Integrand size = 88, antiderivative size = 25 \[ \int \frac {e^4 \left (-3+2 x+x^2+4 \log (x)-\log ^2(x)\right ) \left (-12-6 x-4 x^2-6 x^3-4 x^4+\left (6-6 x^2\right ) \log (x)+2 x^2 \log ^2(x)\right )}{3 x-2 x^2-x^3-4 x \log (x)+x \log ^2(x)} \, dx=e^4 \left (3+x^2\right ) (1-x-\log (x)) (-3-x+\log (x)) \] Output:
(x^2+3)*exp(ln((ln(x)-3-x)*(1-ln(x)-x))+4)
Leaf count is larger than twice the leaf count of optimal. \(60\) vs. \(2(25)=50\).
Time = 0.01 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.40 \[ \int \frac {e^4 \left (-3+2 x+x^2+4 \log (x)-\log ^2(x)\right ) \left (-12-6 x-4 x^2-6 x^3-4 x^4+\left (6-6 x^2\right ) \log (x)+2 x^2 \log ^2(x)\right )}{3 x-2 x^2-x^3-4 x \log (x)+x \log ^2(x)} \, dx=6 e^4 x+2 e^4 x^3+e^4 x^4+12 e^4 \log (x)+4 e^4 x^2 \log (x)-3 e^4 \log ^2(x)-e^4 x^2 \log ^2(x) \] Input:
Integrate[(E^4*(-3 + 2*x + x^2 + 4*Log[x] - Log[x]^2)*(-12 - 6*x - 4*x^2 - 6*x^3 - 4*x^4 + (6 - 6*x^2)*Log[x] + 2*x^2*Log[x]^2))/(3*x - 2*x^2 - x^3 - 4*x*Log[x] + x*Log[x]^2),x]
Output:
6*E^4*x + 2*E^4*x^3 + E^4*x^4 + 12*E^4*Log[x] + 4*E^4*x^2*Log[x] - 3*E^4*L og[x]^2 - E^4*x^2*Log[x]^2
Time = 0.51 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.96, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {27, 27, 7239, 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.
\(\displaystyle \int \frac {e^4 \left (x^2+2 x-\log ^2(x)+4 \log (x)-3\right ) \left (-4 x^4-6 x^3-4 x^2+2 x^2 \log ^2(x)+\left (6-6 x^2\right ) \log (x)-6 x-12\right )}{-x^3-2 x^2+3 x+x \log ^2(x)-4 x \log (x)} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle e^4 \int \frac {2 \left (-x^2-2 x+\log ^2(x)-4 \log (x)+3\right ) \left (2 x^4+3 x^3-\log ^2(x) x^2+2 x^2+3 x-3 \left (1-x^2\right ) \log (x)+6\right )}{-x^3-2 x^2+\log ^2(x) x-4 \log (x) x+3 x}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 e^4 \int \frac {\left (-x^2-2 x+\log ^2(x)-4 \log (x)+3\right ) \left (2 x^4+3 x^3-\log ^2(x) x^2+2 x^2+3 x-3 \left (1-x^2\right ) \log (x)+6\right )}{-x^3-2 x^2+\log ^2(x) x-4 \log (x) x+3 x}dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle 2 e^4 \int \left (2 x^3+3 x^2-\log ^2(x) x+2 x+3+\frac {3 \left (x^2-1\right ) \log (x)}{x}+\frac {6}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 e^4 \left (\frac {x^4}{2}+x^3-\frac {1}{2} x^2 \log ^2(x)+2 x^2 \log (x)+3 x-\frac {3 \log ^2(x)}{2}+6 \log (x)\right )\) |
Input:
Int[(E^4*(-3 + 2*x + x^2 + 4*Log[x] - Log[x]^2)*(-12 - 6*x - 4*x^2 - 6*x^3 - 4*x^4 + (6 - 6*x^2)*Log[x] + 2*x^2*Log[x]^2))/(3*x - 2*x^2 - x^3 - 4*x* Log[x] + x*Log[x]^2),x]
Output:
2*E^4*(3*x + x^3 + x^4/2 + 6*Log[x] + 2*x^2*Log[x] - (3*Log[x]^2)/2 - (x^2 *Log[x]^2)/2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 6.85 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68
