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

Optimal. Leaf size=18 \[ \tan ^{-1}\left (\frac {\log (x)}{\sqrt {a^2-\log ^2(x)}}\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {217, 203} \[ \tan ^{-1}\left (\frac {\log (x)}{\sqrt {a^2-\log ^2(x)}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[a^2 - Log[x]^2]),x]

[Out]

ArcTan[Log[x]/Sqrt[a^2 - Log[x]^2]]

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 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 18, normalized size = 1.00 \[ \tan ^{-1}\left (\frac {\log (x)}{\sqrt {a^2-\log ^2(x)}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[Log[x]/Sqrt[a^2 - Log[x]^2]]

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

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [A]  time = 0.91, size = 25, normalized size = 1.39 \[ -2 \, \arctan \left (-\frac {a - \sqrt {a^{2} - \log \relax (x)^{2}}}{\log \relax (x)}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.77, size = 10, normalized size = 0.56 \[ \arcsin \left (\frac {\log \relax (x)}{a}\right ) \mathrm {sgn}\relax (a) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(log(x)/a)*sgn(a)

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maple [A]  time = 0.02, size = 17, normalized size = 0.94

method	result	size
derivativedivides	\(\arctan \left (\frac {\ln \relax (x )}{\sqrt {a^{2}-\ln \relax (x )^{2}}}\right )\)	\(17\)
default	\(\arctan \left (\frac {\ln \relax (x )}{\sqrt {a^{2}-\ln \relax (x )^{2}}}\right )\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.00, size = 7, normalized size = 0.39 \[ \arcsin \left (\frac {\log \relax (x)}{a}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(log(x)/a)

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mupad [B]  time = 0.61, size = 16, normalized size = 0.89 \[ \mathrm {atan}\left (\frac {\ln \relax (x)}{\sqrt {a^2-{\ln \relax (x)}^2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a^2 - log(x)^2)^(1/2)),x)

[Out]

atan(log(x)/(a^2 - log(x)^2)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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