {3 \left (\ln \left (5\right )^{2} \ln \left (x \right )^{2}-x^{3}\right ) \ln \left (3+x \right )}{\ln \left (5\right )^{2} x \ln \left (x \right )^{2}}-\frac {\ln \left (5\right )^{2} \ln \left (3+x \right ) x \ln \left (x \right )^{2}-6 \ln \left (5\right )^{2} \ln \left (x \right )^{2}+6 x^{3}}{\ln \left (5\right )^{2} x \ln \left (x \right )^{2}}\) | \(77\) |
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,method=_RETURNVERBOSE)
-1/ln(5)^2*(ln(5)^2*ln(3+x)*x*ln(x)^2+3*ln(5)^2*ln(3+x)*ln(x)^2-6*ln(5)^2* ln(x)^2-3*ln(3+x)*x^3+6*x^3)/x/ln(x)^2
Time = 0.24 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.86 \[ \int \frac {36 x^3+12 x^4+\left (-36 x^3-9 x^4\right ) \log (x)+\left (-18-9 x-x^2\right ) \log ^2(5) \log ^3(x)+\left (-18 x^3-6 x^4+\left (18 x^3+6 x^4\right ) \log (x)+(9+3 x) \log ^2(5) \log ^3(x)\right ) \log (3+x)}{\left (3 x^2+x^3\right ) \log ^2(5) \log ^3(x)} \, dx=\frac {6 \, \log \left (5\right )^{2} \log \left (x\right )^{2} - 6 \, x^{3} - {\left ({\left (x + 3\right )} \log \left (5\right )^{2} \log \left (x\right )^{2} - 3 \, x^{3}\right )} \log \left (x + 3\right )}{x \log \left (5\right )^{2} \log \left (x\right )^{2}} \]
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=\
(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. \[ \int \frac {36 x^3+12 x^4+\left (-36 x^3-9 x^4\right ) \log (x)+\left (-18-9 x-x^2\right ) \log ^2(5) \log ^3(x)+\left (-18 x^3-6 x^4+\left (18 x^3+6 x^4\right ) \log (x)+(9+3 x) \log ^2(5) \log ^3(x)\right ) \log (3+x)}{\left (3 x^2+x^3\right ) \log ^2(5) \log ^3(x)} \, dx=\text {Exception raised: TypeError} \]
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)
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (27) = 54\).
Time = 0.30 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.07 \[ \int \frac {36 x^3+12 x^4+\left (-36 x^3-9 x^4\right ) \log (x)+\left (-18-9 x-x^2\right ) \log ^2(5) \log ^3(x)+\left (-18 x^3-6 x^4+\left (18 x^3+6 x^4\right ) \log (x)+(9+3 x) \log ^2(5) \log ^3(x)\right ) \log (3+x)}{\left (3 x^2+x^3\right ) \log ^2(5) \log ^3(x)} \, dx=\frac {6 \, \log \left (5\right )^{2} \log \left (x\right )^{2} - 6 \, x^{3} + {\left (3 \, x^{3} - {\left (x \log \left (5\right )^{2} + 3 \, \log \left (5\right )^{2}\right )} \log \left (x\right )^{2}\right )} \log \left (x + 3\right )}{x \log \left (5\right )^{2} \log \left (x\right )^{2}} \]
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=\
(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 = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.07 \[ \int \frac {36 x^3+12 x^4+\left (-36 x^3-9 x^4\right ) \log (x)+\left (-18-9 x-x^2\right ) \log ^2(5) \log ^3(x)+\left (-18 x^3-6 x^4+\left (18 x^3+6 x^4\right ) \log (x)+(9+3 x) \log ^2(5) \log ^3(x)\right ) \log (3+x)}{\left (3 x^2+x^3\right ) \log ^2(5) \log ^3(x)} \, dx=-\frac {\log \left (5\right )^{2} \log \left (x + 3\right ) + 3 \, {\left (\frac {\log \left (5\right )^{2}}{x} - \frac {x^{2}}{\log \left (x\right )^{2}}\right )} \log \left (x + 3\right ) - \frac {6 \, \log \left (5\right )^{2}}{x} + \frac {6 \, x^{2}}{\log \left (x\right )^{2}}}{\log \left (5\right )^{2}} \]
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=\
-(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 = 13.92 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.82 \[ \int \frac {36 x^3+12 x^4+\left (-36 x^3-9 x^4\right ) \log (x)+\left (-18-9 x-x^2\right ) \log ^2(5) \log ^3(x)+\left (-18 x^3-6 x^4+\left (18 x^3+6 x^4\right ) \log (x)+(9+3 x) \log ^2(5) \log ^3(x)\right ) \log (3+x)}{\left (3 x^2+x^3\right ) \log ^2(5) \log ^3(x)} \, dx=\frac {6}{x}-\frac {3\,\ln \left (x+3\right )}{x}-\ln \left (x+3\right )-\frac {6\,x^2}{{\ln \left (5\right )}^2\,{\ln \left (x\right )}^2}+\frac {3\,x^2\,\ln \left (x+3\right )}{{\ln \left (5\right )}^2\,{\ln \left (x\right )}^2} \]
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)