\(\int \frac {-2+\log (-x^2)}{\log ^2(-x^2)} \, dx\) [4844]

Optimal result

Integrand size = 17, antiderivative size = 17 \[ \int \frac {-2+\log \left (-x^2\right )}{\log ^2\left (-x^2\right )} \, dx=4+\frac {2}{e^4}+\frac {x}{\log \left (-x^2\right )} \]

[Out]

2/exp(4)+x/ln(-x^2)+4

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.59, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2407, 2334, 2337, 2209} \[ \int \frac {-2+\log \left (-x^2\right )}{\log ^2\left (-x^2\right )} \, dx=\frac {x}{\log \left (-x^2\right )} \]

[In]

Int[(-2 + Log[-x^2])/Log[-x^2]^2,x]

[Out]

x/Log[-x^2]

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2334

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

Rule 2337

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[E^(x/n)*(a +
b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2407

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(Log[(c_.)*(x_)^(n_.)]*(e_.) + (d_))^(q_.), x_Symbol] :> Int[E
xpandIntegrand[(a + b*Log[c*x^n])^p*(d + e*Log[c*x^n])^q, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IntegerQ[p
] && IntegerQ[q]

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.59 \[ \int \frac {-2+\log \left (-x^2\right )}{\log ^2\left (-x^2\right )} \, dx=\frac {x}{\log \left (-x^2\right )} \]

[In]

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

[Out]

x/Log[-x^2]

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65

method	result	size
norman	\(\frac {x}{\ln \left (-x^{2}\right )}\)	\(11\)
risch	\(\frac {x}{\ln \left (-x^{2}\right )}\)	\(11\)
parallelrisch	\(\frac {x}{\ln \left (-x^{2}\right )}\)	\(11\)

[In]

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

[Out]

x/ln(-x^2)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.59 \[ \int \frac {-2+\log \left (-x^2\right )}{\log ^2\left (-x^2\right )} \, dx=\frac {x}{\log \left (-x^{2}\right )} \]

[In]

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

[Out]

x/log(-x^2)

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.41 \[ \int \frac {-2+\log \left (-x^2\right )}{\log ^2\left (-x^2\right )} \, dx=\frac {x}{\log {\left (- x^{2} \right )}} \]

[In]

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

[Out]

x/log(-x**2)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.71 \[ \int \frac {-2+\log \left (-x^2\right )}{\log ^2\left (-x^2\right )} \, dx=\frac {x}{i \, \pi + 2 \, \log \left (x\right )} \]

[In]

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

[Out]

x/(I*pi + 2*log(x))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.59 \[ \int \frac {-2+\log \left (-x^2\right )}{\log ^2\left (-x^2\right )} \, dx=\frac {x}{\log \left (-x^{2}\right )} \]

[In]

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

[Out]

x/log(-x^2)

Mupad [B] (verification not implemented)

Time = 10.84 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.59 \[ \int \frac {-2+\log \left (-x^2\right )}{\log ^2\left (-x^2\right )} \, dx=\frac {x}{\ln \left (-x^2\right )} \]

[In]

int((log(-x^2) - 2)/log(-x^2)^2,x)

[Out]

x/log(-x^2)