Integrand size = 129, antiderivative size = 30 \[ \int \frac {e^2 \left (8-8 x+3750 x^2-400 x^3+10 x^4\right )-4 e^2 \log \left (\frac {3}{x}\right )}{4 x^2-8 x^3+2504 x^4-2700 x^5+390829 x^6-62504 x^7+3750 x^8-100 x^9+x^{10}+\left (-8 x^2+8 x^3-2500 x^4+200 x^5-4 x^6\right ) \log \left (\frac {3}{x}\right )+4 x^2 \log ^2\left (\frac {3}{x}\right )} \, dx=\frac {e^2}{x \left (-1+x-\frac {1}{2} (-25+x)^2 x^2+\log \left (\frac {3}{x}\right )\right )} \] Output:
exp(2)/x/(x-1/2*(x-25)^2*x^2-1+ln(3/x))
Time = 0.15 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27 \[ \int \frac {e^2 \left (8-8 x+3750 x^2-400 x^3+10 x^4\right )-4 e^2 \log \left (\frac {3}{x}\right )}{4 x^2-8 x^3+2504 x^4-2700 x^5+390829 x^6-62504 x^7+3750 x^8-100 x^9+x^{10}+\left (-8 x^2+8 x^3-2500 x^4+200 x^5-4 x^6\right ) \log \left (\frac {3}{x}\right )+4 x^2 \log ^2\left (\frac {3}{x}\right )} \, dx=\frac {2 e^2}{x \left (-2+2 x-625 x^2+50 x^3-x^4+2 \log \left (\frac {3}{x}\right )\right )} \] Input:
Integrate[(E^2*(8 - 8*x + 3750*x^2 - 400*x^3 + 10*x^4) - 4*E^2*Log[3/x])/( 4*x^2 - 8*x^3 + 2504*x^4 - 2700*x^5 + 390829*x^6 - 62504*x^7 + 3750*x^8 - 100*x^9 + x^10 + (-8*x^2 + 8*x^3 - 2500*x^4 + 200*x^5 - 4*x^6)*Log[3/x] + 4*x^2*Log[3/x]^2),x]
Output:
(2*E^2)/(x*(-2 + 2*x - 625*x^2 + 50*x^3 - x^4 + 2*Log[3/x]))
Time = 1.22 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {7239, 27, 7238}
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.
\(\displaystyle \int \frac {e^2 \left (10 x^4-400 x^3+3750 x^2-8 x+8\right )-4 e^2 \log \left (\frac {3}{x}\right )}{x^{10}-100 x^9+3750 x^8-62504 x^7+390829 x^6-2700 x^5+2504 x^4-8 x^3+4 x^2+4 x^2 \log ^2\left (\frac {3}{x}\right )+\left (-4 x^6+200 x^5-2500 x^4+8 x^3-8 x^2\right ) \log \left (\frac {3}{x}\right )} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {2 e^2 \left (5 x^4-200 x^3+1875 x^2-4 x-2 \log \left (\frac {3}{x}\right )+4\right )}{x^2 \left (x^4-50 x^3+625 x^2-2 x-2 \log \left (\frac {3}{x}\right )+2\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 e^2 \int \frac {5 x^4-200 x^3+1875 x^2-4 x-2 \log \left (\frac {3}{x}\right )+4}{x^2 \left (x^4-50 x^3+625 x^2-2 x-2 \log \left (\frac {3}{x}\right )+2\right )^2}dx\) |
\(\Big \downarrow \) 7238 |
\(\displaystyle -\frac {2 e^2}{x \left (x^4-50 x^3+625 x^2-2 x-2 \log \left (\frac {3}{x}\right )+2\right )}\) |
Input:
Int[(E^2*(8 - 8*x + 3750*x^2 - 400*x^3 + 10*x^4) - 4*E^2*Log[3/x])/(4*x^2 - 8*x^3 + 2504*x^4 - 2700*x^5 + 390829*x^6 - 62504*x^7 + 3750*x^8 - 100*x^ 9 + x^10 + (-8*x^2 + 8*x^3 - 2500*x^4 + 200*x^5 - 4*x^6)*Log[3/x] + 4*x^2* Log[3/x]^2),x]
Output:
(-2*E^2)/(x*(2 - 2*x + 625*x^2 - 50*x^3 + x^4 - 2*Log[3/x]))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*(y_)^(m_.)*(z_)^(n_.), x_Symbol] :> With[{q = DerivativeDivides[y* z, u*z^(n - m), x]}, Simp[q*y^(m + 1)*(z^(m + 1)/(m + 1)), x] /; !FalseQ[q ]] /; FreeQ[{m, n}, x] && NeQ[m, -1]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.50 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20
method | result | size |
