\(\int \frac {e^{\frac {1}{3} (-e^{-4+\sqrt [4]{x}}-3 x \log (3))} (-e^{-4+\sqrt [4]{x}} \sqrt [4]{x}-12 x \log (3))}{48 x} \, dx\) [2446]

Optimal result

Integrand size = 52, antiderivative size = 25 \[ \int \frac {e^{\frac {1}{3} \left (-e^{-4+\sqrt [4]{x}}-3 x \log (3)\right )} \left (-e^{-4+\sqrt [4]{x}} \sqrt [4]{x}-12 x \log (3)\right )}{48 x} \, dx=\frac {1}{4} e^{-\frac {1}{3} e^{-4+\sqrt [4]{x}}-x \log (3)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {12, 6838} \[ \int \frac {e^{\frac {1}{3} \left (-e^{-4+\sqrt [4]{x}}-3 x \log (3)\right )} \left (-e^{-4+\sqrt [4]{x}} \sqrt [4]{x}-12 x \log (3)\right )}{48 x} \, dx=\frac {1}{4} 3^{-x} e^{-\frac {1}{3} e^{\sqrt [4]{x}-4}} \]

[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 6838

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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {e^{\frac {1}{3} \left (-e^{-4+\sqrt [4]{x}}-3 x \log (3)\right )} \left (-e^{-4+\sqrt [4]{x}} \sqrt [4]{x}-12 x \log (3)\right )}{48 x} \, dx=\frac {1}{4} 3^{-x} e^{-\frac {1}{3} e^{-4+\sqrt [4]{x}}} \]

[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))

Maple [F]

\[\int \frac {\left (-x^{\frac {1}{4}} {\mathrm e}^{x^{\frac {1}{4}}-4}-12 x \ln \left (3\right )\right ) {\mathrm e}^{-\frac {{\mathrm e}^{x^{\frac {1}{4}}-4}}{3}-x \ln \left (3\right )}}{48 x}d x\]

[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)

Fricas [F(-2)]

Exception generated. \[ \int \frac {e^{\frac {1}{3} \left (-e^{-4+\sqrt [4]{x}}-3 x \log (3)\right )} \left (-e^{-4+\sqrt [4]{x}} \sqrt [4]{x}-12 x \log (3)\right )}{48 x} \, dx=\text {Exception raised: TypeError} \]

[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

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{\frac {1}{3} \left (-e^{-4+\sqrt [4]{x}}-3 x \log (3)\right )} \left (-e^{-4+\sqrt [4]{x}} \sqrt [4]{x}-12 x \log (3)\right )}{48 x} \, dx=\text {Timed out} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68 \[ \int \frac {e^{\frac {1}{3} \left (-e^{-4+\sqrt [4]{x}}-3 x \log (3)\right )} \left (-e^{-4+\sqrt [4]{x}} \sqrt [4]{x}-12 x \log (3)\right )}{48 x} \, dx=\frac {1}{4} \, e^{\left (-x \log \left (3\right ) - \frac {1}{3} \, e^{\left (x^{\frac {1}{4}} - 4\right )}\right )} \]

[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))

Giac [F]

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

[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]

integrate(-1/48*(12*x*log(3) + x^(1/4)*e^(x^(1/4) - 4))*e^(-x*log(3) - 1/3*e^(x^(1/4) - 4))/x, x)

Mupad [B] (verification not implemented)

Time = 9.51 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.64 \[ \int \frac {e^{\frac {1}{3} \left (-e^{-4+\sqrt [4]{x}}-3 x \log (3)\right )} \left (-e^{-4+\sqrt [4]{x}} \sqrt [4]{x}-12 x \log (3)\right )}{48 x} \, dx=\frac {{\mathrm {e}}^{-\frac {{\mathrm {e}}^{x^{1/4}}\,{\mathrm {e}}^{-4}}{3}}}{4\,3^x} \]

[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)