Integrand size = 93, antiderivative size = 21 \[ \int \frac {-3+3 e^2-6 x-3 \log \left (9 e^6\right )}{x^2+e^4 x^2+2 x^3+x^4+e^2 \left (-2 x^2-2 x^3\right )+\left (2 x^2-2 e^2 x^2+2 x^3\right ) \log \left (9 e^6\right )+x^2 \log ^2\left (9 e^6\right )} \, dx=\frac {3}{x \left (1-e^2+x+\log \left (9 e^6\right )\right )} \]
Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {-3+3 e^2-6 x-3 \log \left (9 e^6\right )}{x^2+e^4 x^2+2 x^3+x^4+e^2 \left (-2 x^2-2 x^3\right )+\left (2 x^2-2 e^2 x^2+2 x^3\right ) \log \left (9 e^6\right )+x^2 \log ^2\left (9 e^6\right )} \, dx=\frac {3}{x \left (7-e^2+x+\log (9)\right )} \]
Integrate[(-3 + 3*E^2 - 6*x - 3*Log[9*E^6])/(x^2 + E^4*x^2 + 2*x^3 + x^4 + E^2*(-2*x^2 - 2*x^3) + (2*x^2 - 2*E^2*x^2 + 2*x^3)*Log[9*E^6] + x^2*Log[9 *E^6]^2),x]
Time = 0.23 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {6, 6, 2026, 1184, 27, 83}
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.
\(\displaystyle \int \frac {-6 x+3 e^2-3-3 \log \left (9 e^6\right )}{x^4+2 x^3+e^4 x^2+x^2+x^2 \log ^2\left (9 e^6\right )+e^2 \left (-2 x^3-2 x^2\right )+\left (2 x^3-2 e^2 x^2+2 x^2\right ) \log \left (9 e^6\right )} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {-6 x+3 e^2-3-3 \log \left (9 e^6\right )}{x^4+2 x^3+\left (1+e^4\right ) x^2+x^2 \log ^2\left (9 e^6\right )+e^2 \left (-2 x^3-2 x^2\right )+\left (2 x^3-2 e^2 x^2+2 x^2\right ) \log \left (9 e^6\right )}dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {-6 x+3 e^2-3-3 \log \left (9 e^6\right )}{x^4+2 x^3+x^2 \left (1+e^4+\log ^2\left (9 e^6\right )\right )+e^2 \left (-2 x^3-2 x^2\right )+\left (2 x^3-2 e^2 x^2+2 x^2\right ) \log \left (9 e^6\right )}dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {-6 x+3 e^2-3-3 \log \left (9 e^6\right )}{x^2 \left (x^2+2 x \left (7-e^2+\log (9)\right )+\left (7-e^2+\log (9)\right )^2\right )}dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \int -\frac {3 \left (2 x-e^2+7+\log (9)\right )}{x^2 \left (x-e^2+7+\log (9)\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -3 \int \frac {2 x+\log (9)-e^2+7}{x^2 \left (x+\log (9)-e^2+7\right )^2}dx\) |
\(\Big \downarrow \) 83 |
\(\displaystyle \frac {3}{x \left (x-e^2+7+\log (9)\right )}\) |
Int[(-3 + 3*E^2 - 6*x - 3*Log[9*E^6])/(x^2 + E^4*x^2 + 2*x^3 + x^4 + E^2*( -2*x^2 - 2*x^3) + (2*x^2 - 2*E^2*x^2 + 2*x^3)*Log[9*E^6] + x^2*Log[9*E^6]^ 2),x]
3.11.44.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && EqQ[a*d*f *(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
