3.95.54 \(\int e^{-6-x-x^2} (2 e^{6+x+x^2} x+(2+7 x+2 x^2) \log (\log (5))) \, dx\)

Optimal. Leaf size=24 \[ x^2-e^{-6-x-x^2} (3+x) \log (\log (5)) \]

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Rubi [A]  time = 0.27, antiderivative size = 39, normalized size of antiderivative = 1.62, number of steps used = 15, number of rules used = 5, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {6742, 2234, 2205, 2240, 2241} \begin {gather*} x^2-e^{-x^2-x-6} x \log (\log (5))-3 e^{-x^2-x-6} \log (\log (5)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(-6 - x - x^2)*(2*E^(6 + x + x^2)*x + (2 + 7*x + 2*x^2)*Log[Log[5]]),x]

[Out]

x^2 - 3*E^(-6 - x - x^2)*Log[Log[5]] - E^(-6 - x - x^2)*x*Log[Log[5]]

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 2234

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 2240

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] - Dist[(b*e - 2*c*d)/(2*c), Int[F^(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e}, x] &&
NeQ[b*e - 2*c*d, 0]

Rule 2241

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_))^(m_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*F^(a + b*x + c*x^2))/(2*c*Log[F]), x] + (-Dist[(b*e - 2*c*d)/(2*c), Int[(d + e*x)^(m - 1)*F^(a + b*x + c*x^2)
, x], x] - Dist[((m - 1)*e^2)/(2*c*Log[F]), Int[(d + e*x)^(m - 2)*F^(a + b*x + c*x^2), x], x]) /; FreeQ[{F, a,
b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && GtQ[m, 1]

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2 x+e^{-6-x-x^2} \left (2+7 x+2 x^2\right ) \log (\log (5))\right ) \, dx\\ &=x^2+\log (\log (5)) \int e^{-6-x-x^2} \left (2+7 x+2 x^2\right ) \, dx\\ &=x^2+\log (\log (5)) \int \left (2 e^{-6-x-x^2}+7 e^{-6-x-x^2} x+2 e^{-6-x-x^2} x^2\right ) \, dx\\ &=x^2+(2 \log (\log (5))) \int e^{-6-x-x^2} \, dx+(2 \log (\log (5))) \int e^{-6-x-x^2} x^2 \, dx+(7 \log (\log (5))) \int e^{-6-x-x^2} x \, dx\\ &=x^2-\frac {7}{2} e^{-6-x-x^2} \log (\log (5))-e^{-6-x-x^2} x \log (\log (5))+\log (\log (5)) \int e^{-6-x-x^2} \, dx-\log (\log (5)) \int e^{-6-x-x^2} x \, dx-\frac {1}{2} (7 \log (\log (5))) \int e^{-6-x-x^2} \, dx+\frac {(2 \log (\log (5))) \int e^{-\frac {1}{4} (-1-2 x)^2} \, dx}{e^{23/4}}\\ &=x^2-3 e^{-6-x-x^2} \log (\log (5))-e^{-6-x-x^2} x \log (\log (5))-\frac {\sqrt {\pi } \text {erf}\left (\frac {1}{2} (-1-2 x)\right ) \log (\log (5))}{e^{23/4}}+\frac {1}{2} \log (\log (5)) \int e^{-6-x-x^2} \, dx+\frac {\log (\log (5)) \int e^{-\frac {1}{4} (-1-2 x)^2} \, dx}{e^{23/4}}-\frac {(7 \log (\log (5))) \int e^{-\frac {1}{4} (-1-2 x)^2} \, dx}{2 e^{23/4}}\\ &=x^2-3 e^{-6-x-x^2} \log (\log (5))-e^{-6-x-x^2} x \log (\log (5))+\frac {\sqrt {\pi } \text {erf}\left (\frac {1}{2} (-1-2 x)\right ) \log (\log (5))}{4 e^{23/4}}+\frac {\log (\log (5)) \int e^{-\frac {1}{4} (-1-2 x)^2} \, dx}{2 e^{23/4}}\\ &=x^2-3 e^{-6-x-x^2} \log (\log (5))-e^{-6-x-x^2} x \log (\log (5))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 31, normalized size = 1.29 \begin {gather*} x^2+e^{-x-x^2} \left (-\frac {3}{e^6}-\frac {x}{e^6}\right ) \log (\log (5)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-6 - x - x^2)*(2*E^(6 + x + x^2)*x + (2 + 7*x + 2*x^2)*Log[Log[5]]),x]

[Out]

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

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fricas [A]  time = 0.57, size = 32, normalized size = 1.33 \begin {gather*} {\left (x^{2} e^{\left (x^{2} + x + 6\right )} - {\left (x + 3\right )} \log \left (\log \relax (5)\right )\right )} e^{\left (-x^{2} - x - 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+7*x+2)*log(log(5))+2*x*exp(x)*exp(x^2+6))/exp(x)/exp(x^2+6),x, algorithm="fricas")

[Out]

(x^2*e^(x^2 + x + 6) - (x + 3)*log(log(5)))*e^(-x^2 - x - 6)

