3.92.46 \(\int -2 5^{2-x^2} x \log (5) \, dx\)

Optimal. Leaf size=9 \[ 5^{2-x^2} \]

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Rubi [A]  time = 0.01, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 2209} \begin {gather*} 5^{2-x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-2*5^(2 - x^2)*x*Log[5],x]

[Out]

5^(2 - 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 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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\left ((2 \log (5)) \int 5^{2-x^2} x \, dx\right )\\ &=5^{2-x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 9, normalized size = 1.00 \begin {gather*} 5^{2-x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-2*5^(2 - x^2)*x*Log[5],x]

[Out]

5^(2 - x^2)

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fricas [A]  time = 0.59, size = 9, normalized size = 1.00 \begin {gather*} 5^{-x^{2} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5^(-x^2 + 2)

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giac [A]  time = 0.14, size = 9, normalized size = 1.00 \begin {gather*} 5^{-x^{2} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5^(-x^2 + 2)

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maple [A]  time = 0.04, size = 10, normalized size = 1.11

method	result	size
risch	\(5^{-x^{2}+2}\)	\(10\)
gosper	\({\mathrm e}^{-\left (x^{2}-2\right ) \ln \relax (5)}\)	\(11\)
derivativedivides	\({\mathrm e}^{\left (-x^{2}+2\right ) \ln \relax (5)}\)	\(12\)
default	\({\mathrm e}^{\left (-x^{2}+2\right ) \ln \relax (5)}\)	\(12\)
norman	\({\mathrm e}^{\left (-x^{2}+2\right ) \ln \relax (5)}\)	\(12\)
meijerg	\(-25+25 \,{\mathrm e}^{-x^{2} \ln \relax (5)}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-2*x*ln(5)*exp((-x^2+2)*ln(5)),x,method=_RETURNVERBOSE)

[Out]

5^(-x^2+2)

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maxima [A]  time = 0.36, size = 9, normalized size = 1.00 \begin {gather*} 5^{-x^{2} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5^(-x^2 + 2)

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mupad [B]  time = 5.16, size = 9, normalized size = 1.00 \begin {gather*} \frac {25}{5^{x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/5^(x^2)

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sympy [A]  time = 0.10, size = 8, normalized size = 0.89 \begin {gather*} e^{\left (2 - x^{2}\right ) \log {\relax (5 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2*x*ln(5)*exp((-x**2+2)*ln(5)),x)

[Out]

exp((2 - x**2)*log(5))

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