3.49.44 \(\int \frac {-2+\log (-x^2)}{\log ^2(-x^2)} \, dx\)

Optimal. Leaf size=17 \[ 4+\frac {2}{e^4}+\frac {x}{\log \left (-x^2\right )} \]

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Rubi [A]  time = 0.05, antiderivative size = 10, normalized size of antiderivative = 0.59, number of steps used = 7, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2360, 2297, 2300, 2178} \begin {gather*} \frac {x}{\log \left (-x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2 + Log[-x^2])/Log[-x^2]^2,x]

[Out]

x/Log[-x^2]

Rule 2178

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

Rule 2297

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 2300

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 2360

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 {gather*} \begin {aligned} \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 \operatorname {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 \operatorname {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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 10, normalized size = 0.59 \begin {gather*} \frac {x}{\log \left (-x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/Log[-x^2]

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fricas [A]  time = 0.62, size = 10, normalized size = 0.59 \begin {gather*} \frac {x}{\log \left (-x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/log(-x^2)

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giac [A]  time = 0.16, size = 10, normalized size = 0.59 \begin {gather*} \frac {x}{\log \left (-x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/log(-x^2)

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maple [A]  time = 0.02, size = 11, normalized size = 0.65

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/ln(-x^2)

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maxima [C]  time = 0.37, size = 12, normalized size = 0.71 \begin {gather*} \frac {x}{i \, \pi + 2 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(I*pi + 2*log(x))

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mupad [B]  time = 3.46, size = 10, normalized size = 0.59 \begin {gather*} \frac {x}{\ln \left (-x^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(-x^2) - 2)/log(-x^2)^2,x)

[Out]

x/log(-x^2)

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sympy [A]  time = 0.08, size = 7, normalized size = 0.41 \begin {gather*} \frac {x}{\log {\left (- x^{2} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/log(-x**2)

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