3.48.47 \(\int \frac {6-3 x-3 \log (\frac {e^x}{27 x^2})}{10 x^2 \log ^2(\frac {e^x}{27 x^2})} \, dx\)

Optimal. Leaf size=20 \[ \frac {3}{10 x \log \left (\frac {e^x}{27 x^2}\right )} \]

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Rubi [A]  time = 0.08, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {12, 6687} \begin {gather*} \frac {3}{10 x \log \left (\frac {e^x}{27 x^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(6 - 3*x - 3*Log[E^x/(27*x^2)])/(10*x^2*Log[E^x/(27*x^2)]^2),x]

[Out]

3/(10*x*Log[E^x/(27*x^2)])

Rule 12

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

Rule 6687

Int[(u_)*(y_)^(m_.)*(z_)^(n_.), x_Symbol] :> With[{q = DerivativeDivides[y*z, u*z^(n - m), x]}, Simp[(q*y^(m +
1)*z^(m + 1))/(m + 1), x] /;  !FalseQ[q]] /; FreeQ[{m, n}, x] && NeQ[m, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{10} \int \frac {6-3 x-3 \log \left (\frac {e^x}{27 x^2}\right )}{x^2 \log ^2\left (\frac {e^x}{27 x^2}\right )} \, dx\\ &=\frac {3}{10 x \log \left (\frac {e^x}{27 x^2}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.96, size = 20, normalized size = 1.00 \begin {gather*} \frac {3}{10 x \log \left (\frac {e^x}{27 x^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6 - 3*x - 3*Log[E^x/(27*x^2)])/(10*x^2*Log[E^x/(27*x^2)]^2),x]

[Out]

3/(10*x*Log[E^x/(27*x^2)])

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fricas [A]  time = 0.75, size = 15, normalized size = 0.75 \begin {gather*} \frac {3}{10 \, x \log \left (\frac {e^{x}}{27 \, x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/10*(-3*log(1/27*exp(x)/x^2)-3*x+6)/x^2/log(1/27*exp(x)/x^2)^2,x, algorithm="fricas")

[Out]

3/10/(x*log(1/27*e^x/x^2))

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giac [A]  time = 0.19, size = 17, normalized size = 0.85 \begin {gather*} \frac {3}{10 \, {\left (x^{2} - x \log \left (27 \, x^{2}\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/10*(-3*log(1/27*exp(x)/x^2)-3*x+6)/x^2/log(1/27*exp(x)/x^2)^2,x, algorithm="giac")

[Out]

3/10/(x^2 - x*log(27*x^2))

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maple [A]  time = 0.17, size = 16, normalized size = 0.80

method	result	size
default	\(\frac {3}{10 x \ln \left (\frac {{\mathrm e}^{x}}{27 x^{2}}\right )}\)	\(16\)
norman	\(\frac {3}{10 x \ln \left (\frac {{\mathrm e}^{x}}{27 x^{2}}\right )}\)	\(16\)
risch	\(-\frac {3 i}{5 x \left (\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x^{2}}\right )+\pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x^{2}}\right )^{2}+\pi \,\mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x^{2}}\right )^{2}-\pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x^{2}}\right )^{3}+6 i \ln \relax (3)+4 i \ln \relax (x )-2 i \ln \left ({\mathrm e}^{x}\right )\right )}\)	\(148\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/10*(-3*ln(1/27*exp(x)/x^2)-3*x+6)/x^2/ln(1/27*exp(x)/x^2)^2,x,method=_RETURNVERBOSE)

[Out]

3/10/x/ln(1/27*exp(x)/x^2)

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maxima [A]  time = 0.47, size = 18, normalized size = 0.90 \begin {gather*} \frac {3}{10 \, {\left (x^{2} - 3 \, x \log \relax (3) - 2 \, x \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/10*(-3*log(1/27*exp(x)/x^2)-3*x+6)/x^2/log(1/27*exp(x)/x^2)^2,x, algorithm="maxima")

[Out]

3/10/(x^2 - 3*x*log(3) - 2*x*log(x))

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mupad [B]  time = 3.25, size = 15, normalized size = 0.75 \begin {gather*} \frac {3}{10\,x\,\left (x+\ln \left (\frac {1}{27\,x^2}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((3*x)/10 + (3*log(exp(x)/(27*x^2)))/10 - 3/5)/(x^2*log(exp(x)/(27*x^2))^2),x)

[Out]

3/(10*x*(x + log(1/(27*x^2))))

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sympy [A]  time = 0.12, size = 14, normalized size = 0.70 \begin {gather*} \frac {3}{10 x \log {\left (\frac {e^{x}}{27 x^{2}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/10*(-3*ln(1/27*exp(x)/x**2)-3*x+6)/x**2/ln(1/27*exp(x)/x**2)**2,x)

[Out]

3/(10*x*log(exp(x)/(27*x**2)))

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