3.25.46 \(\int \frac {e^{\frac {1}{3} (-e^{-4+\sqrt [4]{x}}-3 x \log (3))} (-e^{-4+\sqrt [4]{x}} \sqrt [4]{x}-12 x \log (3))}{48 x} \, dx\)

Optimal. Leaf size=25 \[ \frac {1}{4} e^{-\frac {1}{3} e^{-4+\sqrt [4]{x}}-x \log (3)} \]

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Rubi [A]  time = 0.48, antiderivative size = 24, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 2, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {12, 6706} \begin {gather*} \frac {1}{4} 3^{-x} e^{-\frac {1}{3} e^{\sqrt [4]{x}-4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((-E^(-4 + x^(1/4)) - 3*x*Log[3])/3)*(-(E^(-4 + x^(1/4))*x^(1/4)) - 12*x*Log[3]))/(48*x),x]

[Out]

1/(4*3^x*E^(E^(-4 + x^(1/4))/3))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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} &=\frac {1}{48} \int \frac {e^{\frac {1}{3} \left (-e^{-4+\sqrt [4]{x}}-3 x \log (3)\right )} \left (-e^{-4+\sqrt [4]{x}} \sqrt [4]{x}-12 x \log (3)\right )}{x} \, dx\\ &=\frac {1}{4} 3^{-x} e^{-\frac {1}{3} e^{-4+\sqrt [4]{x}}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 24, normalized size = 0.96 \begin {gather*} \frac {1}{4} 3^{-x} e^{-\frac {1}{3} e^{-4+\sqrt [4]{x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-E^(-4 + x^(1/4)) - 3*x*Log[3])/3)*(-(E^(-4 + x^(1/4))*x^(1/4)) - 12*x*Log[3]))/(48*x),x]

[Out]

1/(4*3^x*E^(E^(-4 + x^(1/4))/3))

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/48*(-x^(1/4)*exp(x^(1/4)-4)-12*x*log(3))*exp(-1/3*exp(x^(1/4)-4)-x*log(3))/x,x, algorithm="fricas"
)

[Out]

Exception raised: TypeError >>  Error detected within library code:   alglogextint: unimplemented

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giac [B]  time = 0.40, size = 58, normalized size = 2.32 \begin {gather*} \frac {1}{4} \, e^{\left (-{\left (x^{\frac {1}{4}} - 4\right )}^{4} \log \relax (3) - 16 \, {\left (x^{\frac {1}{4}} - 4\right )}^{3} \log \relax (3) - 96 \, {\left (x^{\frac {1}{4}} - 4\right )}^{2} \log \relax (3) - 256 \, {\left (x^{\frac {1}{4}} - 4\right )} \log \relax (3) - \frac {1}{3} \, e^{\left (x^{\frac {1}{4}} - 4\right )} - 256 \, \log \relax (3)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/48*(-x^(1/4)*exp(x^(1/4)-4)-12*x*log(3))*exp(-1/3*exp(x^(1/4)-4)-x*log(3))/x,x, algorithm="giac")

[Out]

1/4*e^(-(x^(1/4) - 4)^4*log(3) - 16*(x^(1/4) - 4)^3*log(3) - 96*(x^(1/4) - 4)^2*log(3) - 256*(x^(1/4) - 4)*log
(3) - 1/3*e^(x^(1/4) - 4) - 256*log(3))

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (-x^{\frac {1}{4}} {\mathrm e}^{x^{\frac {1}{4}}-4}-12 x \ln \relax (3)\right ) {\mathrm e}^{-\frac {{\mathrm e}^{x^{\frac {1}{4}}-4}}{3}-x \ln \relax (3)}}{48 x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/48*(-x^(1/4)*exp(x^(1/4)-4)-12*x*ln(3))*exp(-1/3*exp(x^(1/4)-4)-x*ln(3))/x,x)

[Out]

int(1/48*(-x^(1/4)*exp(x^(1/4)-4)-12*x*ln(3))*exp(-1/3*exp(x^(1/4)-4)-x*ln(3))/x,x)

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maxima [A]  time = 0.65, size = 17, normalized size = 0.68 \begin {gather*} \frac {1}{4} \, e^{\left (-x \log \relax (3) - \frac {1}{3} \, e^{\left (x^{\frac {1}{4}} - 4\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/48*(-x^(1/4)*exp(x^(1/4)-4)-12*x*log(3))*exp(-1/3*exp(x^(1/4)-4)-x*log(3))/x,x, algorithm="maxima"
)

[Out]

1/4*e^(-x*log(3) - 1/3*e^(x^(1/4) - 4))

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mupad [B]  time = 1.60, size = 16, normalized size = 0.64 \begin {gather*} \frac {{\mathrm {e}}^{-\frac {{\mathrm {e}}^{x^{1/4}}\,{\mathrm {e}}^{-4}}{3}}}{4\,3^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(- exp(x^(1/4) - 4)/3 - x*log(3))*(12*x*log(3) + x^(1/4)*exp(x^(1/4) - 4)))/(48*x),x)

[Out]

exp(-(exp(x^(1/4))*exp(-4))/3)/(4*3^x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/48*(-x**(1/4)*exp(x**(1/4)-4)-12*x*ln(3))*exp(-1/3*exp(x**(1/4)-4)-x*ln(3))/x,x)

[Out]

Timed out

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