\(\int \frac {-9 x^2+x^2 \log ^2(36)+2 \log (x)-3 \log ^2(x)}{9 x^3-x^3 \log ^2(36)+x \log ^2(x)} \, dx\) [3338]

Optimal result

Integrand size = 48, antiderivative size = 21 \[ \int \frac {-9 x^2+x^2 \log ^2(36)+2 \log (x)-3 \log ^2(x)}{9 x^3-x^3 \log ^2(36)+x \log ^2(x)} \, dx=\log \left (\frac {9-\log ^2(36)+\frac {\log ^2(x)}{x^2}}{x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6, 2641, 6874, 2621} \[ \int \frac {-9 x^2+x^2 \log ^2(36)+2 \log (x)-3 \log ^2(x)}{9 x^3-x^3 \log ^2(36)+x \log ^2(x)} \, dx=\log \left (x^2 \left (9-\log ^2(36)\right )+\log ^2(x)\right )-3 \log (x) \]

[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 2621

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 2641

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 6874

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

Rubi steps \begin{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {-9 x^2+x^2 \log ^2(36)+2 \log (x)-3 \log ^2(x)}{9 x^3-x^3 \log ^2(36)+x \log ^2(x)} \, dx=-3 \log (x)+\log \left (x^2 \left (-9+\log ^2(36)\right )-\log ^2(x)\right ) \]

[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]

Maple [A] (verified)

Time = 3.92 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33

method	result	size
default	\(-3 \ln \left (x \right )+\ln \left (4 x^{2} \ln \left (6\right )^{2}-9 x^{2}-\ln \left (x \right )^{2}\right )\)	\(28\)
norman	\(-3 \ln \left (x \right )+\ln \left (4 x^{2} \ln \left (6\right )^{2}-9 x^{2}-\ln \left (x \right )^{2}\right )\)	\(28\)
parallelrisch	\(\ln \left (\frac {4 x^{2} \ln \left (6\right )^{2}-9 x^{2}-\ln \left (x \right )^{2}}{4 \ln \left (6\right )^{2}-9}\right )-3 \ln \left (x \right )\)	\(39\)
risch	\(-3 \ln \left (x \right )+\ln \left (-4 x^{2} \ln \left (2\right )^{2}-8 \ln \left (2\right ) \ln \left (3\right ) x^{2}-4 x^{2} \ln \left (3\right )^{2}+9 x^{2}+\ln \left (x \right )^{2}\right )\)	\(44\)

[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)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {-9 x^2+x^2 \log ^2(36)+2 \log (x)-3 \log ^2(x)}{9 x^3-x^3 \log ^2(36)+x \log ^2(x)} \, dx=\log \left (-4 \, x^{2} \log \left (6\right )^{2} + 9 \, x^{2} + \log \left (x\right )^{2}\right ) - 3 \, \log \left (x\right ) \]

[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)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int \frac {-9 x^2+x^2 \log ^2(36)+2 \log (x)-3 \log ^2(x)}{9 x^3-x^3 \log ^2(36)+x \log ^2(x)} \, dx=- 3 \log {\left (x \right )} + \log {\left (- 4 x^{2} \log {\left (6 \right )}^{2} + 9 x^{2} + \log {\left (x \right )}^{2} \right )} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.71 \[ \int \frac {-9 x^2+x^2 \log ^2(36)+2 \log (x)-3 \log ^2(x)}{9 x^3-x^3 \log ^2(36)+x \log ^2(x)} \, dx=\log \left (-{\left (4 \, \log \left (3\right )^{2} + 8 \, \log \left (3\right ) \log \left (2\right ) + 4 \, \log \left (2\right )^{2} - 9\right )} x^{2} + \log \left (x\right )^{2}\right ) - 3 \, \log \left (x\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29 \[ \int \frac {-9 x^2+x^2 \log ^2(36)+2 \log (x)-3 \log ^2(x)}{9 x^3-x^3 \log ^2(36)+x \log ^2(x)} \, dx=\log \left (4 \, x^{2} \log \left (6\right )^{2} - 9 \, x^{2} - \log \left (x\right )^{2}\right ) - 3 \, \log \left (x\right ) \]

[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)

Mupad [B] (verification not implemented)

Time = 8.66 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {-9 x^2+x^2 \log ^2(36)+2 \log (x)-3 \log ^2(x)}{9 x^3-x^3 \log ^2(36)+x \log ^2(x)} \, dx=\ln \left ({\ln \left (x\right )}^2-4\,x^2\,{\ln \left (6\right )}^2+9\,x^2\right )-3\,\ln \left (x\right ) \]

[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)