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 $?=0$ that we need, the integral diverges:
Integrate[Exp[-n x]/x^2 x^? , {x, 0, ?}]
And the sum can be done in closed form for general $?$:
Sum[n^(1 - ?) Gamma[-1 + ?] Cos[(n ?)/2] (2 + n^2), {n, ?}]
Expanding the result around $?=0$ gives a finite leading term:
Series[%, {?, 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, ?, ?}]
The leading terms are:
Series[(E^(-n ?)/? - n Gamma[0, n ?]), {?, 0, 2}, Assumptions -> ? > 0]
The n term does give no sum contribution:
Sum[n Cos[(n ?)/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 ?)/2] (2 + n^2) n^?, {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:
