3.45.6 \(\int \frac {3-2 e^{x/3} x}{3 x} \, dx\)

Optimal. Leaf size=19 \[ \log \left (e^{-2 e^{x/3}} \left (\frac {5}{86}+e\right ) x\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 12, normalized size of antiderivative = 0.63, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {12, 14, 2194} \begin {gather*} \log (x)-2 e^{x/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 - 2*E^(x/3)*x)/(3*x),x]

[Out]

-2*E^(x/3) + Log[x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 12, normalized size = 0.63 \begin {gather*} -2 e^{x/3}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 - 2*E^(x/3)*x)/(3*x),x]

[Out]

-2*E^(x/3) + Log[x]

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fricas [A]  time = 0.54, size = 9, normalized size = 0.47 \begin {gather*} -2 \, e^{\left (\frac {1}{3} \, x\right )} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-2*x*exp(1/3*x)+3)/x,x, algorithm="fricas")

[Out]

-2*e^(1/3*x) + log(x)

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giac [A]  time = 0.17, size = 9, normalized size = 0.47 \begin {gather*} -2 \, e^{\left (\frac {1}{3} \, x\right )} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-2*x*exp(1/3*x)+3)/x,x, algorithm="giac")

[Out]

-2*e^(1/3*x) + log(x)

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maple [A]  time = 0.02, size = 10, normalized size = 0.53

method	result	size
norman	\(-2 \,{\mathrm e}^{\frac {x}{3}}+\ln \relax (x )\)	\(10\)
risch	\(-2 \,{\mathrm e}^{\frac {x}{3}}+\ln \relax (x )\)	\(10\)
derivativedivides	\(\ln \left (\frac {x}{3}\right )-2 \,{\mathrm e}^{\frac {x}{3}}\)	\(12\)
default	\(\ln \left (\frac {x}{3}\right )-2 \,{\mathrm e}^{\frac {x}{3}}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(-2*x*exp(1/3*x)+3)/x,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.35, size = 9, normalized size = 0.47 \begin {gather*} -2 \, e^{\left (\frac {1}{3} \, x\right )} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-2*x*exp(1/3*x)+3)/x,x, algorithm="maxima")

[Out]

-2*e^(1/3*x) + log(x)

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mupad [B]  time = 0.04, size = 9, normalized size = 0.47 \begin {gather*} \ln \relax (x)-2\,{\mathrm {e}}^{x/3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((2*x*exp(x/3))/3 - 1)/x,x)

[Out]

log(x) - 2*exp(x/3)

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sympy [A]  time = 0.08, size = 8, normalized size = 0.42 \begin {gather*} - 2 e^{\frac {x}{3}} + \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-2*x*exp(1/3*x)+3)/x,x)

[Out]

-2*exp(x/3) + log(x)

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