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

Optimal. Leaf size=24 \[ -1-\frac {3}{e^5}+\frac {2}{-e^x+e^{4 x^2}} \]

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Rubi [A]  time = 0.14, antiderivative size = 17, normalized size of antiderivative = 0.71, number of steps used = 3, number of rules used = 3, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {6688, 12, 6686} \begin {gather*} -\frac {2}{e^x-e^{4 x^2}} \end {gather*}

Antiderivative was successfully verified.

[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 6686

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 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 17, normalized size = 0.71 \begin {gather*} -\frac {2}{e^x-e^{4 x^2}} \end {gather*}

Antiderivative was successfully verified.

[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))

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fricas [A]  time = 0.87, size = 27, normalized size = 1.12 \begin {gather*} \frac {2 \, e^{\left (4 \, x^{2}\right )}}{e^{\left (8 \, x^{2}\right )} - e^{\left (4 \, x^{2} + x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 0.32, size = 15, normalized size = 0.62 \begin {gather*} \frac {2}{e^{\left (4 \, x^{2}\right )} - e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.04, size = 16, normalized size = 0.67

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

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maxima [A]  time = 0.39, size = 15, normalized size = 0.62 \begin {gather*} \frac {2}{e^{\left (4 \, x^{2}\right )} - e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 6.71, size = 15, normalized size = 0.62 \begin {gather*} \frac {2}{{\mathrm {e}}^{4\,x^2}-{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 0.12, size = 10, normalized size = 0.42 \begin {gather*} \frac {2}{- e^{x} + e^{4 x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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