3.28.7 \(\int \frac {e^{\frac {48-4 x^2+x^3-3 x^4 \log ^2(x)}{x}} (-48-4 x^2+2 x^3-6 x^4 \log (x)-9 x^4 \log ^2(x))}{x^2} \, dx\)

Optimal. Leaf size=24 \[ e^{x \left (-4+x+3 \left (\frac {16}{x^2}-x^2 \log ^2(x)\right )\right )} \]

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Rubi [A]  time = 0.53, antiderivative size = 25, normalized size of antiderivative = 1.04, number of steps used = 1, number of rules used = 1, integrand size = 57, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {6706} \begin {gather*} e^{\frac {-3 x^4 \log ^2(x)+x^3-4 x^2+48}{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((48 - 4*x^2 + x^3 - 3*x^4*Log[x]^2)/x)*(-48 - 4*x^2 + 2*x^3 - 6*x^4*Log[x] - 9*x^4*Log[x]^2))/x^2,x]

[Out]

E^((48 - 4*x^2 + x^3 - 3*x^4*Log[x]^2)/x)

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} &=e^{\frac {48-4 x^2+x^3-3 x^4 \log ^2(x)}{x}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.29, size = 23, normalized size = 0.96 \begin {gather*} e^{\frac {48}{x}-4 x+x^2-3 x^3 \log ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((48 - 4*x^2 + x^3 - 3*x^4*Log[x]^2)/x)*(-48 - 4*x^2 + 2*x^3 - 6*x^4*Log[x] - 9*x^4*Log[x]^2))/x^
2,x]

[Out]

E^(48/x - 4*x + x^2 - 3*x^3*Log[x]^2)

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fricas [A]  time = 0.60, size = 27, normalized size = 1.12 \begin {gather*} e^{\left (-\frac {3 \, x^{4} \log \relax (x)^{2} - x^{3} + 4 \, x^{2} - 48}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*x^4*log(x)^2-6*x^4*log(x)+2*x^3-4*x^2-48)*exp((-3*x^4*log(x)^2+x^3-4*x^2+48)/x)/x^2,x, algorithm
="fricas")

[Out]

e^(-(3*x^4*log(x)^2 - x^3 + 4*x^2 - 48)/x)

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giac [A]  time = 0.26, size = 22, normalized size = 0.92 \begin {gather*} e^{\left (-3 \, x^{3} \log \relax (x)^{2} + x^{2} - 4 \, x + \frac {48}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*x^4*log(x)^2-6*x^4*log(x)+2*x^3-4*x^2-48)*exp((-3*x^4*log(x)^2+x^3-4*x^2+48)/x)/x^2,x, algorithm
="giac")

[Out]

e^(-3*x^3*log(x)^2 + x^2 - 4*x + 48/x)

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maple [A]  time = 0.01, size = 28, normalized size = 1.17

method	result	size
risch	\({\mathrm e}^{-\frac {3 x^{4} \ln \relax (x )^{2}-x^{3}+4 x^{2}-48}{x}}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-9*x^4*ln(x)^2-6*x^4*ln(x)+2*x^3-4*x^2-48)*exp((-3*x^4*ln(x)^2+x^3-4*x^2+48)/x)/x^2,x,method=_RETURNVERBO
SE)

[Out]

exp(-(3*x^4*ln(x)^2-x^3+4*x^2-48)/x)

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maxima [A]  time = 0.70, size = 22, normalized size = 0.92 \begin {gather*} e^{\left (-3 \, x^{3} \log \relax (x)^{2} + x^{2} - 4 \, x + \frac {48}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*x^4*log(x)^2-6*x^4*log(x)+2*x^3-4*x^2-48)*exp((-3*x^4*log(x)^2+x^3-4*x^2+48)/x)/x^2,x, algorithm
="maxima")

[Out]

e^(-3*x^3*log(x)^2 + x^2 - 4*x + 48/x)

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mupad [B]  time = 1.62, size = 25, normalized size = 1.04 \begin {gather*} {\mathrm {e}}^{-4\,x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{48/x}\,{\mathrm {e}}^{-3\,x^3\,{\ln \relax (x)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(3*x^4*log(x)^2 + 4*x^2 - x^3 - 48)/x)*(6*x^4*log(x) + 9*x^4*log(x)^2 + 4*x^2 - 2*x^3 + 48))/x^2,x)

[Out]

exp(-4*x)*exp(x^2)*exp(48/x)*exp(-3*x^3*log(x)^2)

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sympy [A]  time = 0.34, size = 22, normalized size = 0.92 \begin {gather*} e^{\frac {- 3 x^{4} \log {\relax (x )}^{2} + x^{3} - 4 x^{2} + 48}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*x**4*ln(x)**2-6*x**4*ln(x)+2*x**3-4*x**2-48)*exp((-3*x**4*ln(x)**2+x**3-4*x**2+48)/x)/x**2,x)

[Out]

exp((-3*x**4*log(x)**2 + x**3 - 4*x**2 + 48)/x)

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