\(\int \frac {-2-7 e^{2 x}+e^x (-2-14 x)-7 x^2}{e^{2 x}+2 e^x x+x^2} \, dx\) [6570]

Optimal result

Integrand size = 41, antiderivative size = 13 \[ \int \frac {-2-7 e^{2 x}+e^x (-2-14 x)-7 x^2}{e^{2 x}+2 e^x x+x^2} \, dx=-7 x+\frac {2}{e^x+x} \]

[Out]

-7*x+2/(exp(x)+x)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {6873, 6874, 2305} \[ \int \frac {-2-7 e^{2 x}+e^x (-2-14 x)-7 x^2}{e^{2 x}+2 e^x x+x^2} \, dx=\frac {2}{x+e^x}-7 x \]

[In]

Int[(-2 - 7*E^(2*x) + E^x*(-2 - 14*x) - 7*x^2)/(E^(2*x) + 2*E^x*x + x^2),x]

[Out]

-7*x + 2/(E^x + x)

Rule 2305

Int[(x_)^(m_.)*(E^(x_) + (x_)^(m_.))^(n_), x_Symbol] :> Simp[-(E^x + x^m)^(n + 1)/(n + 1), x] + (Dist[m, Int[x
^(m - 1)*(E^x + x^m)^n, x], x] + Int[(E^x + x^m)^(n + 1), x]) /; RationalQ[m, n] && GtQ[m, 0] && LtQ[n, 0] &&
NeQ[n, -1]

Rule 6873

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-2-7 e^{2 x}+e^x (-2-14 x)-7 x^2}{\left (e^x+x\right )^2} \, dx \\ & = \int \left (-7+\frac {2 (-1+x)}{\left (e^x+x\right )^2}-\frac {2}{e^x+x}\right ) \, dx \\ & = -7 x+2 \int \frac {-1+x}{\left (e^x+x\right )^2} \, dx-2 \int \frac {1}{e^x+x} \, dx \\ & = -7 x-2 \int \frac {1}{e^x+x} \, dx+2 \int \left (-\frac {1}{\left (e^x+x\right )^2}+\frac {x}{\left (e^x+x\right )^2}\right ) \, dx \\ & = -7 x-2 \int \frac {1}{\left (e^x+x\right )^2} \, dx+2 \int \frac {x}{\left (e^x+x\right )^2} \, dx-2 \int \frac {1}{e^x+x} \, dx \\ & = -7 x+\frac {2}{e^x+x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.69 \[ \int \frac {-2-7 e^{2 x}+e^x (-2-14 x)-7 x^2}{e^{2 x}+2 e^x x+x^2} \, dx=-\frac {-2+7 e^x x+7 x^2}{e^x+x} \]

[In]

Integrate[(-2 - 7*E^(2*x) + E^x*(-2 - 14*x) - 7*x^2)/(E^(2*x) + 2*E^x*x + x^2),x]

[Out]

-((-2 + 7*E^x*x + 7*x^2)/(E^x + x))

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00

method	result	size
risch	\(-7 x +\frac {2}{{\mathrm e}^{x}+x}\)	\(13\)
norman	\(\frac {2-7 x^{2}-7 \,{\mathrm e}^{x} x}{{\mathrm e}^{x}+x}\)	\(20\)
parallelrisch	\(-\frac {7 \,{\mathrm e}^{x} x +7 x^{2}-2}{{\mathrm e}^{x}+x}\)	\(21\)

[In]

int((-7*exp(x)^2+(-14*x-2)*exp(x)-7*x^2-2)/(exp(x)^2+2*exp(x)*x+x^2),x,method=_RETURNVERBOSE)

[Out]

-7*x+2/(exp(x)+x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.54 \[ \int \frac {-2-7 e^{2 x}+e^x (-2-14 x)-7 x^2}{e^{2 x}+2 e^x x+x^2} \, dx=-\frac {7 \, x^{2} + 7 \, x e^{x} - 2}{x + e^{x}} \]

[In]

integrate((-7*exp(x)^2+(-14*x-2)*exp(x)-7*x^2-2)/(exp(x)^2+2*exp(x)*x+x^2),x, algorithm="fricas")

[Out]

-(7*x^2 + 7*x*e^x - 2)/(x + e^x)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.62 \[ \int \frac {-2-7 e^{2 x}+e^x (-2-14 x)-7 x^2}{e^{2 x}+2 e^x x+x^2} \, dx=- 7 x + \frac {2}{x + e^{x}} \]

[In]

integrate((-7*exp(x)**2+(-14*x-2)*exp(x)-7*x**2-2)/(exp(x)**2+2*exp(x)*x+x**2),x)

[Out]

-7*x + 2/(x + exp(x))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.54 \[ \int \frac {-2-7 e^{2 x}+e^x (-2-14 x)-7 x^2}{e^{2 x}+2 e^x x+x^2} \, dx=-\frac {7 \, x^{2} + 7 \, x e^{x} - 2}{x + e^{x}} \]

[In]

integrate((-7*exp(x)^2+(-14*x-2)*exp(x)-7*x^2-2)/(exp(x)^2+2*exp(x)*x+x^2),x, algorithm="maxima")

[Out]

-(7*x^2 + 7*x*e^x - 2)/(x + e^x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.54 \[ \int \frac {-2-7 e^{2 x}+e^x (-2-14 x)-7 x^2}{e^{2 x}+2 e^x x+x^2} \, dx=-\frac {7 \, x^{2} + 7 \, x e^{x} - 2}{x + e^{x}} \]

[In]

integrate((-7*exp(x)^2+(-14*x-2)*exp(x)-7*x^2-2)/(exp(x)^2+2*exp(x)*x+x^2),x, algorithm="giac")

[Out]

-(7*x^2 + 7*x*e^x - 2)/(x + e^x)

Mupad [B] (verification not implemented)

Time = 12.65 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {-2-7 e^{2 x}+e^x (-2-14 x)-7 x^2}{e^{2 x}+2 e^x x+x^2} \, dx=\frac {2}{x+{\mathrm {e}}^x}-7\,x \]

[In]

int(-(7*exp(2*x) + exp(x)*(14*x + 2) + 7*x^2 + 2)/(exp(2*x) + 2*x*exp(x) + x^2),x)

[Out]

2/(x + exp(x)) - 7*x