3.627 \(\int \frac {1}{x \log (x) \sqrt {-a^2+\log ^2(x)}} \, dx\)

Optimal. Leaf size=23 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {\log ^2(x)-a^2}}{a}\right )}{a} \]

________________________________________________________________________________________

Rubi [A]  time = 0.10, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {266, 63, 203} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {\log ^2(x)-a^2}}{a}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Log[x]*Sqrt[-a^2 + Log[x]^2]),x]

[Out]

ArcTan[Sqrt[-a^2 + Log[x]^2]/a]/a

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {1}{x \log (x) \sqrt {-a^2+\log ^2(x)}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{x \sqrt {-a^2+x^2}} \, dx,x,\log (x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {-a^2+x}} \, dx,x,\log ^2(x)\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{a^2+x^2} \, dx,x,\sqrt {-a^2+\log ^2(x)}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {-a^2+\log ^2(x)}}{a}\right )}{a}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 23, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {\log ^2(x)-a^2}}{a}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Log[x]*Sqrt[-a^2 + Log[x]^2]),x]

[Out]

ArcTan[Sqrt[-a^2 + Log[x]^2]/a]/a

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \log (x) \sqrt {-a^2+\log ^2(x)}} \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[1/(x*Log[x]*Sqrt[-a^2 + Log[x]^2]),x]

[Out]

Could not integrate

________________________________________________________________________________________

fricas [A]  time = 0.87, size = 27, normalized size = 1.17 \[ \frac {2 \, \arctan \left (\frac {\sqrt {-a^{2} + \log \relax (x)^{2}} - \log \relax (x)}{a}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/log(x)/(-a^2+log(x)^2)^(1/2),x, algorithm="fricas")

[Out]

2*arctan((sqrt(-a^2 + log(x)^2) - log(x))/a)/a

________________________________________________________________________________________

giac [A]  time = 1.63, size = 21, normalized size = 0.91 \[ \frac {\arctan \left (\frac {\sqrt {-a^{2} + \log \relax (x)^{2}}}{a}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/log(x)/(-a^2+log(x)^2)^(1/2),x, algorithm="giac")

[Out]

arctan(sqrt(-a^2 + log(x)^2)/a)/a

________________________________________________________________________________________

maple [A]  time = 0.02, size = 43, normalized size = 1.87

method	result	size
derivativedivides	\(-\frac {\ln \left (\frac {-2 a^{2}+2 \sqrt {-a^{2}}\, \sqrt {-a^{2}+\ln \relax (x )^{2}}}{\ln \relax (x )}\right )}{\sqrt {-a^{2}}}\)	\(43\)
default	\(-\frac {\ln \left (\frac {-2 a^{2}+2 \sqrt {-a^{2}}\, \sqrt {-a^{2}+\ln \relax (x )^{2}}}{\ln \relax (x )}\right )}{\sqrt {-a^{2}}}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/ln(x)/(-a^2+ln(x)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/(-a^2)^(1/2)*ln((-2*a^2+2*(-a^2)^(1/2)*(-a^2+ln(x)^2)^(1/2))/ln(x))

________________________________________________________________________________________

maxima [A]  time = 0.95, size = 13, normalized size = 0.57 \[ -\frac {\arcsin \left (\frac {a}{{\left | \log \relax (x) \right |}}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/log(x)/(-a^2+log(x)^2)^(1/2),x, algorithm="maxima")

[Out]

-arcsin(a/abs(log(x)))/a

________________________________________________________________________________________

mupad [B]  time = 0.52, size = 25, normalized size = 1.09 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {{\ln \relax (x)}^2-a^2}}{\sqrt {a^2}}\right )}{\sqrt {a^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*log(x)*(log(x)^2 - a^2)^(1/2)),x)

[Out]

atan((log(x)^2 - a^2)^(1/2)/(a^2)^(1/2))/(a^2)^(1/2)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {- \left (a - \log {\relax (x )}\right ) \left (a + \log {\relax (x )}\right )} \log {\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/ln(x)/(-a**2+ln(x)**2)**(1/2),x)

[Out]

Integral(1/(x*sqrt(-(a - log(x))*(a + log(x)))*log(x)), x)

________________________________________________________________________________________