Integrand size = 99, antiderivative size = 26 \[ \int \frac {-x+x^3+e^{12} \left (-x+x^3\right )+e^6 \left (-2 x+2 x^3\right )+\left (-1+2 x^2+e^{12} \left (-1+2 x^2\right )+e^6 \left (-2+4 x^2\right )\right ) \log (3)+\left (x+2 e^6 x+e^{12} x\right ) \log (x)}{x^3+2 x^2 \log (3)+x \log ^2(3)} \, dx=\left (1+e^6\right )^2 \left (1-\frac {-x^2+\log (x)}{x+\log (3)}\right ) \]
Time = 0.07 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {-x+x^3+e^{12} \left (-x+x^3\right )+e^6 \left (-2 x+2 x^3\right )+\left (-1+2 x^2+e^{12} \left (-1+2 x^2\right )+e^6 \left (-2+4 x^2\right )\right ) \log (3)+\left (x+2 e^6 x+e^{12} x\right ) \log (x)}{x^3+2 x^2 \log (3)+x \log ^2(3)} \, dx=\frac {\left (1+e^6\right )^2 \left (x^2+x \log (3)+\log ^2(3)-\log (x)\right )}{x+\log (3)} \]
Integrate[(-x + x^3 + E^12*(-x + x^3) + E^6*(-2*x + 2*x^3) + (-1 + 2*x^2 + E^12*(-1 + 2*x^2) + E^6*(-2 + 4*x^2))*Log[3] + (x + 2*E^6*x + E^12*x)*Log [x])/(x^3 + 2*x^2*Log[3] + x*Log[3]^2),x]
Leaf count is larger than twice the leaf count of optimal. \(58\) vs. \(2(26)=52\).
Time = 0.74 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.23, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2026, 2007, 7239, 27, 25, 7293, 2009}
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.
\(\displaystyle \int \frac {x^3+e^{12} \left (x^3-x\right )+e^6 \left (2 x^3-2 x\right )+\left (2 x^2+e^{12} \left (2 x^2-1\right )+e^6 \left (4 x^2-2\right )-1\right ) \log (3)-x+\left (e^{12} x+2 e^6 x+x\right ) \log (x)}{x^3+2 x^2 \log (3)+x \log ^2(3)} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {x^3+e^{12} \left (x^3-x\right )+e^6 \left (2 x^3-2 x\right )+\left (2 x^2+e^{12} \left (2 x^2-1\right )+e^6 \left (4 x^2-2\right )-1\right ) \log (3)-x+\left (e^{12} x+2 e^6 x+x\right ) \log (x)}{x \left (x^2+2 x \log (3)+\log ^2(3)\right )}dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {x^3+e^{12} \left (x^3-x\right )+e^6 \left (2 x^3-2 x\right )+\left (2 x^2+e^{12} \left (2 x^2-1\right )+e^6 \left (4 x^2-2\right )-1\right ) \log (3)-x+\left (e^{12} x+2 e^6 x+x\right ) \log (x)}{x (x+\log (3))^2}dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {\left (1+e^6\right )^2 \left (x^3+x^2 \log (9)-x+x \log (x)-\log (3)\right )}{x (x+\log (3))^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \left (1+e^6\right )^2 \int -\frac {-x^3-\log (9) x^2-\log (x) x+x+\log (3)}{x (x+\log (3))^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\left (1+e^6\right )^2 \int \frac {-x^3-\log (9) x^2-\log (x) x+x+\log (3)}{x (x+\log (3))^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\left (1+e^6\right )^2 \int \left (-\frac {x^2}{(x+\log (3))^2}-\frac {\log (9) x}{(x+\log (3))^2}-\frac {\log (x)}{(x+\log (3))^2}+\frac {1}{(x+\log (3))^2}+\frac {\log (3)}{(x+\log (3))^2 x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\left (1+e^6\right )^2 \left (-x+\frac {\log ^2(3)}{x+\log (3)}-\frac {x \log (x)}{\log (3) (x+\log (3))}+\frac {\log (x)}{\log (3)}-\frac {\log (3) \log (9)}{x+\log (3)}\right )\) |