method | result | size |
default | \({\mathrm e}^{4} \left (-x^{2} \ln \left (x \right )^{2}+4 x^{2} \ln \left (x \right )+x^{4}+2 x^{3}-3 \ln \left (x \right )^{2}+6 x +12 \ln \left (x \right )\right )\) | \(42\) |
risch | \({\mathrm e}^{4} \left (-x^{2}-3\right ) \ln \left (x \right )^{2}+4 \,{\mathrm e}^{4} x^{2} \ln \left (x \right )+x^{4} {\mathrm e}^{4}+2 x^{3} {\mathrm e}^{4}+6 x \,{\mathrm e}^{4}+12 \ln \left (x \right ) {\mathrm e}^{4}\) | \(49\) |
parallelrisch | \(\frac {-1368 \,{\mathrm e}^{\ln \left (-\ln \left (x \right )^{2}+4 \ln \left (x \right )+x^{2}+2 x -3\right )+4}+768 \,{\mathrm e}^{\ln \left (-\ln \left (x \right )^{2}+4 \ln \left (x \right )+x^{2}+2 x -3\right )+4} x -192 \,{\mathrm e}^{\ln \left (-\ln \left (x \right )^{2}+4 \ln \left (x \right )+x^{2}+2 x -3\right )+4} \ln \left (x \right )^{3}+24 \,{\mathrm e}^{\ln \left (-\ln \left (x \right )^{2}+4 \ln \left (x \right )+x^{2}+2 x -3\right )+4} \ln \left (x \right )^{4}+8 \,{\mathrm e}^{\ln \left (-\ln \left (x \right )^{2}+4 \ln \left (x \right )+x^{2}+2 x -3\right )+4} x^{6}+8 \,{\mathrm e}^{\ln \left (-\ln \left (x \right )^{2}+4 \ln \left (x \right )+x^{2}+2 x -3\right )+4} x^{4}+32 \,{\mathrm e}^{\ln \left (-\ln \left (x \right )^{2}+4 \ln \left (x \right )+x^{2}+2 x -3\right )+4} x^{5}+552 x^{2} {\mathrm e}^{\ln \left (-\ln \left (x \right )^{2}+4 \ln \left (x \right )+x^{2}+2 x -3\right )+4}+1536 \,{\mathrm e}^{\ln \left (-\ln \left (x \right )^{2}+4 \ln \left (x \right )+x^{2}+2 x -3\right )+4} \ln \left (x \right )+384 \,{\mathrm e}^{\ln \left (-\ln \left (x \right )^{2}+4 \ln \left (x \right )+x^{2}+2 x -3\right )+4} \ln \left (x \right ) x +128 \,{\mathrm e}^{\ln \left (-\ln \left (x \right )^{2}+4 \ln \left (x \right )+x^{2}+2 x -3\right )+4} x^{3} \ln \left (x \right )+64 \,{\mathrm e}^{\ln \left (-\ln \left (x \right )^{2}+4 \ln \left (x \right )+x^{2}+2 x -3\right )+4} \ln \left (x \right ) x^{4}-32 \,{\mathrm e}^{\ln \left (-\ln \left (x \right )^{2}+4 \ln \left (x \right )+x^{2}+2 x -3\right )+4} x^{3} \ln \left (x \right )^{2}-64 \,{\mathrm e}^{\ln \left (-\ln \left (x \right )^{2}+4 \ln \left (x \right )+x^{2}+2 x -3\right )+4} x^{2} \ln \left (x \right )^{3}+128 x^{2} \ln \left (x \right )^{2} {\mathrm e}^{\ln \left (-\ln \left (x \right )^{2}+4 \ln \left (x \right )+x^{2}+2 x -3\right )+4}-96 \,{\mathrm e}^{\ln \left (-\ln \left (x \right )^{2}+4 \ln \left (x \right )+x^{2}+2 x -3\right )+4} \ln \left (x \right )^{2} x -16 \,{\mathrm e}^{\ln \left (-\ln \left (x \right )^{2}+4 \ln \left (x \right )+x^{2}+2 x -3\right )+4} x^{4} \ln \left (x \right )^{2}+8 \,{\mathrm e}^{\ln \left (-\ln \left (x \right )^{2}+4 \ln \left (x \right )+x^{2}+2 x -3\right )+4} \ln \left (x \right )^{4} x^{2}}{8 \left (-\ln \left (x \right )^{2}+4 \ln \left (x \right )+x^{2}+2 x -3\right )^{2}}\) | \(532\) |
