3.71.23 \(\int \frac {x^3+e^{\frac {-2 x^2+\log (4)}{x^3}} (2 x^2-3 \log (4))}{x^4} \, dx\)

Optimal. Leaf size=20 \[ 2+e^{\frac {-2 x+\frac {\log (4)}{x}}{x^2}}+\log (x) \]

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Rubi [A]  time = 0.10, antiderivative size = 16, normalized size of antiderivative = 0.80, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {14, 2288} \begin {gather*} 4^{\frac {1}{x^3}} e^{-2/x}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

4^x^(-3)/E^(2/x) + Log[x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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} &=\int \left (\frac {1}{x}+\frac {4^{\frac {1}{x^3}} e^{-2/x} \left (2 x^2-3 \log (4)\right )}{x^4}\right ) \, dx\\ &=\log (x)+\int \frac {4^{\frac {1}{x^3}} e^{-2/x} \left (2 x^2-3 \log (4)\right )}{x^4} \, dx\\ &=4^{\frac {1}{x^3}} e^{-2/x}+\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 16, normalized size = 0.80 \begin {gather*} 4^{\frac {1}{x^3}} e^{-2/x}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

4^x^(-3)/E^(2/x) + Log[x]

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fricas [A]  time = 0.67, size = 17, normalized size = 0.85 \begin {gather*} e^{\left (-\frac {2 \, {\left (x^{2} - \log \relax (2)\right )}}{x^{3}}\right )} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-2*(x^2 - log(2))/x^3) + log(x)

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giac [A]  time = 0.14, size = 17, normalized size = 0.85 \begin {gather*} e^{\left (-\frac {2}{x} + \frac {2 \, \log \relax (2)}{x^{3}}\right )} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-2/x + 2*log(2)/x^3) + log(x)

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maple [A]  time = 0.16, size = 16, normalized size = 0.80

method	result	size
risch	\(4^{\frac {1}{x^{3}}} {\mathrm e}^{-\frac {2}{x}}+\ln \relax (x )\)	\(16\)
norman	\({\mathrm e}^{\frac {2 \ln \relax (2)-2 x^{2}}{x^{3}}}+\ln \relax (x )\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-6*ln(2)+2*x^2)*exp((2*ln(2)-2*x^2)/x^3)+x^3)/x^4,x,method=_RETURNVERBOSE)

[Out]

4^(1/x^3)*exp(-2/x)+ln(x)

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maxima [A]  time = 0.54, size = 17, normalized size = 0.85 \begin {gather*} e^{\left (-\frac {2}{x} + \frac {2 \, \log \relax (2)}{x^{3}}\right )} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-2/x + 2*log(2)/x^3) + log(x)

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mupad [B]  time = 4.18, size = 17, normalized size = 0.85 \begin {gather*} \ln \relax (x)+2^{\frac {2}{x^3}}\,{\mathrm {e}}^{-\frac {2}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) + 2^(2/x^3)*exp(-2/x)

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sympy [A]  time = 0.18, size = 17, normalized size = 0.85 \begin {gather*} e^{\frac {- 2 x^{2} + 2 \log {\relax (2 )}}{x^{3}}} + \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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