Int[(-x + x^3 + E^12*(-x + x^3) + E^6*(-2*x + 2*x^3) + (-1 + 2*x^2 + E^12* (-1 + 2*x^2) + E^6*(-2 + 4*x^2))*Log[3] + (x + 2*E^6*x + E^12*x)*Log[x])/( x^3 + 2*x^2*Log[3] + x*Log[3]^2),x]
-((1 + E^6)^2*(-x + Log[3]^2/(x + Log[3]) - (Log[3]*Log[9])/(x + Log[3]) + Log[x]/Log[3] - (x*Log[x])/(Log[3]*(x + Log[3]))))
3.13.99.3.1 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[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
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])
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.89 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62
method | result | size |
norman | \(\frac {\left ({\mathrm e}^{12}+2 \,{\mathrm e}^{6}+1\right ) x^{2}+\left (-{\mathrm e}^{12}-2 \,{\mathrm e}^{6}-1\right ) \ln \left (x \right )}{\ln \left (3\right )+x}\) | \(42\) |
risch | \(-\frac {\left ({\mathrm e}^{12}+2 \,{\mathrm e}^{6}+1\right ) \ln \left (x \right )}{\ln \left (3\right )+x}+\frac {\left ({\mathrm e}^{12}+2 \,{\mathrm e}^{6}+1\right ) \left (\ln \left (3\right )^{2}+x \ln \left (3\right )+x^{2}\right )}{\ln \left (3\right )+x}\) | \(47\) |
parallelrisch | \(\frac {x^{2} {\mathrm e}^{12}-\ln \left (x \right ) {\mathrm e}^{12}+2 x^{2} {\mathrm e}^{6}-2 \,{\mathrm e}^{6} \ln \left (x \right )+x^{2}-\ln \left (x \right )}{\ln \left (3\right )+x}\) | \(49\) |
default | \(\left ({\mathrm e}^{12}+2 \,{\mathrm e}^{6}+1\right ) \left (-\frac {\ln \left (\ln \left (3\right )+x \right )}{\ln \left (3\right )}+\frac {\ln \left (x \right ) x}{\ln \left (3\right ) \left (\ln \left (3\right )+x \right )}\right )+\left ({\mathrm e}^{12}+2 \,{\mathrm e}^{6}+1\right ) \left (x -\frac {\ln \left (x \right )}{\ln \left (3\right )}+\frac {\ln \left (\ln \left (3\right )+x \right )}{\ln \left (3\right )}+\frac {\ln \left (3\right )^{2}}{\ln \left (3\right )+x}\right )\) | \(85\) |
parts | \(\left ({\mathrm e}^{12}+2 \,{\mathrm e}^{6}+1\right ) \left (-\frac {\ln \left (\ln \left (3\right )+x \right )}{\ln \left (3\right )}+\frac {\ln \left (x \right ) x}{\ln \left (3\right ) \left (\ln \left (3\right )+x \right )}\right )+\left ({\mathrm e}^{12}+2 \,{\mathrm e}^{6}+1\right ) \left (x -\frac {\ln \left (x \right )}{\ln \left (3\right )}+\frac {\ln \left (\ln \left (3\right )+x \right )}{\ln \left (3\right )}+\frac {\ln \left (3\right )^{2}}{\ln \left (3\right )+x}\right )\) | \(85\) |
int(((x*exp(3)^4+2*x*exp(3)^2+x)*ln(x)+((2*x^2-1)*exp(3)^4+(4*x^2-2)*exp(3 )^2+2*x^2-1)*ln(3)+(x^3-x)*exp(3)^4+(2*x^3-2*x)*exp(3)^2+x^3-x)/(x*ln(3)^2 +2*x^2*ln(3)+x^3),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (24) = 48\).