Input:
int((2*x^2*ln(x)^2+(-6*x^2+6)*ln(x)-4*x^4-6*x^3-4*x^2-6*x-12)*exp(ln(-ln(x )^2+4*ln(x)+x^2+2*x-3)+4)/(x*ln(x)^2-4*x*ln(x)-x^3-2*x^2+3*x),x,method=_RE TURNVERBOSE)
Output:
exp(4)*(-x^2*ln(x)^2+4*x^2*ln(x)+x^4+2*x^3-3*ln(x)^2+6*x+12*ln(x))
Time = 0.09 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.60 \[ \int \frac {e^4 \left (-3+2 x+x^2+4 \log (x)-\log ^2(x)\right ) \left (-12-6 x-4 x^2-6 x^3-4 x^4+\left (6-6 x^2\right ) \log (x)+2 x^2 \log ^2(x)\right )}{3 x-2 x^2-x^3-4 x \log (x)+x \log ^2(x)} \, dx=-{\left (x^{2} + 3\right )} e^{4} \log \left (x\right )^{2} + 4 \, {\left (x^{2} + 3\right )} e^{4} \log \left (x\right ) + {\left (x^{4} + 2 \, x^{3} + 6 \, x\right )} e^{4} \] Input:
integrate((2*x^2*log(x)^2+(-6*x^2+6)*log(x)-4*x^4-6*x^3-4*x^2-6*x-12)*exp( log(-log(x)^2+4*log(x)+x^2+2*x-3)+4)/(x*log(x)^2-4*x*log(x)-x^3-2*x^2+3*x) ,x, algorithm="fricas")
Output:
-(x^2 + 3)*e^4*log(x)^2 + 4*(x^2 + 3)*e^4*log(x) + (x^4 + 2*x^3 + 6*x)*e^4
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (20) = 40\).
Time = 0.13 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.40 \[ \int \frac {e^4 \left (-3+2 x+x^2+4 \log (x)-\log ^2(x)\right ) \left (-12-6 x-4 x^2-6 x^3-4 x^4+\left (6-6 x^2\right ) \log (x)+2 x^2 \log ^2(x)\right )}{3 x-2 x^2-x^3-4 x \log (x)+x \log ^2(x)} \, dx=x^{4} e^{4} + 2 x^{3} e^{4} + 4 x^{2} e^{4} \log {\left (x \right )} + 6 x e^{4} + \left (- x^{2} e^{4} - 3 e^{4}\right ) \log {\left (x \right )}^{2} + 12 e^{4} \log {\left (x \right )} \] Input:
integrate((2*x**2*ln(x)**2+(-6*x**2+6)*ln(x)-4*x**4-6*x**3-4*x**2-6*x-12)* exp(ln(-ln(x)**2+4*ln(x)+x**2+2*x-3)+4)/(x*ln(x)**2-4*x*ln(x)-x**3-2*x**2+ 3*x),x)
Output:
x**4*exp(4) + 2*x**3*exp(4) + 4*x**2*exp(4)*log(x) + 6*x*exp(4) + (-x**2*e xp(4) - 3*exp(4))*log(x)**2 + 12*exp(4)*log(x)
Time = 0.08 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \frac {e^4 \left (-3+2 x+x^2+4 \log (x)-\log ^2(x)\right ) \left (-12-6 x-4 x^2-6 x^3-4 x^4+\left (6-6 x^2\right ) \log (x)+2 x^2 \log ^2(x)\right )}{3 x-2 x^2-x^3-4 x \log (x)+x \log ^2(x)} \, dx={\left (x^{4} + 2 \, x^{3} - {\left (x^{2} + 3\right )} \log \left (x\right )^{2} + 4 \, {\left (x^{2} + 3\right )} \log \left (x\right ) + 6 \, x\right )} e^{4} \] Input:
integrate((2*x^2*log(x)^2+(-6*x^2+6)*log(x)-4*x^4-6*x^3-4*x^2-6*x-12)*exp( log(-log(x)^2+4*log(x)+x^2+2*x-3)+4)/(x*log(x)^2-4*x*log(x)-x^3-2*x^2+3*x) ,x, algorithm="maxima")
Output:
(x^4 + 2*x^3 - (x^2 + 3)*log(x)^2 + 4*(x^2 + 3)*log(x) + 6*x)*e^4
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (23) = 46\).
