3.56.17 \(\int \frac {-2+\log (x^2)}{\log ^2(x^2)} \, dx\) [5517]

Optimal. Leaf size=8 \[ \frac {x}{\log \left (x^2\right )} \]

[Out]

x/ln(x^2)

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2407, 2334, 2337, 2209} \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 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 {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 \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 {\log \left (x^2\right )}{2}\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 {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 8, normalized size = 1.00 \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]

________________________________________________________________________________________

Maple [A]
time = 2.59, size = 9, normalized size = 1.12

method	result	size
norman	\(\frac {x}{\ln \left (x^{2}\right )}\)	\(9\)
risch	\(\frac {x}{\ln \left (x^{2}\right )}\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/ln(x^2)

________________________________________________________________________________________

Maxima [A]
time = 0.30, size = 7, normalized size = 0.88 \begin {gather*} \frac {x}{2 \, \log \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x/log(x)

________________________________________________________________________________________

Fricas [A]
time = 0.33, size = 8, normalized size = 1.00 \begin {gather*} \frac {x}{\log \left (x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/log(x^2)

________________________________________________________________________________________

Sympy [A]
time = 0.02, size = 5, normalized size = 0.62 \begin {gather*} \frac {x}{\log {\left (x^{2} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/log(x**2)

________________________________________________________________________________________

Giac [A]
time = 0.41, size = 8, normalized size = 1.00 \begin {gather*} \frac {x}{\log \left (x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/log(x^2)

________________________________________________________________________________________

Mupad [B]
time = 3.61, size = 8, normalized size = 1.00 \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)

________________________________________________________________________________________