3.45.64 \(\int \frac {2+x+6 \log (\frac {x^3}{e^2})}{x} \, dx\)

Optimal. Leaf size=18 \[ x+\log \left (4 x^2\right )+\log ^2\left (\frac {x^3}{e^2}\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 16, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {14, 43, 2301} \begin {gather*} \left (2-\log \left (x^3\right )\right )^2+x+2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + x + 6*Log[x^3/E^2])/x,x]

[Out]

x + 2*Log[x] + (2 - Log[x^3])^2

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2+x}{x}+\frac {6 \left (-2+\log \left (x^3\right )\right )}{x}\right ) \, dx\\ &=6 \int \frac {-2+\log \left (x^3\right )}{x} \, dx+\int \frac {2+x}{x} \, dx\\ &=\left (2-\log \left (x^3\right )\right )^2+\int \left (1+\frac {2}{x}\right ) \, dx\\ &=x+2 \log (x)+\left (2-\log \left (x^3\right )\right )^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 12, normalized size = 0.67 \begin {gather*} x-10 \log (x)+\log ^2\left (x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + x + 6*Log[x^3/E^2])/x,x]

[Out]

x - 10*Log[x] + Log[x^3]^2

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fricas [A]  time = 0.52, size = 20, normalized size = 1.11 \begin {gather*} \log \left (x^{3} e^{\left (-2\right )}\right )^{2} + x + \frac {2}{3} \, \log \left (x^{3} e^{\left (-2\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*log(x^3/exp(1)^2)+2+x)/x,x, algorithm="fricas")

[Out]

log(x^3*e^(-2))^2 + x + 2/3*log(x^3*e^(-2))

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giac [A]  time = 0.21, size = 12, normalized size = 0.67 \begin {gather*} \log \left (x^{3}\right )^{2} + x - 10 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*log(x^3/exp(1)^2)+2+x)/x,x, algorithm="giac")

[Out]

log(x^3)^2 + x - 10*log(x)

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maple [A]  time = 0.02, size = 13, normalized size = 0.72

method	result	size
default	\(x -10 \ln \relax (x )+\ln \left (x^{3}\right )^{2}\)	\(13\)
risch	\(x +\ln \left (x^{3} {\mathrm e}^{-2}\right )^{2}+2 \ln \relax (x )\)	\(16\)
norman	\(x +\ln \left (x^{3} {\mathrm e}^{-2}\right )^{2}+2 \ln \relax (x )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*ln(x^3/exp(1)^2)+2+x)/x,x,method=_RETURNVERBOSE)

[Out]

x-10*ln(x)+ln(x^3)^2

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maxima [A]  time = 0.35, size = 15, normalized size = 0.83 \begin {gather*} \log \left (x^{3} e^{\left (-2\right )}\right )^{2} + x + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*log(x^3/exp(1)^2)+2+x)/x,x, algorithm="maxima")

[Out]

log(x^3*e^(-2))^2 + x + 2*log(x)

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mupad [B]  time = 3.09, size = 14, normalized size = 0.78 \begin {gather*} {\ln \left (x^3\right )}^2-\frac {10\,\ln \left (x^3\right )}{3}+x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 6*log(x^3*exp(-2)) + 2)/x,x)

[Out]

x - (10*log(x^3))/3 + log(x^3)^2

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sympy [A]  time = 0.11, size = 15, normalized size = 0.83 \begin {gather*} x + 2 \log {\relax (x )} + \log {\left (\frac {x^{3}}{e^{2}} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*ln(x**3/exp(1)**2)+2+x)/x,x)

[Out]

x + 2*log(x) + log(x**3*exp(-2))**2

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