Time = 0.12 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.12 \[ \int \frac {e^4 \left (-3+2 x+x^2+4 \log (x)-\log ^2(x)\right ) \left (-12-6 x-4 x^2-6 x^3-4 x^4+\left (6-6 x^2\right ) \log (x)+2 x^2 \log ^2(x)\right )}{3 x-2 x^2-x^3-4 x \log (x)+x \log ^2(x)} \, dx=x^{4} e^{4} - x^{2} e^{4} \log \left (x\right )^{2} + 2 \, x^{3} e^{4} + 4 \, x^{2} e^{4} \log \left (x\right ) - 3 \, e^{4} \log \left (x\right )^{2} + 6 \, x e^{4} + 12 \, e^{4} \log \left (x\right ) \] Input:
integrate((2*x^2*log(x)^2+(-6*x^2+6)*log(x)-4*x^4-6*x^3-4*x^2-6*x-12)*exp( log(-log(x)^2+4*log(x)+x^2+2*x-3)+4)/(x*log(x)^2-4*x*log(x)-x^3-2*x^2+3*x) ,x, algorithm="giac")
Output:
x^4*e^4 - x^2*e^4*log(x)^2 + 2*x^3*e^4 + 4*x^2*e^4*log(x) - 3*e^4*log(x)^2 + 6*x*e^4 + 12*e^4*log(x)
Time = 7.92 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.64 \[ \int \frac {e^4 \left (-3+2 x+x^2+4 \log (x)-\log ^2(x)\right ) \left (-12-6 x-4 x^2-6 x^3-4 x^4+\left (6-6 x^2\right ) \log (x)+2 x^2 \log ^2(x)\right )}{3 x-2 x^2-x^3-4 x \log (x)+x \log ^2(x)} \, dx={\mathrm {e}}^4\,\left (x^4+2\,x^3-x^2\,{\ln \left (x\right )}^2+4\,x^2\,\ln \left (x\right )+6\,x-3\,{\ln \left (x\right )}^2+12\,\ln \left (x\right )\right ) \] Input:
int((exp(log(2*x + 4*log(x) - log(x)^2 + x^2 - 3) + 4)*(6*x - 2*x^2*log(x) ^2 + 4*x^2 + 6*x^3 + 4*x^4 + log(x)*(6*x^2 - 6) + 12))/(4*x*log(x) - x*log (x)^2 - 3*x + 2*x^2 + x^3),x)
Output:
exp(4)*(6*x + 12*log(x) + 4*x^2*log(x) - 3*log(x)^2 - x^2*log(x)^2 + 2*x^3 + x^4)
Time = 0.17 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \[ \int \frac {e^4 \left (-3+2 x+x^2+4 \log (x)-\log ^2(x)\right ) \left (-12-6 x-4 x^2-6 x^3-4 x^4+\left (6-6 x^2\right ) \log (x)+2 x^2 \log ^2(x)\right )}{3 x-2 x^2-x^3-4 x \log (x)+x \log ^2(x)} \, dx=e^{4} \left (-\mathrm {log}\left (x \right )^{2} x^{2}-3 \mathrm {log}\left (x \right )^{2}+4 \,\mathrm {log}\left (x \right ) x^{2}+12 \,\mathrm {log}\left (x \right )+x^{4}+2 x^{3}+6 x \right ) \] Input:
int((2*x^2*log(x)^2+(-6*x^2+6)*log(x)-4*x^4-6*x^3-4*x^2-6*x-12)*exp(log(-l og(x)^2+4*log(x)+x^2+2*x-3)+4)/(x*log(x)^2-4*x*log(x)-x^3-2*x^2+3*x),x)
Output:
e**4*( - log(x)**2*x**2 - 3*log(x)**2 + 4*log(x)*x**2 + 12*log(x) + x**4 + 2*x**3 + 6*x)