Integrand size = 45, antiderivative size = 26 \[ \int \frac {1-36 x-3 x^2+e^x \left (-18-18 x-3 x^2\right )}{36+12 x+x^2+e \left (36+12 x+x^2\right )} \, dx=4-\frac {x \left (\frac {1}{x}+3 \left (e^x+x\right )\right )}{(1+e) (6+x)} \] Output:
4-(1/x+3*exp(x)+3*x)*x/(6+x)/(1+exp(1))
Time = 0.11 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {1-36 x-3 x^2+e^x \left (-18-18 x-3 x^2\right )}{36+12 x+x^2+e \left (36+12 x+x^2\right )} \, dx=-\frac {109+3 \left (6+e^x\right ) x+3 x^2}{(1+e) (6+x)} \] Input:
Integrate[(1 - 36*x - 3*x^2 + E^x*(-18 - 18*x - 3*x^2))/(36 + 12*x + x^2 + E*(36 + 12*x + x^2)),x]
Output:
-((109 + 3*(6 + E^x)*x + 3*x^2)/((1 + E)*(6 + x)))
Time = 0.47 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.77, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2007, 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.
\(\displaystyle \int \frac {-3 x^2+e^x \left (-3 x^2-18 x-18\right )-36 x+1}{x^2+e \left (x^2+12 x+36\right )+12 x+36} \, dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {-3 x^2+e^x \left (-3 x^2-18 x-18\right )-36 x+1}{\left (\sqrt {1+e} x+6 \sqrt {1+e}\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {-3 x^2-36 x+1}{(1+e) (x+6)^2}-\frac {3 e^x \left (x^2+6 x+6\right )}{(1+e) (x+6)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3 x}{1+e}+\frac {18 e^x}{(1+e) (x+6)}-\frac {109}{(1+e) (x+6)}-\frac {3 e^x}{1+e}\) |
Input:
Int[(1 - 36*x - 3*x^2 + E^x*(-18 - 18*x - 3*x^2))/(36 + 12*x + x^2 + E*(36 + 12*x + x^2)),x]
Output:
(-3*E^x)/(1 + E) - (3*x)/(1 + E) - 109/((1 + E)*(6 + x)) + (18*E^x)/((1 + E)*(6 + 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]
Time = 0.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00
method | result | size |
parallelrisch | \(-\frac {3 x^{2}+3 \,{\mathrm e}^{x} x +1}{\left (1+{\mathrm e}\right ) \left (6+x \right )}\) | \(26\) |
norman | \(\frac {-\frac {3 x^{2}}{1+{\mathrm e}}-\frac {3 x \,{\mathrm e}^{x}}{1+{\mathrm e}}-\frac {1}{1+{\mathrm e}}}{6+x}\) | \(38\) |
risch | \(-\frac {3 x}{1+{\mathrm e}}-\frac {109 \,{\mathrm e}}{\left (1+{\mathrm e}\right ) \left (x \,{\mathrm e}+6 \,{\mathrm e}+x +6\right )}-\frac {109}{\left (1+{\mathrm e}\right ) \left (x \,{\mathrm e}+6 \,{\mathrm e}+x +6\right )}-\frac {3 x \,{\mathrm e}^{x}}{\left (1+{\mathrm e}\right ) \left (6+x \right )}\) | \(71\) |
parts | \(-\frac {3 x +\frac {109}{6+x}}{1+{\mathrm e}}-\frac {3 \,{\mathrm e}^{x}}{1+{\mathrm e}}-\frac {18 \left (-\frac {{\mathrm e}^{x}}{6+x}-{\mathrm e}^{-6} \operatorname {expIntegral}_{1}\left (-x -6\right )\right )}{1+{\mathrm e}}-\frac {18 \,{\mathrm e}^{-6} \operatorname {expIntegral}_{1}\left (-x -6\right )}{1+{\mathrm e}}\) | \(77\) |
default | \(-\frac {1}{\left (1+{\mathrm e}\right ) \left (6+x \right )}-\frac {36 \left (\frac {6}{6+x}+\ln \left (6+x \right )\right )}{1+{\mathrm e}}-\frac {3 \left (x -\frac {36}{6+x}-12 \ln \left (6+x \right )\right )}{1+{\mathrm e}}-\frac {18 \left (-\frac {{\mathrm e}^{x}}{6+x}-{\mathrm e}^{-6} \operatorname {expIntegral}_{1}\left (-x -6\right )\right )}{1+{\mathrm e}}-\frac {18 \,{\mathrm e}^{-6} \operatorname {expIntegral}_{1}\left (-x -6\right )}{1+{\mathrm e}}-\frac {3 \,{\mathrm e}^{x}}{1+{\mathrm e}}\) | \(114\) |
