3.45.84 \(\int e^{-10-x^2} (e^{4+x^2} (-e^6-e^{6+x})-2 x \log (4)) \, dx\)

Optimal. Leaf size=21 \[ -e^x-x+e^{-10-x^2} \log (4) \]

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Rubi [A]  time = 0.09, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {6688, 2194, 2209} \begin {gather*} e^{-x^2-10} \log (4)-x-e^x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(-10 - x^2)*(E^(4 + x^2)*(-E^6 - E^(6 + x)) - 2*x*Log[4]),x]

[Out]

-E^x - x + E^(-10 - x^2)*Log[4]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

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 \left (-1-e^x-2 e^{-10-x^2} x \log (4)\right ) \, dx\\ &=-x-(2 \log (4)) \int e^{-10-x^2} x \, dx-\int e^x \, dx\\ &=-e^x-x+e^{-10-x^2} \log (4)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 21, normalized size = 1.00 \begin {gather*} -e^x-x+e^{-10-x^2} \log (4) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-10 - x^2)*(E^(4 + x^2)*(-E^6 - E^(6 + x)) - 2*x*Log[4]),x]

[Out]

-E^x - x + E^(-10 - x^2)*Log[4]

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fricas [A]  time = 0.50, size = 31, normalized size = 1.48 \begin {gather*} -{\left ({\left (x e^{6} + e^{\left (x + 6\right )}\right )} e^{\left (x^{2} + 4\right )} - 2 \, \log \relax (2)\right )} e^{\left (-x^{2} - 10\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((x*e^6 + e^(x + 6))*e^(x^2 + 4) - 2*log(2))*e^(-x^2 - 10)

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giac [A]  time = 0.14, size = 23, normalized size = 1.10 \begin {gather*} -{\left (x e^{10} - 2 \, e^{\left (-x^{2}\right )} \log \relax (2) + e^{\left (x + 10\right )}\right )} e^{\left (-10\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x*e^10 - 2*e^(-x^2)*log(2) + e^(x + 10))*e^(-10)

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maple [A]  time = 0.04, size = 21, normalized size = 1.00

method	result	size
risch	\(-x -{\mathrm e}^{x}+2 \ln \relax (2) {\mathrm e}^{-x^{2}-10}\)	\(21\)
default	\({\mathrm e}^{-6} \left (-{\mathrm e}^{6} {\mathrm e}^{x}+2 \,{\mathrm e}^{-4} \ln \relax (2) {\mathrm e}^{-x^{2}}-x \,{\mathrm e}^{6}\right )\)	\(32\)
norman	\(\left (-x \,{\mathrm e}^{x^{2}+4}+2 \,{\mathrm e}^{-6} \ln \relax (2)-{\mathrm e}^{x} {\mathrm e}^{x^{2}+4}\right ) {\mathrm e}^{-x^{2}-4}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x-exp(x)+2*ln(2)*exp(-x^2-10)

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maxima [A]  time = 0.37, size = 20, normalized size = 0.95 \begin {gather*} 2 \, e^{\left (-x^{2} - 10\right )} \log \relax (2) - x - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e^(-x^2 - 10)*log(2) - x - e^x

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mupad [B]  time = 0.11, size = 20, normalized size = 0.95 \begin {gather*} 2\,{\mathrm {e}}^{-10}\,{\mathrm {e}}^{-x^2}\,\ln \relax (2)-{\mathrm {e}}^x-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-6)*exp(- x^2 - 4)*(4*x*log(2) + exp(x^2 + 4)*(exp(6) + exp(6)*exp(x))),x)

[Out]

2*exp(-10)*exp(-x^2)*log(2) - exp(x) - x

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sympy [A]  time = 0.16, size = 20, normalized size = 0.95 \begin {gather*} - x - e^{x} + \frac {2 e^{- x^{2} - 4} \log {\relax (2 )}}{e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp(6)*exp(x)-exp(6))*exp(x**2+4)-4*x*ln(2))/exp(6)/exp(x**2+4),x)

[Out]

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

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