3.44.81 \(\int \frac {-9-3 e+3 \log (36 x^2-12 x^2 \log (3)+x^2 \log ^2(3))}{x^2} \, dx\)

Optimal. Leaf size=23 \[ \frac {3 \left (1+e-\log \left (x^2 (6-\log (3))^2\right )\right )}{x} \]

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Rubi [A]  time = 0.03, antiderivative size = 29, normalized size of antiderivative = 1.26, number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {2461, 2304} \begin {gather*} \frac {3 \left (-\log \left (x^2 (6-\log (3))^2\right )+e+3\right )}{x}-\frac {6}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-9 - 3*E + 3*Log[36*x^2 - 12*x^2*Log[3] + x^2*Log[3]^2])/x^2,x]

[Out]

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

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2461

Int[((a_.) + Log[(c_.)*(v_)^(p_.)]*(b_.))^(q_.)*((f_.)*(x_))^(m_.), x_Symbol] :> Int[(f*x)^m*(a + b*Log[c*Expa
ndToSum[v, x]^p])^q, x] /; FreeQ[{a, b, c, f, m, p, q}, x] && BinomialQ[v, x] &&  !BinomialMatchQ[v, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-9-3 e+3 \log \left (x^2 (6-\log (3))^2\right )}{x^2} \, dx\\ &=-\frac {6}{x}+\frac {3 \left (3+e-\log \left (x^2 (6-\log (3))^2\right )\right )}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 28, normalized size = 1.22 \begin {gather*} \frac {3}{x}+\frac {3 e}{x}-\frac {3 \log \left (x^2 (-6+\log (3))^2\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9 - 3*E + 3*Log[36*x^2 - 12*x^2*Log[3] + x^2*Log[3]^2])/x^2,x]

[Out]

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

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fricas [A]  time = 0.57, size = 33, normalized size = 1.43 \begin {gather*} \frac {3 \, {\left (e - \log \left (x^{2} \log \relax (3)^{2} - 12 \, x^{2} \log \relax (3) + 36 \, x^{2}\right ) + 1\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*log(x^2*log(3)^2-12*x^2*log(3)+36*x^2)-3*exp(1)-9)/x^2,x, algorithm="fricas")

[Out]

3*(e - log(x^2*log(3)^2 - 12*x^2*log(3) + 36*x^2) + 1)/x

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giac [A]  time = 0.12, size = 33, normalized size = 1.43 \begin {gather*} \frac {3 \, {\left (e - \log \left (x^{2} \log \relax (3)^{2} - 12 \, x^{2} \log \relax (3) + 36 \, x^{2}\right ) + 1\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*log(x^2*log(3)^2-12*x^2*log(3)+36*x^2)-3*exp(1)-9)/x^2,x, algorithm="giac")

[Out]

3*(e - log(x^2*log(3)^2 - 12*x^2*log(3) + 36*x^2) + 1)/x

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maple [A]  time = 0.05, size = 35, normalized size = 1.52

method	result	size
norman	\(\frac {3-3 \ln \left (x^{2} \ln \relax (3)^{2}-12 x^{2} \ln \relax (3)+36 x^{2}\right )+3 \,{\mathrm e}}{x}\)	\(35\)
risch	\(-\frac {3 \ln \left (x^{2} \ln \relax (3)^{2}-12 x^{2} \ln \relax (3)+36 x^{2}\right )}{x}+\frac {3 \,{\mathrm e}}{x}+\frac {3}{x}\)	\(41\)
default	\(-\frac {3 \ln \left (x^{2} \ln \relax (3)^{2}-12 x^{2} \ln \relax (3)+36 x^{2}\right )}{x}-\frac {6}{x}-\frac {3 \left (-3-{\mathrm e}\right )}{x}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*ln(x^2*ln(3)^2-12*x^2*ln(3)+36*x^2)-3*exp(1)-9)/x^2,x,method=_RETURNVERBOSE)

[Out]

(3-3*ln(x^2*ln(3)^2-12*x^2*ln(3)+36*x^2)+3*exp(1))/x

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maxima [A]  time = 0.42, size = 40, normalized size = 1.74 \begin {gather*} \frac {3 \, e}{x} - \frac {3 \, \log \left (x^{2} \log \relax (3)^{2} - 12 \, x^{2} \log \relax (3) + 36 \, x^{2}\right )}{x} + \frac {3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*log(x^2*log(3)^2-12*x^2*log(3)+36*x^2)-3*exp(1)-9)/x^2,x, algorithm="maxima")

[Out]

3*e/x - 3*log(x^2*log(3)^2 - 12*x^2*log(3) + 36*x^2)/x + 3/x

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mupad [B]  time = 3.24, size = 38, normalized size = 1.65 \begin {gather*} \frac {3\,\mathrm {e}+3}{x}-\frac {3\,\ln \left (x^2\,{\ln \relax (3)}^2-12\,x^2\,\ln \relax (3)+36\,x^2\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*exp(1) - 3*log(x^2*log(3)^2 - 12*x^2*log(3) + 36*x^2) + 9)/x^2,x)

[Out]

(3*exp(1) + 3)/x - (3*log(x^2*log(3)^2 - 12*x^2*log(3) + 36*x^2))/x

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sympy [A]  time = 0.12, size = 37, normalized size = 1.61 \begin {gather*} - \frac {3 \log {\left (- 12 x^{2} \log {\relax (3 )} + x^{2} \log {\relax (3 )}^{2} + 36 x^{2} \right )}}{x} - \frac {- 3 e - 3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*ln(x**2*ln(3)**2-12*x**2*ln(3)+36*x**2)-3*exp(1)-9)/x**2,x)

[Out]

-3*log(-12*x**2*log(3) + x**2*log(3)**2 + 36*x**2)/x - (-3*E - 3)/x

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