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giac [A]  time = 0.14, size = 32, normalized size = 1.33 \begin {gather*} x^{2} - \frac {1}{2} \, {\left ({\left (2 \, x + 1\right )} \log \left (\log \relax (5)\right ) + 5 \, \log \left (\log \relax (5)\right )\right )} e^{\left (-x^{2} - x - 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+7*x+2)*log(log(5))+2*x*exp(x)*exp(x^2+6))/exp(x)/exp(x^2+6),x, algorithm="giac")

[Out]

x^2 - 1/2*((2*x + 1)*log(log(5)) + 5*log(log(5)))*e^(-x^2 - x - 6)

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maple [A]  time = 0.07, size = 24, normalized size = 1.00

method	result	size
risch	\(x^{2}-\ln \left (\ln \relax (5)\right ) \left (3+x \right ) {\mathrm e}^{-x^{2}-x -6}\)	\(24\)
default	\(x^{2}-3 \ln \left (\ln \relax (5)\right ) {\mathrm e}^{-x^{2}-x -6}-x \ln \left (\ln \relax (5)\right ) {\mathrm e}^{-x^{2}-x -6}\)	\(38\)
norman	\(\left (x^{2} {\mathrm e}^{x} {\mathrm e}^{x^{2}+6}-x \ln \left (\ln \relax (5)\right )-3 \ln \left (\ln \relax (5)\right )\right ) {\mathrm e}^{-x} {\mathrm e}^{-x^{2}-6}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2+7*x+2)*ln(ln(5))+2*x*exp(x)*exp(x^2+6))/exp(x)/exp(x^2+6),x,method=_RETURNVERBOSE)

[Out]

x^2-ln(ln(5))*(3+x)*exp(-x^2-x-6)

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maxima [C]  time = 0.41, size = 152, normalized size = 6.33 \begin {gather*} \sqrt {\pi } \operatorname {erf}\left (x + \frac {1}{2}\right ) e^{\left (-\frac {23}{4}\right )} \log \left (\log \relax (5)\right ) - \frac {1}{4} i \, {\left (-\frac {4 i \, {\left (2 \, x + 1\right )}^{3} \Gamma \left (\frac {3}{2}, \frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}{{\left ({\left (2 \, x + 1\right )}^{2}\right )}^{\frac {3}{2}}} + \frac {i \, \sqrt {\pi } {\left (2 \, x + 1\right )} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {{\left (2 \, x + 1\right )}^{2}}\right ) - 1\right )}}{\sqrt {{\left (2 \, x + 1\right )}^{2}}} + 4 i \, e^{\left (-\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}\right )} e^{\left (-\frac {23}{4}\right )} \log \left (\log \relax (5)\right ) - \frac {7}{4} i \, {\left (-\frac {i \, \sqrt {\pi } {\left (2 \, x + 1\right )} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {{\left (2 \, x + 1\right )}^{2}}\right ) - 1\right )}}{\sqrt {{\left (2 \, x + 1\right )}^{2}}} - 2 i \, e^{\left (-\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}\right )} e^{\left (-\frac {23}{4}\right )} \log \left (\log \relax (5)\right ) + x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+7*x+2)*log(log(5))+2*x*exp(x)*exp(x^2+6))/exp(x)/exp(x^2+6),x, algorithm="maxima")

[Out]

sqrt(pi)*erf(x + 1/2)*e^(-23/4)*log(log(5)) - 1/4*I*(-4*I*(2*x + 1)^3*gamma(3/2, 1/4*(2*x + 1)^2)/((2*x + 1)^2
)^(3/2) + I*sqrt(pi)*(2*x + 1)*(erf(1/2*sqrt((2*x + 1)^2)) - 1)/sqrt((2*x + 1)^2) + 4*I*e^(-1/4*(2*x + 1)^2))*
e^(-23/4)*log(log(5)) - 7/4*I*(-I*sqrt(pi)*(2*x + 1)*(erf(1/2*sqrt((2*x + 1)^2)) - 1)/sqrt((2*x + 1)^2) - 2*I*
e^(-1/4*(2*x + 1)^2))*e^(-23/4)*log(log(5)) + x^2

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mupad [B]  time = 0.18, size = 39, normalized size = 1.62 \begin {gather*} x^2-3\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-6}\,{\mathrm {e}}^{-x^2}\,\ln \left (\ln \relax (5)\right )-x\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-6}\,{\mathrm {e}}^{-x^2}\,\ln \left (\ln \relax (5)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-x)*exp(- x^2 - 6)*(log(log(5))*(7*x + 2*x^2 + 2) + 2*x*exp(x^2 + 6)*exp(x)),x)

[Out]

x^2 - 3*exp(-x)*exp(-6)*exp(-x^2)*log(log(5)) - x*exp(-x)*exp(-6)*exp(-x^2)*log(log(5))

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sympy [A]  time = 0.23, size = 32, normalized size = 1.33 \begin {gather*} x^{2} + \left (- x e^{- x} \log {\left (\log {\relax (5 )} \right )} - 3 e^{- x} \log {\left (\log {\relax (5 )} \right )}\right ) e^{- x^{2} - 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2+7*x+2)*ln(ln(5))+2*x*exp(x)*exp(x**2+6))/exp(x)/exp(x**2+6),x)

[Out]

x**2 + (-x*exp(-x)*log(log(5)) - 3*exp(-x)*log(log(5)))*exp(-x**2 - 6)

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