3.7.25 \(\int \frac {1}{x \log (x) \sqrt {a^2+\log ^2(x)}} \, dx\) [625]

Optimal. Leaf size=22 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a^2+\log ^2(x)}}{a}\right )}{a} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {272, 65, 213} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a^2+\log ^2(x)}}{a}\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(ArcTanh[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 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[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*} \int \frac {1}{x \log (x) \sqrt {a^2+\log ^2(x)}} \, dx &=\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 {\tanh ^{-1}\left (\frac {\sqrt {a^2+\log ^2(x)}}{a}\right )}{a}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 22, normalized size = 1.00 \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a^2+\log ^2(x)}}{a}\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(ArcTanh[Sqrt[a^2 + Log[x]^2]/a]/a)

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Maple [A]
time = 0.01, size = 37, normalized size = 1.68

method	result	size
derivativedivides	\(-\frac {\ln \left (\frac {2 a^{2}+2 \sqrt {a^{2}}\, \sqrt {a^{2}+\ln \left (x \right )^{2}}}{\ln \left (x \right )}\right )}{\sqrt {a^{2}}}\)	\(37\)
default	\(-\frac {\ln \left (\frac {2 a^{2}+2 \sqrt {a^{2}}\, \sqrt {a^{2}+\ln \left (x \right )^{2}}}{\ln \left (x \right )}\right )}{\sqrt {a^{2}}}\)	\(37\)

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))

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Maxima [A]
time = 2.38, size = 13, normalized size = 0.59 \begin {gather*} -\frac {\operatorname {arsinh}\left (\frac {a}{{\left | \log \left (x\right ) \right |}}\right )}{a} \end {gather*}

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]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (20) = 40\).
time = 0.66, size = 44, normalized size = 2.00 \begin {gather*} -\frac {\log \left (a + \sqrt {a^{2} + \log \left (x\right )^{2}} - \log \left (x\right )\right ) - \log \left (-a + \sqrt {a^{2} + \log \left (x\right )^{2}} - \log \left (x\right )\right )}{a} \end {gather*}

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]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt {a^{2} + \log {\left (x \right )}^{2}} \log {\left (x \right )}}\, dx \end {gather*}

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**2 + log(x)**2)*log(x)), x)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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]

Timed out

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Mupad [B]
time = 0.58, size = 27, normalized size = 1.23 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {a^2+{\ln \left (x\right )}^2}}{\sqrt {-a^2}}\right )}{\sqrt {-a^2}} \end {gather*}

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)

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