I see in An Unpublished Manuscript of Ramanujan on identities of B.C. Berndt a series similar to the following and I try to sum using Abel-Plana sum formula but I do not sure it is right could you show it. First, define $a_k=(4k+1)\pi$.
I. Let $u_1=1+\sqrt{2}$
$$A=\sum _{k=0}^{\infty } \frac{64 \left(u_1 \sinh \left(a_k\sin \frac{\pi}{8}\right)-\sin \left(a_k\cos \frac{\pi}{8}\right)\right)}{\pi ^5 (4 k+1)^{11} \left(\cos \left(a_k\cos \frac{\pi}{8}\right)+\cosh \left(a_k\sin \frac{\pi}{8}\right)\right)}$$
then
$$A=\frac{17 \pi ^6 \sqrt{\frac{1}{2} \left(2-\sqrt{2}\right)}}{60480}+\frac{169 \pi ^6 \sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)}}{302400}$$
II. Let $v_1 = 2+\sqrt{2}$ and $v_2 =-2+\sqrt{2-\sqrt{2}}$
$$B=\sum _{k=0}^{\infty } \frac{32 \sqrt{2} \left(\sqrt{v_1} \sinh \left(a_k\sin \frac{\pi}{16}\right)-v_2 \sin \left(a_k\cos \frac{\pi}{16}\right)\right)}{\pi ^5 (4 k+1)^{11} \left(\cos \left(a_k\cos \frac{\pi}{16}\right)+\cosh \left(a_k\sin \frac{\pi}{16}\right)\right)}$$
then
$${B=}\frac{ \pi^6}{8} \left(-\frac{11 \sqrt{\frac{1}{2} \left(2-\sqrt{v_1}\right)}}{22680}+\frac{ \left(\sqrt{2-\sqrt{v_1}}+\sqrt{v_1 \left(2-\sqrt{v_1}\right)}\right)}{450 \sqrt{2}}+\\ \frac{31 \sqrt{\frac{1}{2} \left(2+\sqrt{v_1}\right)}}{11340}-\frac{17 \left(\sqrt{2+\sqrt{v_1}}\right)-\sqrt{v_1 \left(2+\sqrt{v_1}\right)}}{7560 \sqrt{2}}\right)$$