Time = 0.24 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.42 \[ \int \frac {-x+x^3+e^{12} \left (-x+x^3\right )+e^6 \left (-2 x+2 x^3\right )+\left (-1+2 x^2+e^{12} \left (-1+2 x^2\right )+e^6 \left (-2+4 x^2\right )\right ) \log (3)+\left (x+2 e^6 x+e^{12} x\right ) \log (x)}{x^3+2 x^2 \log (3)+x \log ^2(3)} \, dx=\frac {x^{2} e^{12} + 2 \, x^{2} e^{6} + {\left (e^{12} + 2 \, e^{6} + 1\right )} \log \left (3\right )^{2} + x^{2} + {\left (x e^{12} + 2 \, x e^{6} + x\right )} \log \left (3\right ) - {\left (e^{12} + 2 \, e^{6} + 1\right )} \log \left (x\right )}{x + \log \left (3\right )} \]
integrate(((x*exp(3)^4+2*x*exp(3)^2+x)*log(x)+((2*x^2-1)*exp(3)^4+(4*x^2-2 )*exp(3)^2+2*x^2-1)*log(3)+(x^3-x)*exp(3)^4+(2*x^3-2*x)*exp(3)^2+x^3-x)/(x *log(3)^2+2*x^2*log(3)+x^3),x, algorithm=\
(x^2*e^12 + 2*x^2*e^6 + (e^12 + 2*e^6 + 1)*log(3)^2 + x^2 + (x*e^12 + 2*x* e^6 + x)*log(3) - (e^12 + 2*e^6 + 1)*log(x))/(x + log(3))
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (19) = 38\).
Time = 0.17 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.31 \[ \int \frac {-x+x^3+e^{12} \left (-x+x^3\right )+e^6 \left (-2 x+2 x^3\right )+\left (-1+2 x^2+e^{12} \left (-1+2 x^2\right )+e^6 \left (-2+4 x^2\right )\right ) \log (3)+\left (x+2 e^6 x+e^{12} x\right ) \log (x)}{x^3+2 x^2 \log (3)+x \log ^2(3)} \, dx=x \left (1 + 2 e^{6} + e^{12}\right ) + \frac {\left (- e^{12} - 2 e^{6} - 1\right ) \log {\left (x \right )}}{x + \log {\left (3 \right )}} + \frac {\log {\left (3 \right )}^{2} + 2 e^{6} \log {\left (3 \right )}^{2} + e^{12} \log {\left (3 \right )}^{2}}{x + \log {\left (3 \right )}} \]
integrate(((x*exp(3)**4+2*x*exp(3)**2+x)*ln(x)+((2*x**2-1)*exp(3)**4+(4*x* *2-2)*exp(3)**2+2*x**2-1)*ln(3)+(x**3-x)*exp(3)**4+(2*x**3-2*x)*exp(3)**2+ x**3-x)/(x*ln(3)**2+2*x**2*ln(3)+x**3),x)
x*(1 + 2*exp(6) + exp(12)) + (-exp(12) - 2*exp(6) - 1)*log(x)/(x + log(3)) + (log(3)**2 + 2*exp(6)*log(3)**2 + exp(12)*log(3)**2)/(x + log(3))
Leaf count of result is larger than twice the leaf count of optimal. 369 vs. \(2 (24) = 48\).