orering | \(\frac {\left (3 x^{4}+21 x^{3}+73 x^{2}+114 x +190\right ) \left (\left (-3 x^{2}-18 x -18\right ) {\mathrm e}^{x}-3 x^{2}-36 x +1\right )}{\left (3 x^{3}+36 x^{2}+35 x -38\right ) \left (\left (x^{2}+12 x +36\right ) {\mathrm e}+x^{2}+12 x +36\right )}-\frac {\left (3 x^{3}+19 x +19\right ) \left (6+x \right ) \left (\frac {\left (-6 x -18\right ) {\mathrm e}^{x}+\left (-3 x^{2}-18 x -18\right ) {\mathrm e}^{x}-6 x -36}{\left (x^{2}+12 x +36\right ) {\mathrm e}+x^{2}+12 x +36}-\frac {\left (\left (-3 x^{2}-18 x -18\right ) {\mathrm e}^{x}-3 x^{2}-36 x +1\right ) \left (\left (2 x +12\right ) {\mathrm e}+2 x +12\right )}{{\left (\left (x^{2}+12 x +36\right ) {\mathrm e}+x^{2}+12 x +36\right )}^{2}}\right )}{3 x^{3}+36 x^{2}+35 x -38}\) | \(224\) |
Input:
int(((-3*x^2-18*x-18)*exp(x)-3*x^2-36*x+1)/((x^2+12*x+36)*exp(1)+x^2+12*x+ 36),x,method=_RETURNVERBOSE)
Output:
-(3*x^2+3*exp(x)*x+1)/(1+exp(1))/(6+x)
Time = 0.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {1-36 x-3 x^2+e^x \left (-18-18 x-3 x^2\right )}{36+12 x+x^2+e \left (36+12 x+x^2\right )} \, dx=-\frac {3 \, x^{2} + 3 \, x e^{x} + 18 \, x + 109}{{\left (x + 6\right )} e + x + 6} \] Input:
integrate(((-3*x^2-18*x-18)*exp(x)-3*x^2-36*x+1)/((x^2+12*x+36)*exp(1)+x^2 +12*x+36),x, algorithm="fricas")
Output:
-(3*x^2 + 3*x*e^x + 18*x + 109)/((x + 6)*e + x + 6)
Time = 0.16 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69 \[ \int \frac {1-36 x-3 x^2+e^x \left (-18-18 x-3 x^2\right )}{36+12 x+x^2+e \left (36+12 x+x^2\right )} \, dx=- \frac {3 x}{1 + e} - \frac {3 x e^{x}}{x + e x + 6 + 6 e} - \frac {109}{x \left (1 + e\right ) + 6 + 6 e} \] Input:
integrate(((-3*x**2-18*x-18)*exp(x)-3*x**2-36*x+1)/((x**2+12*x+36)*exp(1)+ x**2+12*x+36),x)
Output:
-3*x/(1 + E) - 3*x*exp(x)/(x + E*x + 6 + 6*E) - 109/(x*(1 + E) + 6 + 6*E)
\[ \int \frac {1-36 x-3 x^2+e^x \left (-18-18 x-3 x^2\right )}{36+12 x+x^2+e \left (36+12 x+x^2\right )} \, dx=\int { -\frac {3 \, x^{2} + 3 \, {\left (x^{2} + 6 \, x + 6\right )} e^{x} + 36 \, x - 1}{x^{2} + {\left (x^{2} + 12 \, x + 36\right )} e + 12 \, x + 36} \,d x } \] Input:
integrate(((-3*x^2-18*x-18)*exp(x)-3*x^2-36*x+1)/((x^2+12*x+36)*exp(1)+x^2 +12*x+36),x, algorithm="maxima")
Output:
-3*x*e^x/(x*(e + 1) + 6*e + 6) - 3*x/(e + 1) + 18*e^(-6)*exp_integral_e(2, -x - 6)/((x + 6)*(e + 1)) - 109/(x*(e + 1) + 6*e + 6) + 18*integrate(e^x/ (x^2*(e + 1) + 12*x*(e + 1) + 36*e + 36), x)
Time = 0.11 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {1-36 x-3 x^2+e^x \left (-18-18 x-3 x^2\right )}{36+12 x+x^2+e \left (36+12 x+x^2\right )} \, dx=-\frac {3 \, x^{2} + 3 \, x e^{x} + 18 \, x + 109}{x e + x + 6 \, e + 6} \] Input:
integrate(((-3*x^2-18*x-18)*exp(x)-3*x^2-36*x+1)/((x^2+12*x+36)*exp(1)+x^2 +12*x+36),x, algorithm="giac")
Output:
-(3*x^2 + 3*x*e^x + 18*x + 109)/(x*e + x + 6*e + 6)
Time = 0.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {1-36 x-3 x^2+e^x \left (-18-18 x-3 x^2\right )}{36+12 x+x^2+e \left (36+12 x+x^2\right )} \, dx=-\frac {x\,\left (18\,x+18\,{\mathrm {e}}^x-1\right )}{6\,\left (\mathrm {e}+1\right )\,\left (x+6\right )} \] Input:
int(-(36*x + exp(x)*(18*x + 3*x^2 + 18) + 3*x^2 - 1)/(12*x + exp(1)*(12*x + x^2 + 36) + x^2 + 36),x)
Output:
-(x*(18*x + 18*exp(x) - 1))/(6*(exp(1) + 1)*(x + 6))
Time = 0.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {1-36 x-3 x^2+e^x \left (-18-18 x-3 x^2\right )}{36+12 x+x^2+e \left (36+12 x+x^2\right )} \, dx=\frac {x \left (-18 e^{x}-18 x +1\right )}{6 e x +36 e +6 x +36} \] Input:
int(((-3*x^2-18*x-18)*exp(x)-3*x^2-36*x+1)/((x^2+12*x+36)*exp(1)+x^2+12*x+ 36),x)
Output:
(x*( - 18*e**x - 18*x + 1))/(6*(e*x + 6*e + x + 6))