risch | \(-\frac {2 \,{\mathrm e}^{2}}{x \left (x^{4}-50 x^{3}+625 x^{2}-2 x -2 \ln \left (\frac {3}{x}\right )+2\right )}\) | \(36\) |
parallelrisch | \(-\frac {2 \,{\mathrm e}^{2}}{x \left (x^{4}-50 x^{3}+625 x^{2}-2 x -2 \ln \left (\frac {3}{x}\right )+2\right )}\) | \(36\) |
derivativedivides | \(\frac {{\mathrm e}^{2} \left (\frac {486 \ln \left (\frac {3}{x}\right )}{x^{5}}-\frac {486}{x^{5}}\right )}{3 \left (\frac {162 \ln \left (\frac {3}{x}\right )}{x^{4}}-\frac {162}{x^{4}}+\frac {162}{x^{3}}-\frac {50625}{x^{2}}+\frac {4050}{x}-81\right ) \left (\ln \left (\frac {3}{x}\right )-1\right )}\) | \(67\) |
default | \(\frac {{\mathrm e}^{2} \left (\frac {486 \ln \left (\frac {3}{x}\right )}{x^{5}}-\frac {486}{x^{5}}\right )}{3 \left (\frac {162 \ln \left (\frac {3}{x}\right )}{x^{4}}-\frac {162}{x^{4}}+\frac {162}{x^{3}}-\frac {50625}{x^{2}}+\frac {4050}{x}-81\right ) \left (\ln \left (\frac {3}{x}\right )-1\right )}\) | \(67\) |
Input:
int((-4*exp(2)*ln(3/x)+(10*x^4-400*x^3+3750*x^2-8*x+8)*exp(2))/(4*x^2*ln(3 /x)^2+(-4*x^6+200*x^5-2500*x^4+8*x^3-8*x^2)*ln(3/x)+x^10-100*x^9+3750*x^8- 62504*x^7+390829*x^6-2700*x^5+2504*x^4-8*x^3+4*x^2),x,method=_RETURNVERBOS E)
Output:
-2*exp(2)/x/(x^4-50*x^3+625*x^2-2*x-2*ln(3/x)+2)
Time = 0.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {e^2 \left (8-8 x+3750 x^2-400 x^3+10 x^4\right )-4 e^2 \log \left (\frac {3}{x}\right )}{4 x^2-8 x^3+2504 x^4-2700 x^5+390829 x^6-62504 x^7+3750 x^8-100 x^9+x^{10}+\left (-8 x^2+8 x^3-2500 x^4+200 x^5-4 x^6\right ) \log \left (\frac {3}{x}\right )+4 x^2 \log ^2\left (\frac {3}{x}\right )} \, dx=-\frac {2 \, e^{2}}{x^{5} - 50 \, x^{4} + 625 \, x^{3} - 2 \, x^{2} - 2 \, x \log \left (\frac {3}{x}\right ) + 2 \, x} \] Input:
integrate((-4*exp(2)*log(3/x)+(10*x^4-400*x^3+3750*x^2-8*x+8)*exp(2))/(4*x ^2*log(3/x)^2+(-4*x^6+200*x^5-2500*x^4+8*x^3-8*x^2)*log(3/x)+x^10-100*x^9+ 3750*x^8-62504*x^7+390829*x^6-2700*x^5+2504*x^4-8*x^3+4*x^2),x, algorithm= "fricas")
Output:
-2*e^2/(x^5 - 50*x^4 + 625*x^3 - 2*x^2 - 2*x*log(3/x) + 2*x)
Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {e^2 \left (8-8 x+3750 x^2-400 x^3+10 x^4\right )-4 e^2 \log \left (\frac {3}{x}\right )}{4 x^2-8 x^3+2504 x^4-2700 x^5+390829 x^6-62504 x^7+3750 x^8-100 x^9+x^{10}+\left (-8 x^2+8 x^3-2500 x^4+200 x^5-4 x^6\right ) \log \left (\frac {3}{x}\right )+4 x^2 \log ^2\left (\frac {3}{x}\right )} \, dx=\frac {2 e^{2}}{- x^{5} + 50 x^{4} - 625 x^{3} + 2 x^{2} + 2 x \log {\left (\frac {3}{x} \right )} - 2 x} \] Input:
integrate((-4*exp(2)*ln(3/x)+(10*x**4-400*x**3+3750*x**2-8*x+8)*exp(2))/(4 *x**2*ln(3/x)**2+(-4*x**6+200*x**5-2500*x**4+8*x**3-8*x**2)*ln(3/x)+x**10- 100*x**9+3750*x**8-62504*x**7+390829*x**6-2700*x**5+2504*x**4-8*x**3+4*x** 2),x)
Output:
2*exp(2)/(-x**5 + 50*x**4 - 625*x**3 + 2*x**2 + 2*x*log(3/x) - 2*x)