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])
Time = 0.98 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90
method | result | size |
norman | \(-\frac {3}{x \left ({\mathrm e}^{2}-2 \ln \left (3\right )-x -7\right )}\) | \(19\) |
risch | \(-\frac {3}{x \left ({\mathrm e}^{2}-2 \ln \left (3\right )-x -7\right )}\) | \(19\) |
gosper | \(-\frac {3}{x \left (-x -1+{\mathrm e}^{2}-\ln \left (9 \,{\mathrm e}^{6}\right )\right )}\) | \(24\) |
parallelrisch | \(-\frac {3}{x \left (-x -1+{\mathrm e}^{2}-\ln \left (9 \,{\mathrm e}^{6}\right )\right )}\) | \(24\) |
int((-3*ln(9*exp(3)^2)+3*exp(2)-6*x-3)/(x^2*ln(9*exp(3)^2)^2+(-2*x^2*exp(2 )+2*x^3+2*x^2)*ln(9*exp(3)^2)+x^2*exp(2)^2+(-2*x^3-2*x^2)*exp(2)+x^4+2*x^3 +x^2),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {-3+3 e^2-6 x-3 \log \left (9 e^6\right )}{x^2+e^4 x^2+2 x^3+x^4+e^2 \left (-2 x^2-2 x^3\right )+\left (2 x^2-2 e^2 x^2+2 x^3\right ) \log \left (9 e^6\right )+x^2 \log ^2\left (9 e^6\right )} \, dx=\frac {3}{x^{2} - x e^{2} + 2 \, x \log \left (3\right ) + 7 \, x} \]
integrate((-3*log(9*exp(3)^2)+3*exp(2)-6*x-3)/(x^2*log(9*exp(3)^2)^2+(-2*x ^2*exp(2)+2*x^3+2*x^2)*log(9*exp(3)^2)+x^2*exp(2)^2+(-2*x^3-2*x^2)*exp(2)+ x^4+2*x^3+x^2),x, algorithm=\
Time = 0.41 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \frac {-3+3 e^2-6 x-3 \log \left (9 e^6\right )}{x^2+e^4 x^2+2 x^3+x^4+e^2 \left (-2 x^2-2 x^3\right )+\left (2 x^2-2 e^2 x^2+2 x^3\right ) \log \left (9 e^6\right )+x^2 \log ^2\left (9 e^6\right )} \, dx=\frac {3}{x^{2} + x \left (- e^{2} + 2 \log {\left (3 \right )} + 7\right )} \]
integrate((-3*ln(9*exp(3)**2)+3*exp(2)-6*x-3)/(x**2*ln(9*exp(3)**2)**2+(-2 *x**2*exp(2)+2*x**3+2*x**2)*ln(9*exp(3)**2)+x**2*exp(2)**2+(-2*x**3-2*x**2 )*exp(2)+x**4+2*x**3+x**2),x)
Time = 0.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \frac {-3+3 e^2-6 x-3 \log \left (9 e^6\right )}{x^2+e^4 x^2+2 x^3+x^4+e^2 \left (-2 x^2-2 x^3\right )+\left (2 x^2-2 e^2 x^2+2 x^3\right ) \log \left (9 e^6\right )+x^2 \log ^2\left (9 e^6\right )} \, dx=\frac {3}{x^{2} - x {\left (e^{2} - \log \left (9 \, e^{6}\right ) - 1\right )}} \]
integrate((-3*log(9*exp(3)^2)+3*exp(2)-6*x-3)/(x^2*log(9*exp(3)^2)^2+(-2*x ^2*exp(2)+2*x^3+2*x^2)*log(9*exp(3)^2)+x^2*exp(2)^2+(-2*x^3-2*x^2)*exp(2)+ x^4+2*x^3+x^2),x, algorithm=\
Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {-3+3 e^2-6 x-3 \log \left (9 e^6\right )}{x^2+e^4 x^2+2 x^3+x^4+e^2 \left (-2 x^2-2 x^3\right )+\left (2 x^2-2 e^2 x^2+2 x^3\right ) \log \left (9 e^6\right )+x^2 \log ^2\left (9 e^6\right )} \, dx=\frac {3}{x^{2} - x e^{2} + x \log \left (9 \, e^{6}\right ) + x} \]
integrate((-3*log(9*exp(3)^2)+3*exp(2)-6*x-3)/(x^2*log(9*exp(3)^2)^2+(-2*x ^2*exp(2)+2*x^3+2*x^2)*log(9*exp(3)^2)+x^2*exp(2)^2+(-2*x^3-2*x^2)*exp(2)+ x^4+2*x^3+x^2),x, algorithm=\
Time = 27.58 (sec) , antiderivative size = 16488, normalized size of antiderivative = 785.14 \[ \int \frac {-3+3 e^2-6 x-3 \log \left (9 e^6\right )}{x^2+e^4 x^2+2 x^3+x^4+e^2 \left (-2 x^2-2 x^3\right )+\left (2 x^2-2 e^2 x^2+2 x^3\right ) \log \left (9 e^6\right )+x^2 \log ^2\left (9 e^6\right )} \, dx=\text {Too large to display} \]
int(-(6*x + 3*log(9*exp(6)) - 3*exp(2) + 3)/(x^2*exp(4) - exp(2)*(2*x^2 + 2*x^3) + x^2*log(9*exp(6))^2 + x^2 + 2*x^3 + x^4 + log(9*exp(6))*(2*x^2 - 2*x^2*exp(2) + 2*x^3)),x)
(log(729) - 3*exp(2) + 21)/(x*(exp(4) - 14*exp(2) + 14*log(9) - 2*exp(2)*l og(9) + log(9)^2 + 49)) - (log((441*log(81) - 1764*log(9) + 294*log(729) + 504*exp(2)*log(9) - 36*exp(4)*log(9) - 126*exp(2)*log(81) + 9*exp(4)*log( 81) - 84*exp(2)*log(729) + 6*exp(4)*log(729) - 84*log(9)*log(729) + 42*log (81)*log(729) + 18*exp(2)*log(9)^2 - 2*exp(2)*log(729)^2 - 6*log(9)^2*log( 729) + log(81)*log(729)^2 - 126*log(9)^2 + 14*log(729)^2 + 12*exp(2)*log(9 )*log(729) - 6*exp(2)*log(81)*log(729))/(294*exp(4) - 1372*exp(2) - 28*exp (6) + exp(8) + 1372*log(9) - 588*exp(2)*log(9) + 84*exp(4)*log(9) - 4*exp( 6)*log(9) - 84*exp(2)*log(9)^2 - 4*exp(2)*log(9)^3 + 6*exp(4)*log(9)^2 + 2 94*log(9)^2 + 28*log(9)^3 + log(9)^4 + 2401) + (x*(9*exp(4) - 126*exp(2) + 42*log(729) - 6*exp(2)*log(729) + log(729)^2 + 441))/(294*exp(4) - 1372*e xp(2) - 28*exp(6) + exp(8) + 1372*log(9) - 588*exp(2)*log(9) + 84*exp(4)*l og(9) - 4*exp(6)*log(9) - 84*exp(2)*log(9)^2 - 4*exp(2)*log(9)^3 + 6*exp(4 )*log(9)^2 + 294*log(9)^2 + 28*log(9)^3 + log(9)^4 + 2401) - (((36015*exp( 2) - 10290*exp(4) + 1470*exp(6) - 105*exp(8) + 3*exp(10) - 86436*log(9) + 14406*log(81) + 7203*log(729) + 49392*exp(2)*log(9) - 10584*exp(4)*log(9) + 1008*exp(6)*log(9) - 36*exp(8)*log(9) - 8232*exp(2)*log(81) + 1764*exp(4 )*log(81) - 168*exp(6)*log(81) + 6*exp(8)*log(81) - 4116*exp(2)*log(729) + 882*exp(4)*log(729) - 84*exp(6)*log(729) + 3*exp(8)*log(729) + 1372*log(9 )*log(729) + 1372*log(81)*log(729) + 11466*exp(2)*log(9)^2 + 840*exp(2)...