3.102.37 \(\int \frac {-6 \log (x)-3 \log ^2(x)}{-7+e^2} \, dx\)

Optimal. Leaf size=16 \[ \frac {3 x \log ^2(x)}{7-e^2} \]

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {12, 2295, 2296} \begin {gather*} \frac {3 x \log ^2(x)}{7-e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-6*Log[x] - 3*Log[x]^2)/(-7 + E^2),x]

[Out]

(3*x*Log[x]^2)/(7 - E^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 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (-6 \log (x)-3 \log ^2(x)\right ) \, dx}{-7+e^2}\\ &=\frac {3 \int \log ^2(x) \, dx}{7-e^2}+\frac {6 \int \log (x) \, dx}{7-e^2}\\ &=-\frac {6 x}{7-e^2}+\frac {6 x \log (x)}{7-e^2}+\frac {3 x \log ^2(x)}{7-e^2}-\frac {6 \int \log (x) \, dx}{7-e^2}\\ &=\frac {3 x \log ^2(x)}{7-e^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 14, normalized size = 0.88 \begin {gather*} -\frac {3 x \log ^2(x)}{-7+e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6*Log[x] - 3*Log[x]^2)/(-7 + E^2),x]

[Out]

(-3*x*Log[x]^2)/(-7 + E^2)

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fricas [A]  time = 0.53, size = 13, normalized size = 0.81 \begin {gather*} -\frac {3 \, x \log \relax (x)^{2}}{e^{2} - 7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*log(x)^2-6*log(x))/(exp(2)-7),x, algorithm="fricas")

[Out]

-3*x*log(x)^2/(e^2 - 7)

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giac [A]  time = 0.24, size = 13, normalized size = 0.81 \begin {gather*} -\frac {3 \, x \log \relax (x)^{2}}{e^{2} - 7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*log(x)^2-6*log(x))/(exp(2)-7),x, algorithm="giac")

[Out]

-3*x*log(x)^2/(e^2 - 7)

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maple [A]  time = 0.03, size = 14, normalized size = 0.88

method	result	size
default	\(-\frac {3 x \ln \relax (x )^{2}}{{\mathrm e}^{2}-7}\)	\(14\)
norman	\(-\frac {3 x \ln \relax (x )^{2}}{{\mathrm e}^{2}-7}\)	\(14\)
risch	\(-\frac {3 x \ln \relax (x )^{2}}{{\mathrm e}^{2}-7}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3*ln(x)^2-6*ln(x))/(exp(2)-7),x,method=_RETURNVERBOSE)

[Out]

-3/(exp(2)-7)*x*ln(x)^2

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maxima [B]  time = 0.34, size = 29, normalized size = 1.81 \begin {gather*} -\frac {3 \, {\left ({\left (\log \relax (x)^{2} - 2 \, \log \relax (x) + 2\right )} x + 2 \, x \log \relax (x) - 2 \, x\right )}}{e^{2} - 7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*log(x)^2-6*log(x))/(exp(2)-7),x, algorithm="maxima")

[Out]

-3*((log(x)^2 - 2*log(x) + 2)*x + 2*x*log(x) - 2*x)/(e^2 - 7)

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mupad [B]  time = 6.50, size = 13, normalized size = 0.81 \begin {gather*} -\frac {3\,x\,{\ln \relax (x)}^2}{{\mathrm {e}}^2-7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(6*log(x) + 3*log(x)^2)/(exp(2) - 7),x)

[Out]

-(3*x*log(x)^2)/(exp(2) - 7)

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sympy [A]  time = 0.10, size = 14, normalized size = 0.88 \begin {gather*} - \frac {3 x \log {\relax (x )}^{2}}{-7 + e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*ln(x)**2-6*ln(x))/(exp(2)-7),x)

[Out]

-3*x*log(x)**2/(-7 + exp(2))

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