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

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

[Out]

-(ArcTanh[Sqrt[a^2 + Log[x]^2]/a]/a)

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Rubi [A]  time = 0.0879265, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {266, 63, 207} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a^2+\log ^2(x)}}{a}\right )}{a} \]

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 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]]

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 207

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

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{\tanh ^{-1}\left (\frac{\sqrt{a^2+\log ^2(x)}}{a}\right )}{a}\\ \end{align*}

Mathematica [A]  time = 0.0171483, size = 22, normalized size = 1. \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a^2+\log ^2(x)}}{a}\right )}{a} \]

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 = 0.005, size = 37, normalized size = 1.7 \begin{align*} -{\ln \left ({\frac{1}{\ln \left ( x \right ) } \left ( 2\,{a}^{2}+2\,\sqrt{{a}^{2}}\sqrt{{a}^{2}+ \left ( \ln \left ( x \right ) \right ) ^{2}} \right ) } \right ){\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/ln(x)/(a^2+ln(x)^2)^(1/2),x)

[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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [B]  time = 2.3981, size = 117, normalized size = 5.32 \begin{align*} -\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{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{a^{2} + \log{\left (x \right )}^{2}} \log{\left (x \right )}}\, dx \end{align*}

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]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

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]

sage0*x