\(\int \frac {2 e^x-16 e^{4 x^2} x}{e^{2 x}+e^{8 x^2}-2 e^{x+4 x^2}} \, dx\) [8031]

Optimal result

Integrand size = 43, antiderivative size = 24 \[ \int \frac {2 e^x-16 e^{4 x^2} x}{e^{2 x}+e^{8 x^2}-2 e^{x+4 x^2}} \, dx=-1-\frac {3}{e^5}+\frac {2}{-e^x+e^{4 x^2}} \]

[Out]

2/(exp(4*x^2)-exp(x))-3/exp(5)-1

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {6820, 12, 6818} \[ \int \frac {2 e^x-16 e^{4 x^2} x}{e^{2 x}+e^{8 x^2}-2 e^{x+4 x^2}} \, dx=-\frac {2}{e^x-e^{4 x^2}} \]

[In]

Int[(2*E^x - 16*E^(4*x^2)*x)/(E^(2*x) + E^(8*x^2) - 2*E^(x + 4*x^2)),x]

[Out]

-2/(E^x - E^(4*x^2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6818

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*(y^(m + 1)/(m + 1)), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6820

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {2 \left (e^x-8 e^{4 x^2} x\right )}{\left (e^x-e^{4 x^2}\right )^2} \, dx \\ & = 2 \int \frac {e^x-8 e^{4 x^2} x}{\left (e^x-e^{4 x^2}\right )^2} \, dx \\ & = -\frac {2}{e^x-e^{4 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71 \[ \int \frac {2 e^x-16 e^{4 x^2} x}{e^{2 x}+e^{8 x^2}-2 e^{x+4 x^2}} \, dx=-\frac {2}{e^x-e^{4 x^2}} \]

[In]

Integrate[(2*E^x - 16*E^(4*x^2)*x)/(E^(2*x) + E^(8*x^2) - 2*E^(x + 4*x^2)),x]

[Out]

-2/(E^x - E^(4*x^2))

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67

method	result	size
norman	\(-\frac {2}{{\mathrm e}^{x}-{\mathrm e}^{4 x^{2}}}\)	\(16\)
risch	\(-\frac {2}{{\mathrm e}^{x}-{\mathrm e}^{4 x^{2}}}\)	\(16\)
parallelrisch	\(-\frac {2}{{\mathrm e}^{x}-{\mathrm e}^{4 x^{2}}}\)	\(16\)

[In]

int((-16*x*exp(4*x^2)+2*exp(x))/(exp(4*x^2)^2-2*exp(x)*exp(4*x^2)+exp(x)^2),x,method=_RETURNVERBOSE)

[Out]

-2/(exp(x)-exp(4*x^2))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {2 e^x-16 e^{4 x^2} x}{e^{2 x}+e^{8 x^2}-2 e^{x+4 x^2}} \, dx=\frac {2 \, e^{\left (4 \, x^{2}\right )}}{e^{\left (8 \, x^{2}\right )} - e^{\left (4 \, x^{2} + x\right )}} \]

[In]

integrate((-16*x*exp(4*x^2)+2*exp(x))/(exp(4*x^2)^2-2*exp(x)*exp(4*x^2)+exp(x)^2),x, algorithm="fricas")

[Out]

2*e^(4*x^2)/(e^(8*x^2) - e^(4*x^2 + x))

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.42 \[ \int \frac {2 e^x-16 e^{4 x^2} x}{e^{2 x}+e^{8 x^2}-2 e^{x+4 x^2}} \, dx=\frac {2}{- e^{x} + e^{4 x^{2}}} \]

[In]

integrate((-16*x*exp(4*x**2)+2*exp(x))/(exp(4*x**2)**2-2*exp(x)*exp(4*x**2)+exp(x)**2),x)

[Out]

2/(-exp(x) + exp(4*x**2))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.62 \[ \int \frac {2 e^x-16 e^{4 x^2} x}{e^{2 x}+e^{8 x^2}-2 e^{x+4 x^2}} \, dx=\frac {2}{e^{\left (4 \, x^{2}\right )} - e^{x}} \]

[In]

integrate((-16*x*exp(4*x^2)+2*exp(x))/(exp(4*x^2)^2-2*exp(x)*exp(4*x^2)+exp(x)^2),x, algorithm="maxima")

[Out]

2/(e^(4*x^2) - e^x)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.62 \[ \int \frac {2 e^x-16 e^{4 x^2} x}{e^{2 x}+e^{8 x^2}-2 e^{x+4 x^2}} \, dx=\frac {2}{e^{\left (4 \, x^{2}\right )} - e^{x}} \]

[In]

integrate((-16*x*exp(4*x^2)+2*exp(x))/(exp(4*x^2)^2-2*exp(x)*exp(4*x^2)+exp(x)^2),x, algorithm="giac")

[Out]

2/(e^(4*x^2) - e^x)

Mupad [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.62 \[ \int \frac {2 e^x-16 e^{4 x^2} x}{e^{2 x}+e^{8 x^2}-2 e^{x+4 x^2}} \, dx=\frac {2}{{\mathrm {e}}^{4\,x^2}-{\mathrm {e}}^x} \]

[In]

int((2*exp(x) - 16*x*exp(4*x^2))/(exp(2*x) + exp(8*x^2) - 2*exp(4*x^2)*exp(x)),x)

[Out]

2/(exp(4*x^2) - exp(x))