3.57.32 \(\int \frac {e^{\frac {4 x}{12+x^2 \log (3)}} (12-x^2 \log (3))}{1152+192 x^2 \log (3)+8 x^4 \log ^2(3)} \, dx\)

Optimal. Leaf size=21 \[ \frac {1}{32} e^{\frac {x}{3+\frac {1}{4} x^2 \log (3)}} \]

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Rubi [A]  time = 0.25, antiderivative size = 19, normalized size of antiderivative = 0.90, number of steps used = 2, number of rules used = 2, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {28, 6706} \begin {gather*} \frac {1}{32} e^{\frac {4 x}{x^2 \log (3)+12}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((4*x)/(12 + x^2*Log[3]))*(12 - x^2*Log[3]))/(1152 + 192*x^2*Log[3] + 8*x^4*Log[3]^2),x]

[Out]

E^((4*x)/(12 + x^2*Log[3]))/32

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\left (8 \log ^2(3)\right ) \int \frac {e^{\frac {4 x}{12+x^2 \log (3)}} \left (12-x^2 \log (3)\right )}{\left (96 \log (3)+8 x^2 \log ^2(3)\right )^2} \, dx\\ &=\frac {1}{32} e^{\frac {4 x}{12+x^2 \log (3)}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.44, size = 19, normalized size = 0.90 \begin {gather*} \frac {1}{32} e^{\frac {4 x}{12+x^2 \log (3)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((4*x)/(12 + x^2*Log[3]))*(12 - x^2*Log[3]))/(1152 + 192*x^2*Log[3] + 8*x^4*Log[3]^2),x]

[Out]

E^((4*x)/(12 + x^2*Log[3]))/32

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fricas [A]  time = 0.52, size = 16, normalized size = 0.76 \begin {gather*} \frac {1}{32} \, e^{\left (\frac {4 \, x}{x^{2} \log \relax (3) + 12}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2*log(3)+12)*exp(4*x/(x^2*log(3)+12))/(8*x^4*log(3)^2+192*x^2*log(3)+1152),x, algorithm="fricas"
)

[Out]

1/32*e^(4*x/(x^2*log(3) + 12))

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giac [A]  time = 0.18, size = 16, normalized size = 0.76 \begin {gather*} \frac {1}{32} \, e^{\left (\frac {4 \, x}{x^{2} \log \relax (3) + 12}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2*log(3)+12)*exp(4*x/(x^2*log(3)+12))/(8*x^4*log(3)^2+192*x^2*log(3)+1152),x, algorithm="giac")

[Out]

1/32*e^(4*x/(x^2*log(3) + 12))

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maple [A]  time = 0.25, size = 17, normalized size = 0.81

method	result	size
gosper	\(\frac {{\mathrm e}^{\frac {4 x}{x^{2} \ln \relax (3)+12}}}{32}\)	\(17\)
risch	\(\frac {{\mathrm e}^{\frac {4 x}{x^{2} \ln \relax (3)+12}}}{32}\)	\(17\)
norman	\(\frac {\frac {x^{2} \ln \relax (3) {\mathrm e}^{\frac {4 x}{x^{2} \ln \relax (3)+12}}}{32}+\frac {3 \,{\mathrm e}^{\frac {4 x}{x^{2} \ln \relax (3)+12}}}{8}}{x^{2} \ln \relax (3)+12}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^2*ln(3)+12)*exp(4*x/(x^2*ln(3)+12))/(8*x^4*ln(3)^2+192*x^2*ln(3)+1152),x,method=_RETURNVERBOSE)

[Out]

1/32*exp(4*x/(x^2*ln(3)+12))

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maxima [A]  time = 0.55, size = 16, normalized size = 0.76 \begin {gather*} \frac {1}{32} \, e^{\left (\frac {4 \, x}{x^{2} \log \relax (3) + 12}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2*log(3)+12)*exp(4*x/(x^2*log(3)+12))/(8*x^4*log(3)^2+192*x^2*log(3)+1152),x, algorithm="maxima"
)

[Out]

1/32*e^(4*x/(x^2*log(3) + 12))

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mupad [B]  time = 0.30, size = 16, normalized size = 0.76 \begin {gather*} \frac {{\mathrm {e}}^{\frac {4\,x}{\ln \relax (3)\,x^2+12}}}{32} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((4*x)/(x^2*log(3) + 12))*(x^2*log(3) - 12))/(8*x^4*log(3)^2 + 192*x^2*log(3) + 1152),x)

[Out]

exp((4*x)/(x^2*log(3) + 12))/32

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sympy [A]  time = 0.25, size = 14, normalized size = 0.67 \begin {gather*} \frac {e^{\frac {4 x}{x^{2} \log {\relax (3 )} + 12}}}{32} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**2*ln(3)+12)*exp(4*x/(x**2*ln(3)+12))/(8*x**4*ln(3)**2+192*x**2*ln(3)+1152),x)

[Out]

exp(4*x/(x**2*log(3) + 12))/32

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