\(\int \frac {-2-\log ^2(3)}{2 x-x^2+x \log ^2(3)} \, dx\) [7534]

Optimal result

Integrand size = 26, antiderivative size = 14 \[ \int \frac {-2-\log ^2(3)}{2 x-x^2+x \log ^2(3)} \, dx=\log \left (1-\frac {2+\log ^2(3)}{x}\right ) \]

[Out]

ln(1-(2+ln(3)^2)/x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {6, 12, 629} \[ \int \frac {-2-\log ^2(3)}{2 x-x^2+x \log ^2(3)} \, dx=\log \left (-x+2+\log ^2(3)\right )-\log (x) \]

[In]

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

[Out]

-Log[x] + Log[2 - x + Log[3]^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 12

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

Rule 629

Int[((b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[Log[x]/b, x] - Simp[Log[RemoveContent[b + c*x, x]]/b,
x] /; FreeQ[{b, c}, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-2-\log ^2(3)}{-x^2+x \left (2+\log ^2(3)\right )} \, dx \\ & = \left (-2-\log ^2(3)\right ) \int \frac {1}{-x^2+x \left (2+\log ^2(3)\right )} \, dx \\ & = -\log (x)+\log \left (2-x+\log ^2(3)\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int \frac {-2-\log ^2(3)}{2 x-x^2+x \log ^2(3)} \, dx=-\log (x)+\log \left (2-x+\log ^2(3)\right ) \]

[In]

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

[Out]

-Log[x] + Log[2 - x + Log[3]^2]

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14

method	result	size
norman	\(-\ln \left (x \right )+\ln \left (\ln \left (3\right )^{2}-x +2\right )\)	\(16\)
parallelrisch	\(\frac {\left (-\ln \left (3\right )^{2}-2\right ) \left (\ln \left (x \right )-\ln \left (-\ln \left (3\right )^{2}+x -2\right )\right )}{2+\ln \left (3\right )^{2}}\)	\(33\)
default	\(\left (-\ln \left (3\right )^{2}-2\right ) \left (\frac {\ln \left (x \right )}{2+\ln \left (3\right )^{2}}-\frac {\ln \left (-\ln \left (3\right )^{2}+x -2\right )}{2+\ln \left (3\right )^{2}}\right )\)	\(42\)
risch	\(-\frac {\ln \left (x \right ) \ln \left (3\right )^{2}}{2+\ln \left (3\right )^{2}}-\frac {2 \ln \left (x \right )}{2+\ln \left (3\right )^{2}}+\frac {\ln \left (-\ln \left (3\right )^{2}+x -2\right ) \ln \left (3\right )^{2}}{2+\ln \left (3\right )^{2}}+\frac {2 \ln \left (-\ln \left (3\right )^{2}+x -2\right )}{2+\ln \left (3\right )^{2}}\)	\(73\)
meijerg	\(\frac {\ln \left (3\right )^{2} \left (-\ln \left (3\right )^{2}-2\right ) \left (\ln \left (x \right )-\ln \left (2+\ln \left (3\right )^{2}\right )+i \pi -\ln \left (1-\frac {x}{2+\ln \left (3\right )^{2}}\right )\right )}{\left (2+\ln \left (3\right )^{2}\right )^{2}}+\frac {2 \left (-\ln \left (3\right )^{2}-2\right ) \left (\ln \left (x \right )-\ln \left (2+\ln \left (3\right )^{2}\right )+i \pi -\ln \left (1-\frac {x}{2+\ln \left (3\right )^{2}}\right )\right )}{\left (2+\ln \left (3\right )^{2}\right )^{2}}\)	\(105\)

[In]

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

[Out]

-ln(x)+ln(ln(3)^2-x+2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int \frac {-2-\log ^2(3)}{2 x-x^2+x \log ^2(3)} \, dx=\log \left (-\log \left (3\right )^{2} + x - 2\right ) - \log \left (x\right ) \]

[In]

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

[Out]

log(-log(3)^2 + x - 2) - log(x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (10) = 20\).

Time = 0.17 (sec) , antiderivative size = 114, normalized size of antiderivative = 8.14 \[ \int \frac {-2-\log ^2(3)}{2 x-x^2+x \log ^2(3)} \, dx=\left (\frac {\log {\left (x - 1 - \frac {2 \log {\left (3 \right )}^{2}}{\log {\left (3 \right )}^{2} + 2} - \frac {2}{\log {\left (3 \right )}^{2} + 2} - \frac {\log {\left (3 \right )}^{2}}{2} - \frac {\log {\left (3 \right )}^{4}}{2 \left (\log {\left (3 \right )}^{2} + 2\right )} \right )}}{\log {\left (3 \right )}^{2} + 2} - \frac {\log {\left (x - 1 - \frac {\log {\left (3 \right )}^{2}}{2} + \frac {\log {\left (3 \right )}^{4}}{2 \left (\log {\left (3 \right )}^{2} + 2\right )} + \frac {2}{\log {\left (3 \right )}^{2} + 2} + \frac {2 \log {\left (3 \right )}^{2}}{\log {\left (3 \right )}^{2} + 2} \right )}}{\log {\left (3 \right )}^{2} + 2}\right ) \left (\log {\left (3 \right )}^{2} + 2\right ) \]

[In]

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

[Out]

(log(x - 1 - 2*log(3)**2/(log(3)**2 + 2) - 2/(log(3)**2 + 2) - log(3)**2/2 - log(3)**4/(2*(log(3)**2 + 2)))/(l
og(3)**2 + 2) - log(x - 1 - log(3)**2/2 + log(3)**4/(2*(log(3)**2 + 2)) + 2/(log(3)**2 + 2) + 2*log(3)**2/(log
(3)**2 + 2))/(log(3)**2 + 2))*(log(3)**2 + 2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (14) = 28\).

Time = 0.19 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.79 \[ \int \frac {-2-\log ^2(3)}{2 x-x^2+x \log ^2(3)} \, dx={\left (\log \left (3\right )^{2} + 2\right )} {\left (\frac {\log \left (-\log \left (3\right )^{2} + x - 2\right )}{\log \left (3\right )^{2} + 2} - \frac {\log \left (x\right )}{\log \left (3\right )^{2} + 2}\right )} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (14) = 28\).

Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.93 \[ \int \frac {-2-\log ^2(3)}{2 x-x^2+x \log ^2(3)} \, dx={\left (\log \left (3\right )^{2} + 2\right )} {\left (\frac {\log \left ({\left | -\log \left (3\right )^{2} + x - 2 \right |}\right )}{\log \left (3\right )^{2} + 2} - \frac {\log \left ({\left | x \right |}\right )}{\log \left (3\right )^{2} + 2}\right )} \]

[In]

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

[Out]

(log(3)^2 + 2)*(log(abs(-log(3)^2 + x - 2))/(log(3)^2 + 2) - log(abs(x))/(log(3)^2 + 2))

Mupad [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.29 \[ \int \frac {-2-\log ^2(3)}{2 x-x^2+x \log ^2(3)} \, dx=-2\,\mathrm {atanh}\left (\frac {4\,x}{2\,{\ln \left (3\right )}^2+4}-1\right ) \]

[In]

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

[Out]

-2*atanh((4*x)/(2*log(3)^2 + 4) - 1)