NOTE: the notebook with full content is attached at the end of the post.
On Mathoverflow, the question about a closed form for the integral
$\int_0^{\infty}\frac{\tanh^3(x)}{x^2}\,dx$ was raised. Currently, Mathematica can't give a closed form answer for this integral.
Integrate[Tanh[x]^3/x^2, {x, 0, ∞}]
But we can easily get a numerical value.
num = NIntegrate[Tanh[x]^3/x^2, {x, 0, ∞},
WorkingPrecision -> 100, PrecisionGoal -> 30] // N[#, 30] &
1.15478531332317626405907045194
How could one fine a closed form result? With Integrate
, Sum
, SeriesCoefficient
, ... we have a lot of tools at our hands to do some experimental mathematics. A simple manual change of variables does not help.
Integrate[y^3/(ArcTanh[y]^2) Dt[ArcTanh[y], y], {y, 0, 1}]
The next typical trick is expanding the integrand in a series and integrating term by term:
TrigToExp[Tanh[x]^3]
SeriesCoefficient[% /. {Exp[-x] -> g, Exp[x] -> 1/g}, {g, 0, n}] //
FullSimplify[#, n ∈ Integers] &
The resulting integrals do diverge:
Integrate[Exp[-n x]/x^2 , {x, 0, ∞}]
We can make the integral convergent by adding a factor. For the case
$α=0$ that we need, the integral diverges:
Integrate[Exp[-n x]/x^2 x^α , {x, 0, ∞}]
And the sum can be done in closed form for general
$α$:
Sum[n^(1 - α) Gamma[-1 + α] Cos[(n π)/2] (2 + n^2), {n, ∞}]
Expanding the result around
$α=0$ gives a finite leading term:
Series[%, {α, 0, 1}]
And indeed this is the result we were looking for:
N[(-1 - (4 Log[2])/15 + 12 Log[Glaisher] - 120 Derivative[1][Zeta][-3]), 30]
1.15478531332317626405907045194
Alternatively, we can split of the divergent point from the integral:
Integrate[Exp[-n x]/x^2, {x, ε, ∞}]
The leading terms are:
Series[(E^(-n ε)/ε - n Gamma[0, n ε]), {ε, 0, 2}, Assumptions -> ε > 0]
The n term does give no sum contribution:
Sum[n Cos[(n π)/2] (2 + n^2), {n, ∞}, Regularization -> "Dirichlet"]
0
And the
$n \ln(n)$ term gives the same result as above:
Sum[n Log [n] Cos[(n π)/2] (2 + n^2) n^α, {n, ∞}]
Series[%, {\[Alpha], 0, 1}]
By conjecturing the constants involved in the integral, we could have used FindIntegerNullVector
to get a conjecture for a closed form:
FindIntegerNullVector[{1.15478531332317626405907045194,1,
Log[2], EulerGamma, Log[Glaisher], Derivative[1][Zeta][-3]}]
{-15, -15, -4, 0, 180, -1800}
Attachments: