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

Optimal. Leaf size=21 \[ \log \left (\frac {9-\log ^2(36)+\frac {\log ^2(x)}{x^2}}{x}\right ) \]

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Rubi [A]  time = 0.41, antiderivative size = 23, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 4, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6, 2561, 6742, 2541} \begin {gather*} \log \left (x^2 \left (9-\log ^2(36)\right )+\log ^2(x)\right )-3 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-3*Log[x] + Log[x^2*(9 - Log[36]^2) + Log[x]^2]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 2541

Int[(Log[(c_.)*(x_)^(n_.)]^(r_.)*(e_.) + (d_.)*(x_)^(m_.))/((x_)*(Log[(c_.)*(x_)^(n_.)]^(q_)*(b_.) + (a_.)*(x_
)^(m_.))), x_Symbol] :> Simp[(e*Log[a*x^m + b*Log[c*x^n]^q])/(b*n*q), x] /; FreeQ[{a, b, c, d, e, m, n, q, r},
x] && EqQ[r, q - 1] && EqQ[a*e*m - b*d*n*q, 0]

Rule 2561

Int[(u_.)*((a_.)*(x_)^(m_.) + Log[(c_.)*(x_)^(n_.)]^(q_.)*(b_.)*(x_)^(r_.))^(p_.), x_Symbol] :> Int[u*x^(p*r)*
(a*x^(m - r) + b*Log[c*x^n]^q)^p, x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && IntegerQ[p]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^2 \left (-9+\log ^2(36)\right )+2 \log (x)-3 \log ^2(x)}{9 x^3-x^3 \log ^2(36)+x \log ^2(x)} \, dx\\ &=\int \frac {x^2 \left (-9+\log ^2(36)\right )+2 \log (x)-3 \log ^2(x)}{x^3 \left (9-\log ^2(36)\right )+x \log ^2(x)} \, dx\\ &=\int \frac {x^2 \left (-9+\log ^2(36)\right )+2 \log (x)-3 \log ^2(x)}{x \left (x^2 \left (9-\log ^2(36)\right )+\log ^2(x)\right )} \, dx\\ &=\int \left (-\frac {3}{x}+\frac {2 \left (9 x^2 \left (1-\frac {\log ^2(36)}{9}\right )+\log (x)\right )}{x \left (9 x^2 \left (1-\frac {\log ^2(36)}{9}\right )+\log ^2(x)\right )}\right ) \, dx\\ &=-3 \log (x)+2 \int \frac {9 x^2 \left (1-\frac {\log ^2(36)}{9}\right )+\log (x)}{x \left (9 x^2 \left (1-\frac {\log ^2(36)}{9}\right )+\log ^2(x)\right )} \, dx\\ &=-3 \log (x)+\log \left (x^2 \left (9-\log ^2(36)\right )+\log ^2(x)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.20, size = 23, normalized size = 1.10 \begin {gather*} -3 \log (x)+\log \left (x^2 \left (-9+\log ^2(36)\right )-\log ^2(x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-3*Log[x] + Log[x^2*(-9 + Log[36]^2) - Log[x]^2]

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fricas [A]  time = 0.61, size = 25, normalized size = 1.19 \begin {gather*} \log \left (-4 \, x^{2} \log \relax (6)^{2} + 9 \, x^{2} + \log \relax (x)^{2}\right ) - 3 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*log(x)^2+2*log(x)+4*x^2*log(6)^2-9*x^2)/(x*log(x)^2-4*x^3*log(6)^2+9*x^3),x, algorithm="fricas")

[Out]

log(-4*x^2*log(6)^2 + 9*x^2 + log(x)^2) - 3*log(x)

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giac [A]  time = 0.24, size = 27, normalized size = 1.29 \begin {gather*} \log \left (4 \, x^{2} \log \relax (6)^{2} - 9 \, x^{2} - \log \relax (x)^{2}\right ) - 3 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*log(x)^2+2*log(x)+4*x^2*log(6)^2-9*x^2)/(x*log(x)^2-4*x^3*log(6)^2+9*x^3),x, algorithm="giac")

[Out]

log(4*x^2*log(6)^2 - 9*x^2 - log(x)^2) - 3*log(x)

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maple [A]  time = 0.05, size = 28, normalized size = 1.33

method	result	size
norman	\(-3 \ln \relax (x )+\ln \left (4 x^{2} \ln \relax (6)^{2}-9 x^{2}-\ln \relax (x )^{2}\right )\)	\(28\)
risch	\(-3 \ln \relax (x )+\ln \left (-4 x^{2} \ln \relax (2)^{2}-8 \ln \relax (2) \ln \relax (3) x^{2}-4 x^{2} \ln \relax (3)^{2}+9 x^{2}+\ln \relax (x )^{2}\right )\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3*ln(x)^2+2*ln(x)+4*x^2*ln(6)^2-9*x^2)/(x*ln(x)^2-4*x^3*ln(6)^2+9*x^3),x,method=_RETURNVERBOSE)

[Out]

-3*ln(x)+ln(4*x^2*ln(6)^2-9*x^2-ln(x)^2)

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maxima [A]  time = 1.05, size = 36, normalized size = 1.71 \begin {gather*} \log \left (-{\left (4 \, \log \relax (3)^{2} + 8 \, \log \relax (3) \log \relax (2) + 4 \, \log \relax (2)^{2} - 9\right )} x^{2} + \log \relax (x)^{2}\right ) - 3 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*log(x)^2+2*log(x)+4*x^2*log(6)^2-9*x^2)/(x*log(x)^2-4*x^3*log(6)^2+9*x^3),x, algorithm="maxima")

[Out]

log(-(4*log(3)^2 + 8*log(3)*log(2) + 4*log(2)^2 - 9)*x^2 + log(x)^2) - 3*log(x)

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mupad [B]  time = 2.06, size = 25, normalized size = 1.19 \begin {gather*} \ln \left ({\ln \relax (x)}^2-4\,x^2\,{\ln \relax (6)}^2+9\,x^2\right )-3\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*log(x) + 4*x^2*log(6)^2 - 3*log(x)^2 - 9*x^2)/(x*log(x)^2 - 4*x^3*log(6)^2 + 9*x^3),x)

[Out]

log(log(x)^2 - 4*x^2*log(6)^2 + 9*x^2) - 3*log(x)

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sympy [A]  time = 0.19, size = 26, normalized size = 1.24 \begin {gather*} - 3 \log {\relax (x )} + \log {\left (- 4 x^{2} \log {\relax (6 )}^{2} + 9 x^{2} + \log {\relax (x )}^{2} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*ln(x)**2+2*ln(x)+4*x**2*ln(6)**2-9*x**2)/(x*ln(x)**2-4*x**3*ln(6)**2+9*x**3),x)

[Out]

-3*log(x) + log(-4*x**2*log(6)**2 + 9*x**2 + log(x)**2)

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