\(\int \frac {e^{\frac {4}{3 x^3}} (-4+x^3)}{x^3} \, dx\) [5005]

Optimal result

Integrand size = 18, antiderivative size = 13 \[ \int \frac {e^{\frac {4}{3 x^3}} \left (-4+x^3\right )}{x^3} \, dx=-1+e^{\frac {4}{3 x^3}} x \]

[Out]

-1+x/exp(-2/3/x^3)^2

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2326} \[ \int \frac {e^{\frac {4}{3 x^3}} \left (-4+x^3\right )}{x^3} \, dx=e^{\frac {4}{3 x^3}} x \]

[In]

Int[(E^(4/(3*x^3))*(-4 + x^3))/x^3,x]

[Out]

E^(4/(3*x^3))*x

Rule 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = e^{\frac {4}{3 x^3}} x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {e^{\frac {4}{3 x^3}} \left (-4+x^3\right )}{x^3} \, dx=e^{\frac {4}{3 x^3}} x \]

[In]

Integrate[(E^(4/(3*x^3))*(-4 + x^3))/x^3,x]

[Out]

E^(4/(3*x^3))*x

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69

method	result	size
risch	\(x \,{\mathrm e}^{\frac {4}{3 x^{3}}}\)	\(9\)
gosper	\(x \,{\mathrm e}^{\frac {4}{3 x^{3}}}\)	\(11\)
norman	\(x \,{\mathrm e}^{\frac {4}{3 x^{3}}}\)	\(11\)
parallelrisch	\(x \,{\mathrm e}^{\frac {4}{3 x^{3}}}\)	\(11\)
meijerg	\(-\frac {3^{\frac {2}{3}} 4^{\frac {1}{3}} \left (-1\right )^{\frac {1}{3}} \left (-\frac {3 \left (-1\right )^{\frac {2}{3}} \Gamma \left (\frac {2}{3}\right )}{x^{2} \left (-\frac {1}{x^{3}}\right )^{\frac {2}{3}}}+\frac {3 \,3^{\frac {1}{3}} 4^{\frac {2}{3}} x \left (-1\right )^{\frac {2}{3}} {\mathrm e}^{\frac {4}{3 x^{3}}}}{4}+\frac {3 \left (-1\right )^{\frac {2}{3}} \Gamma \left (\frac {2}{3}, -\frac {4}{3 x^{3}}\right )}{x^{2} \left (-\frac {1}{x^{3}}\right )^{\frac {2}{3}}}\right )}{9}-\frac {3^{\frac {2}{3}} 4^{\frac {1}{3}} \left (-1\right )^{\frac {1}{3}} \left (\frac {\left (-1\right )^{\frac {2}{3}} \Gamma \left (\frac {2}{3}\right )}{x^{2} \left (-\frac {1}{x^{3}}\right )^{\frac {2}{3}}}-\frac {\left (-1\right )^{\frac {2}{3}} \Gamma \left (\frac {2}{3}, -\frac {4}{3 x^{3}}\right )}{x^{2} \left (-\frac {1}{x^{3}}\right )^{\frac {2}{3}}}\right )}{3}\)	\(121\)

[In]

int((x^3-4)/x^3/exp(-2/3/x^3)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.62 \[ \int \frac {e^{\frac {4}{3 x^3}} \left (-4+x^3\right )}{x^3} \, dx=x e^{\left (\frac {4}{3 \, x^{3}}\right )} \]

[In]

integrate((x^3-4)/x^3/exp(-2/3/x^3)^2,x, algorithm="fricas")

[Out]

x*e^(4/3/x^3)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.62 \[ \int \frac {e^{\frac {4}{3 x^3}} \left (-4+x^3\right )}{x^3} \, dx=x e^{\frac {4}{3 x^{3}}} \]

[In]

integrate((x**3-4)/x**3/exp(-2/3/x**3)**2,x)

[Out]

x*exp(4/(3*x**3))

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.21 (sec) , antiderivative size = 43, normalized size of antiderivative = 3.31 \[ \int \frac {e^{\frac {4}{3 x^3}} \left (-4+x^3\right )}{x^3} \, dx=\frac {1}{3} \, \left (\frac {4}{3}\right )^{\frac {1}{3}} x \left (-\frac {1}{x^{3}}\right )^{\frac {1}{3}} \Gamma \left (-\frac {1}{3}, -\frac {4}{3 \, x^{3}}\right ) - \frac {\left (\frac {4}{3}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -\frac {4}{3 \, x^{3}}\right )}{x^{2} \left (-\frac {1}{x^{3}}\right )^{\frac {2}{3}}} \]

[In]

integrate((x^3-4)/x^3/exp(-2/3/x^3)^2,x, algorithm="maxima")

[Out]

1/3*(4/3)^(1/3)*x*(-1/x^3)^(1/3)*gamma(-1/3, -4/3/x^3) - (4/3)^(1/3)*gamma(2/3, -4/3/x^3)/(x^2*(-1/x^3)^(2/3))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.62 \[ \int \frac {e^{\frac {4}{3 x^3}} \left (-4+x^3\right )}{x^3} \, dx=x e^{\left (\frac {4}{3 \, x^{3}}\right )} \]

[In]

integrate((x^3-4)/x^3/exp(-2/3/x^3)^2,x, algorithm="giac")

[Out]

x*e^(4/3/x^3)

Mupad [B] (verification not implemented)

Time = 11.64 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.62 \[ \int \frac {e^{\frac {4}{3 x^3}} \left (-4+x^3\right )}{x^3} \, dx=x\,{\mathrm {e}}^{\frac {4}{3\,x^3}} \]

[In]

int((exp(4/(3*x^3))*(x^3 - 4))/x^3,x)

[Out]

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