3.53.13 \(\int 64 e^{4+2 x^2} x \log ^2(4) \, dx\)

Optimal. Leaf size=15 \[ 16 e^{4+2 x^2} \log ^2(4) \]

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Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {12, 2209} \begin {gather*} 16 e^{2 x^2+4} \log ^2(4) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[64*E^(4 + 2*x^2)*x*Log[4]^2,x]

[Out]

16*E^(4 + 2*x^2)*Log[4]^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 (64 \log ^2(4)\right ) \int e^{4+2 x^2} x \, dx\\ &=16 e^{4+2 x^2} \log ^2(4)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} 16 e^{4+2 x^2} \log ^2(4) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[64*E^(4 + 2*x^2)*x*Log[4]^2,x]

[Out]

16*E^(4 + 2*x^2)*Log[4]^2

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fricas [A]  time = 0.66, size = 14, normalized size = 0.93 \begin {gather*} 64 \, e^{\left (2 \, x^{2} + 4\right )} \log \relax (2)^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(256*x*exp(2)^2*log(2)^2*exp(x^2)^2,x, algorithm="fricas")

[Out]

64*e^(2*x^2 + 4)*log(2)^2

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giac [A]  time = 0.33, size = 14, normalized size = 0.93 \begin {gather*} 64 \, e^{\left (2 \, x^{2} + 4\right )} \log \relax (2)^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(256*x*exp(2)^2*log(2)^2*exp(x^2)^2,x, algorithm="giac")

[Out]

64*e^(2*x^2 + 4)*log(2)^2

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maple [A]  time = 0.07, size = 15, normalized size = 1.00

method	result	size
risch	\(64 \ln \relax (2)^{2} {\mathrm e}^{2 x^{2}+4}\)	\(15\)
gosper	\(64 \,{\mathrm e}^{4} {\mathrm e}^{2 x^{2}} \ln \relax (2)^{2}\)	\(17\)
derivativedivides	\(64 \,{\mathrm e}^{4} {\mathrm e}^{2 x^{2}} \ln \relax (2)^{2}\)	\(17\)
default	\(64 \,{\mathrm e}^{4} {\mathrm e}^{2 x^{2}} \ln \relax (2)^{2}\)	\(17\)
norman	\(64 \,{\mathrm e}^{4} {\mathrm e}^{2 x^{2}} \ln \relax (2)^{2}\)	\(17\)
meijerg	\(-64 \,{\mathrm e}^{4} \ln \relax (2)^{2} \left (1-{\mathrm e}^{2 x^{2}}\right )\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(256*x*exp(2)^2*ln(2)^2*exp(x^2)^2,x,method=_RETURNVERBOSE)

[Out]

64*ln(2)^2*exp(2*x^2+4)

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maxima [A]  time = 0.35, size = 14, normalized size = 0.93 \begin {gather*} 64 \, e^{\left (2 \, x^{2} + 4\right )} \log \relax (2)^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(256*x*exp(2)^2*log(2)^2*exp(x^2)^2,x, algorithm="maxima")

[Out]

64*e^(2*x^2 + 4)*log(2)^2

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mupad [B]  time = 0.02, size = 14, normalized size = 0.93 \begin {gather*} 64\,{\mathrm {e}}^4\,{\mathrm {e}}^{2\,x^2}\,{\ln \relax (2)}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(256*x*exp(4)*exp(2*x^2)*log(2)^2,x)

[Out]

64*exp(4)*exp(2*x^2)*log(2)^2

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sympy [A]  time = 0.09, size = 15, normalized size = 1.00 \begin {gather*} 64 e^{4} e^{2 x^{2}} \log {\relax (2 )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(256*x*exp(2)**2*ln(2)**2*exp(x**2)**2,x)

[Out]

64*exp(4)*exp(2*x**2)*log(2)**2

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