3.269 \(\int \frac{\sqrt{1+\log (x)}}{x \log (x)} \, dx\)

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

[Out]

-2*ArcTanh[Sqrt[1 + Log[x]]] + 2*Sqrt[1 + Log[x]]

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Rubi [A]  time = 0.0561296, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2365, 50, 63, 207} \[ 2 \sqrt{\log (x)+1}-2 \tanh ^{-1}\left (\sqrt{\log (x)+1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + Log[x]]/(x*Log[x]),x]

[Out]

-2*ArcTanh[Sqrt[1 + Log[x]]] + 2*Sqrt[1 + Log[x]]

Rule 2365

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(c_.)*(x_)^(n_.)]*(e_.))^(q_.))/(x_), x_Symbol]
:> Dist[1/n, Subst[Int[(a + b*x)^p*(d + e*x)^q, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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

Mathematica [A]  time = 0.021054, size = 22, normalized size = 1. \[ 2 \sqrt{\log (x)+1}-2 \tanh ^{-1}\left (\sqrt{\log (x)+1}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*ArcTanh[Sqrt[1 + Log[x]]] + 2*Sqrt[1 + Log[x]]

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Maple [A]  time = 0.002, size = 30, normalized size = 1.4 \begin{align*} 2\,\sqrt{1+\ln \left ( x \right ) }+\ln \left ( -1+\sqrt{1+\ln \left ( x \right ) } \right ) -\ln \left ( 1+\sqrt{1+\ln \left ( x \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.949866, size = 39, normalized size = 1.77 \begin{align*} 2 \, \sqrt{\log \left (x\right ) + 1} - \log \left (\sqrt{\log \left (x\right ) + 1} + 1\right ) + \log \left (\sqrt{\log \left (x\right ) + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(log(x) + 1) - log(sqrt(log(x) + 1) + 1) + log(sqrt(log(x) + 1) - 1)

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Fricas [A]  time = 1.97212, size = 103, normalized size = 4.68 \begin{align*} 2 \, \sqrt{\log \left (x\right ) + 1} - \log \left (\sqrt{\log \left (x\right ) + 1} + 1\right ) + \log \left (\sqrt{\log \left (x\right ) + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(log(x) + 1) - log(sqrt(log(x) + 1) + 1) + log(sqrt(log(x) + 1) - 1)

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Sympy [A]  time = 1.79135, size = 32, normalized size = 1.45 \begin{align*} 2 \sqrt{\log{\left (x \right )} + 1} + \log{\left (\sqrt{\log{\left (x \right )} + 1} - 1 \right )} - \log{\left (\sqrt{\log{\left (x \right )} + 1} + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(log(x) + 1) + log(sqrt(log(x) + 1) - 1) - log(sqrt(log(x) + 1) + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out