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

Optimal. Leaf size=13 \[ -1+e^{\frac {4}{3 x^3}} x \]

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Rubi [A]  time = 0.04, antiderivative size = 11, normalized size of antiderivative = 0.85, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2288} \begin {gather*} e^{\frac {4}{3 x^3}} x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(4/(3*x^3))*(-4 + x^3))/x^3,x]

[Out]

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

Rule 2288

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 {gather*} \begin {aligned} \text {integral} &=e^{\frac {4}{3 x^3}} x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 11, normalized size = 0.85 \begin {gather*} e^{\frac {4}{3 x^3}} x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.70, size = 8, normalized size = 0.62 \begin {gather*} x e^{\left (\frac {4}{3 \, x^{3}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.13, size = 8, normalized size = 0.62 \begin {gather*} x e^{\left (\frac {4}{3 \, x^{3}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.13, size = 9, normalized size = 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	\(x \,{\mathrm e}^{\frac {4}{3 x^{3}}}\)	\(11\)
meijerg	\(-\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \left (-\frac {3 \left (-1\right )^{\frac {2}{3}} \Gamma \left (\frac {2}{3}\right )}{x^{2} \left (-\frac {1}{x^{3}}\right )^{\frac {2}{3}}}+\frac {3 \,3^{\frac {1}{3}} 4^{\frac {2}{3}} x \left (-1\right )^{\frac {2}{3}} {\mathrm e}^{\frac {4}{3 x^{3}}}}{4}+\frac {3 \left (-1\right )^{\frac {2}{3}} \Gamma \left (\frac {2}{3}, -\frac {4}{3 x^{3}}\right )}{x^{2} \left (-\frac {1}{x^{3}}\right )^{\frac {2}{3}}}\right )}{9}-\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \left (\frac {\left (-1\right )^{\frac {2}{3}} \Gamma \left (\frac {2}{3}\right )}{x^{2} \left (-\frac {1}{x^{3}}\right )^{\frac {2}{3}}}-\frac {\left (-1\right )^{\frac {2}{3}} \Gamma \left (\frac {2}{3}, -\frac {4}{3 x^{3}}\right )}{x^{2} \left (-\frac {1}{x^{3}}\right )^{\frac {2}{3}}}\right )}{3}\)	\(121\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [C]  time = 0.38, size = 43, normalized size = 3.31 \begin {gather*} \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}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 3.20, size = 8, normalized size = 0.62 \begin {gather*} x\,{\mathrm {e}}^{\frac {4}{3\,x^3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4/(3*x^3))*(x^3 - 4))/x^3,x)

[Out]

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

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sympy [A]  time = 0.10, size = 8, normalized size = 0.62 \begin {gather*} x e^{\frac {4}{3 x^{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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