Time = 0.21 (sec) , antiderivative size = 369, normalized size of antiderivative = 14.19 \[ \int \frac {-x+x^3+e^{12} \left (-x+x^3\right )+e^6 \left (-2 x+2 x^3\right )+\left (-1+2 x^2+e^{12} \left (-1+2 x^2\right )+e^6 \left (-2+4 x^2\right )\right ) \log (3)+\left (x+2 e^6 x+e^{12} x\right ) \log (x)}{x^3+2 x^2 \log (3)+x \log ^2(3)} \, dx =\text {Too large to display} \]
integrate(((x*exp(3)^4+2*x*exp(3)^2+x)*log(x)+((2*x^2-1)*exp(3)^4+(4*x^2-2 )*exp(3)^2+2*x^2-1)*log(3)+(x^3-x)*exp(3)^4+(2*x^3-2*x)*exp(3)^2+x^3-x)/(x *log(3)^2+2*x^2*log(3)+x^3),x, algorithm=\
2*(log(3)/(x + log(3)) + log(x + log(3)))*e^12*log(3) - (1/(x*log(3) + log (3)^2) - log(x + log(3))/log(3)^2 + log(x)/log(3)^2)*e^12*log(3) + 4*(log( 3)/(x + log(3)) + log(x + log(3)))*e^6*log(3) - 2*(1/(x*log(3) + log(3)^2) - log(x + log(3))/log(3)^2 + log(x)/log(3)^2)*e^6*log(3) - (2*log(3)*log( x + log(3)) - x + log(3)^2/(x + log(3)))*e^12 - (log(x + log(3))/log(3) - log(x)/log(3))*e^12 - 2*(2*log(3)*log(x + log(3)) - x + log(3)^2/(x + log( 3)))*e^6 - 2*(log(x + log(3))/log(3) - log(x)/log(3))*e^6 + 2*(log(3)/(x + log(3)) + log(x + log(3)))*log(3) - (1/(x*log(3) + log(3)^2) - log(x + lo g(3))/log(3)^2 + log(x)/log(3)^2)*log(3) - 2*log(3)*log(x + log(3)) + x - log(3)^2/(x + log(3)) - e^12*log(x)/(x + log(3)) - 2*e^6*log(x)/(x + log(3 )) + e^12/(x + log(3)) + 2*e^6/(x + log(3)) - log(x + log(3))/log(3) - log (x)/(x + log(3)) + log(x)/log(3) + 1/(x + log(3))
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (24) = 48\).
Time = 0.26 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.92 \[ \int \frac {-x+x^3+e^{12} \left (-x+x^3\right )+e^6 \left (-2 x+2 x^3\right )+\left (-1+2 x^2+e^{12} \left (-1+2 x^2\right )+e^6 \left (-2+4 x^2\right )\right ) \log (3)+\left (x+2 e^6 x+e^{12} x\right ) \log (x)}{x^3+2 x^2 \log (3)+x \log ^2(3)} \, dx=\frac {x^{2} e^{12} + 2 \, x^{2} e^{6} + x e^{12} \log \left (3\right ) + 2 \, x e^{6} \log \left (3\right ) + e^{12} \log \left (3\right )^{2} + 2 \, e^{6} \log \left (3\right )^{2} + x^{2} + x \log \left (3\right ) + \log \left (3\right )^{2} - e^{12} \log \left (x\right ) - 2 \, e^{6} \log \left (x\right ) - \log \left (x\right )}{x + \log \left (3\right )} \]
integrate(((x*exp(3)^4+2*x*exp(3)^2+x)*log(x)+((2*x^2-1)*exp(3)^4+(4*x^2-2 )*exp(3)^2+2*x^2-1)*log(3)+(x^3-x)*exp(3)^4+(2*x^3-2*x)*exp(3)^2+x^3-x)/(x *log(3)^2+2*x^2*log(3)+x^3),x, algorithm=\
(x^2*e^12 + 2*x^2*e^6 + x*e^12*log(3) + 2*x*e^6*log(3) + e^12*log(3)^2 + 2 *e^6*log(3)^2 + x^2 + x*log(3) + log(3)^2 - e^12*log(x) - 2*e^6*log(x) - l og(x))/(x + log(3))
Time = 10.20 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {-x+x^3+e^{12} \left (-x+x^3\right )+e^6 \left (-2 x+2 x^3\right )+\left (-1+2 x^2+e^{12} \left (-1+2 x^2\right )+e^6 \left (-2+4 x^2\right )\right ) \log (3)+\left (x+2 e^6 x+e^{12} x\right ) \log (x)}{x^3+2 x^2 \log (3)+x \log ^2(3)} \, dx=-\frac {\left (\ln \left (x\right )-x^2\right )\,{\left ({\mathrm {e}}^6+1\right )}^2}{x+\ln \left (3\right )} \]