\(\int \frac {e^{-2 x-4 x^2-4 x^4-x^6-\log ^2(x)} (-1-2 x-8 x^2-16 x^4-6 x^6-2 \log (x))}{x^2} \, dx\) [4311]

Optimal result

Integrand size = 55, antiderivative size = 28 \[ \int \frac {e^{-2 x-4 x^2-4 x^4-x^6-\log ^2(x)} \left (-1-2 x-8 x^2-16 x^4-6 x^6-2 \log (x)\right )}{x^2} \, dx=\frac {e^{x-x \left (3+x \left (2+x^2\right )^2\right )-\log ^2(x)}}{x} \]

[Out]

exp(x-ln(x)^2-((x^2+2)^2*x+3)*x)/x

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(73\) vs. \(2(28)=56\).

Time = 0.15 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.61, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {2326} \[ \int \frac {e^{-2 x-4 x^2-4 x^4-x^6-\log ^2(x)} \left (-1-2 x-8 x^2-16 x^4-6 x^6-2 \log (x)\right )}{x^2} \, dx=\frac {e^{-x^6-4 x^4-4 x^2-2 x-\log ^2(x)} \left (3 x^6+8 x^4+4 x^2+x+\log (x)\right )}{x^2 \left (3 x^5+8 x^3+4 x+\frac {\log (x)}{x}+1\right )} \]

[In]

Int[(E^(-2*x - 4*x^2 - 4*x^4 - x^6 - Log[x]^2)*(-1 - 2*x - 8*x^2 - 16*x^4 - 6*x^6 - 2*Log[x]))/x^2,x]

[Out]

(E^(-2*x - 4*x^2 - 4*x^4 - x^6 - Log[x]^2)*(x + 4*x^2 + 8*x^4 + 3*x^6 + Log[x]))/(x^2*(1 + 4*x + 8*x^3 + 3*x^5
+ Log[x]/x))

Rule 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = \frac {e^{-2 x-4 x^2-4 x^4-x^6-\log ^2(x)} \left (x+4 x^2+8 x^4+3 x^6+\log (x)\right )}{x^2 \left (1+4 x+8 x^3+3 x^5+\frac {\log (x)}{x}\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {e^{-2 x-4 x^2-4 x^4-x^6-\log ^2(x)} \left (-1-2 x-8 x^2-16 x^4-6 x^6-2 \log (x)\right )}{x^2} \, dx=\frac {e^{-x \left (2+4 x+4 x^3+x^5\right )-\log ^2(x)}}{x} \]

[In]

Integrate[(E^(-2*x - 4*x^2 - 4*x^4 - x^6 - Log[x]^2)*(-1 - 2*x - 8*x^2 - 16*x^4 - 6*x^6 - 2*Log[x]))/x^2,x]

[Out]

E^(-(x*(2 + 4*x + 4*x^3 + x^5)) - Log[x]^2)/x

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11

method	result	size
risch	\(\frac {{\mathrm e}^{-\ln \left (x \right )^{2}-x^{6}-4 x^{4}-4 x^{2}-2 x}}{x}\)	\(31\)
parallelrisch	\(\frac {{\mathrm e}^{-\ln \left (x \right )^{2}-x^{6}-4 x^{4}-4 x^{2}-2 x}}{x}\)	\(31\)

[In]

int((-2*ln(x)-6*x^6-16*x^4-8*x^2-2*x-1)*exp(-ln(x)^2-x^6-4*x^4-4*x^2-2*x)/x^2,x,method=_RETURNVERBOSE)

[Out]

exp(-ln(x)^2-x^6-4*x^4-4*x^2-2*x)/x

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-2 x-4 x^2-4 x^4-x^6-\log ^2(x)} \left (-1-2 x-8 x^2-16 x^4-6 x^6-2 \log (x)\right )}{x^2} \, dx=\frac {e^{\left (-x^{6} - 4 \, x^{4} - 4 \, x^{2} - \log \left (x\right )^{2} - 2 \, x\right )}}{x} \]

[In]

integrate((-2*log(x)-6*x^6-16*x^4-8*x^2-2*x-1)*exp(-log(x)^2-x^6-4*x^4-4*x^2-2*x)/x^2,x, algorithm="fricas")

[Out]

e^(-x^6 - 4*x^4 - 4*x^2 - log(x)^2 - 2*x)/x

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {e^{-2 x-4 x^2-4 x^4-x^6-\log ^2(x)} \left (-1-2 x-8 x^2-16 x^4-6 x^6-2 \log (x)\right )}{x^2} \, dx=\frac {e^{- x^{6} - 4 x^{4} - 4 x^{2} - 2 x - \log {\left (x \right )}^{2}}}{x} \]

[In]

integrate((-2*ln(x)-6*x**6-16*x**4-8*x**2-2*x-1)*exp(-ln(x)**2-x**6-4*x**4-4*x**2-2*x)/x**2,x)

[Out]

exp(-x**6 - 4*x**4 - 4*x**2 - 2*x - log(x)**2)/x

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-2 x-4 x^2-4 x^4-x^6-\log ^2(x)} \left (-1-2 x-8 x^2-16 x^4-6 x^6-2 \log (x)\right )}{x^2} \, dx=\frac {e^{\left (-x^{6} - 4 \, x^{4} - 4 \, x^{2} - \log \left (x\right )^{2} - 2 \, x\right )}}{x} \]

[In]

integrate((-2*log(x)-6*x^6-16*x^4-8*x^2-2*x-1)*exp(-log(x)^2-x^6-4*x^4-4*x^2-2*x)/x^2,x, algorithm="maxima")

[Out]

e^(-x^6 - 4*x^4 - 4*x^2 - log(x)^2 - 2*x)/x

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-2 x-4 x^2-4 x^4-x^6-\log ^2(x)} \left (-1-2 x-8 x^2-16 x^4-6 x^6-2 \log (x)\right )}{x^2} \, dx=\frac {e^{\left (-x^{6} - 4 \, x^{4} - 4 \, x^{2} - \log \left (x\right )^{2} - 2 \, x\right )}}{x} \]

[In]

integrate((-2*log(x)-6*x^6-16*x^4-8*x^2-2*x-1)*exp(-log(x)^2-x^6-4*x^4-4*x^2-2*x)/x^2,x, algorithm="giac")

[Out]

e^(-x^6 - 4*x^4 - 4*x^2 - log(x)^2 - 2*x)/x

Mupad [B] (verification not implemented)

Time = 10.67 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int \frac {e^{-2 x-4 x^2-4 x^4-x^6-\log ^2(x)} \left (-1-2 x-8 x^2-16 x^4-6 x^6-2 \log (x)\right )}{x^2} \, dx=\frac {{\mathrm {e}}^{-{\ln \left (x\right )}^2}\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{-4\,x^2}\,{\mathrm {e}}^{-x^6}\,{\mathrm {e}}^{-4\,x^4}}{x} \]

[In]

int(-(exp(- 2*x - log(x)^2 - 4*x^2 - 4*x^4 - x^6)*(2*x + 2*log(x) + 8*x^2 + 16*x^4 + 6*x^6 + 1))/x^2,x)

[Out]

(exp(-log(x)^2)*exp(-2*x)*exp(-4*x^2)*exp(-x^6)*exp(-4*x^4))/x