\(\int \frac {1}{x \log (x) \sqrt {-a^2+\log ^2(x)}} \, dx\) [627]

Optimal result

Integrand size = 22, antiderivative size = 23 \[ \int \frac {1}{x \log (x) \sqrt {-a^2+\log ^2(x)}} \, dx=\frac {\arctan \left (\frac {\sqrt {-a^2+\log ^2(x)}}{a}\right )}{a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {272, 65, 209} \[ \int \frac {1}{x \log (x) \sqrt {-a^2+\log ^2(x)}} \, dx=\frac {\arctan \left (\frac {\sqrt {\log ^2(x)-a^2}}{a}\right )}{a} \]

[In]

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

[Out]

ArcTan[Sqrt[-a^2 + Log[x]^2]/a]/a

Rule 65

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 209

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

Rule 272

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \log (x) \sqrt {-a^2+\log ^2(x)}} \, dx=\frac {\arctan \left (\frac {\sqrt {-a^2+\log ^2(x)}}{a}\right )}{a} \]

[In]

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

[Out]

ArcTan[Sqrt[-a^2 + Log[x]^2]/a]/a

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.87

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

[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))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x \log (x) \sqrt {-a^2+\log ^2(x)}} \, dx=\frac {2 \, \arctan \left (\frac {\sqrt {-a^{2} + \log \left (x\right )^{2}} - \log \left (x\right )}{a}\right )}{a} \]

[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

Sympy [F]

\[ \int \frac {1}{x \log (x) \sqrt {-a^2+\log ^2(x)}} \, dx=\int \frac {1}{x \sqrt {- \left (a - \log {\left (x \right )}\right ) \left (a + \log {\left (x \right )}\right )} \log {\left (x \right )}}\, dx \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.57 \[ \int \frac {1}{x \log (x) \sqrt {-a^2+\log ^2(x)}} \, dx=-\frac {\arcsin \left (\frac {a}{{\left | \log \left (x\right ) \right |}}\right )}{a} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x \log (x) \sqrt {-a^2+\log ^2(x)}} \, dx=\frac {\arctan \left (\frac {\sqrt {-a^{2} + \log \left (x\right )^{2}}}{a}\right )}{a} \]

[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

Mupad [B] (verification not implemented)

Time = 0.66 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x \log (x) \sqrt {-a^2+\log ^2(x)}} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {{\ln \left (x\right )}^2-a^2}}{\sqrt {a^2}}\right )}{\sqrt {a^2}} \]

[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)