Time = 0.16 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {e^2 \left (8-8 x+3750 x^2-400 x^3+10 x^4\right )-4 e^2 \log \left (\frac {3}{x}\right )}{4 x^2-8 x^3+2504 x^4-2700 x^5+390829 x^6-62504 x^7+3750 x^8-100 x^9+x^{10}+\left (-8 x^2+8 x^3-2500 x^4+200 x^5-4 x^6\right ) \log \left (\frac {3}{x}\right )+4 x^2 \log ^2\left (\frac {3}{x}\right )} \, dx=-\frac {2 \, e^{2}}{x^{5} - 50 \, x^{4} + 625 \, x^{3} - 2 \, x^{2} - 2 \, x {\left (\log \left (3\right ) - 1\right )} + 2 \, x \log \left (x\right )} \] Input:
integrate((-4*exp(2)*log(3/x)+(10*x^4-400*x^3+3750*x^2-8*x+8)*exp(2))/(4*x ^2*log(3/x)^2+(-4*x^6+200*x^5-2500*x^4+8*x^3-8*x^2)*log(3/x)+x^10-100*x^9+ 3750*x^8-62504*x^7+390829*x^6-2700*x^5+2504*x^4-8*x^3+4*x^2),x, algorithm= "maxima")
Output:
-2*e^2/(x^5 - 50*x^4 + 625*x^3 - 2*x^2 - 2*x*(log(3) - 1) + 2*x*log(x))
Time = 0.14 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {e^2 \left (8-8 x+3750 x^2-400 x^3+10 x^4\right )-4 e^2 \log \left (\frac {3}{x}\right )}{4 x^2-8 x^3+2504 x^4-2700 x^5+390829 x^6-62504 x^7+3750 x^8-100 x^9+x^{10}+\left (-8 x^2+8 x^3-2500 x^4+200 x^5-4 x^6\right ) \log \left (\frac {3}{x}\right )+4 x^2 \log ^2\left (\frac {3}{x}\right )} \, dx=\frac {2 \, e^{2}}{x^{5} {\left (\frac {50}{x} - \frac {625}{x^{2}} + \frac {2}{x^{3}} + \frac {2 \, \log \left (\frac {3}{x}\right )}{x^{4}} - \frac {2}{x^{4}} - 1\right )}} \] Input:
integrate((-4*exp(2)*log(3/x)+(10*x^4-400*x^3+3750*x^2-8*x+8)*exp(2))/(4*x ^2*log(3/x)^2+(-4*x^6+200*x^5-2500*x^4+8*x^3-8*x^2)*log(3/x)+x^10-100*x^9+ 3750*x^8-62504*x^7+390829*x^6-2700*x^5+2504*x^4-8*x^3+4*x^2),x, algorithm= "giac")
Output:
2*e^2/(x^5*(50/x - 625/x^2 + 2/x^3 + 2*log(3/x)/x^4 - 2/x^4 - 1))
Time = 2.98 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {e^2 \left (8-8 x+3750 x^2-400 x^3+10 x^4\right )-4 e^2 \log \left (\frac {3}{x}\right )}{4 x^2-8 x^3+2504 x^4-2700 x^5+390829 x^6-62504 x^7+3750 x^8-100 x^9+x^{10}+\left (-8 x^2+8 x^3-2500 x^4+200 x^5-4 x^6\right ) \log \left (\frac {3}{x}\right )+4 x^2 \log ^2\left (\frac {3}{x}\right )} \, dx=\frac {2\,{\mathrm {e}}^2}{x\,\left (2\,x+2\,\ln \left (\frac {3}{x}\right )-625\,x^2+50\,x^3-x^4-2\right )} \] Input:
int(-(4*exp(2)*log(3/x) - exp(2)*(3750*x^2 - 8*x - 400*x^3 + 10*x^4 + 8))/ (4*x^2*log(3/x)^2 - log(3/x)*(8*x^2 - 8*x^3 + 2500*x^4 - 200*x^5 + 4*x^6) + 4*x^2 - 8*x^3 + 2504*x^4 - 2700*x^5 + 390829*x^6 - 62504*x^7 + 3750*x^8 - 100*x^9 + x^10),x)
Output:
(2*exp(2))/(x*(2*x + 2*log(3/x) - 625*x^2 + 50*x^3 - x^4 - 2))
Time = 0.21 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27 \[ \int \frac {e^2 \left (8-8 x+3750 x^2-400 x^3+10 x^4\right )-4 e^2 \log \left (\frac {3}{x}\right )}{4 x^2-8 x^3+2504 x^4-2700 x^5+390829 x^6-62504 x^7+3750 x^8-100 x^9+x^{10}+\left (-8 x^2+8 x^3-2500 x^4+200 x^5-4 x^6\right ) \log \left (\frac {3}{x}\right )+4 x^2 \log ^2\left (\frac {3}{x}\right )} \, dx=\frac {2 e^{2}}{x \left (2 \,\mathrm {log}\left (\frac {3}{x}\right )-x^{4}+50 x^{3}-625 x^{2}+2 x -2\right )} \] Input:
int((-4*exp(2)*log(3/x)+(10*x^4-400*x^3+3750*x^2-8*x+8)*exp(2))/(4*x^2*log (3/x)^2+(-4*x^6+200*x^5-2500*x^4+8*x^3-8*x^2)*log(3/x)+x^10-100*x^9+3750*x ^8-62504*x^7+390829*x^6-2700*x^5+2504*x^4-8*x^3+4*x^2),x)
Output:
(2*e**2)/(x*(2*log(3/x) - x**4 + 50*x**3 - 625*x**2 + 2*x - 2))