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包含大分数的超长方程

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包含大分数的超长方程

我试图输入这个非常长的方程，但没有成功。我从 Mathematica 中得到这个结果并复制了它。出于某种原因，括号的形状没有根据分数的高度改变。

我尝试过使用自动线制动器，但\usepackage{breqn}没有\begin{dmath}成功。

在此处输入图片描述

-\frac{2 u_g \cosh \left(\frac{h \pi }{D}\right) \sin \left(\frac{h \pi }{D}\right) \sin \left(\frac{\pi  z}{D}\right) \sinh \left(\frac{\pi  z}{D}\right) \cos ^2\left(\frac{h \pi }{D}\right)}{\left(\cos \left(\frac{2 h \pi }{D}\right)+\cosh \left(\frac{2 h \pi }{D}\right)\right) \left(\cos ^2\left(\frac{h \pi }{D}\right) \cosh ^2\left(\frac{h \pi }{D}\right)+\sin ^2\left(\frac{h \pi }{D}\right) \sinh ^2\left(\frac{h \pi }{D}\right)\right) \left(\frac{\cosh \left(\frac{h \pi }{D}\right) \sinh \left(\frac{h \pi }{D}\right)}{\cos ^2\left(\frac{h \pi }{D}\right) \cosh ^2\left(\frac{h \pi }{D}\right)+\sin ^2\left(\frac{h \pi }{D}\right) \sinh ^2\left(\frac{h \pi }{D}\right)}-\frac{\cos \left(\frac{h \pi }{D}\right) \sin \left(\frac{h \pi }{D}\right)}{\cos ^2\left(\frac{h \pi }{D}\right) \cosh ^2\left(\frac{h \pi }{D}\right)+\sin ^2\left(\frac{h \pi }{D}\right) \sinh ^2\left(\frac{h \pi }{D}\right)}\right)}-\frac{4 \pi  \tau_y \cosh ^2\left(\frac{h \pi }{D}\right) \sin \left(\frac{\pi  z}{D}\right) \sinh \left(\frac{\pi  z}{D}\right) \cos ^2\left(\frac{h \pi }{D}\right)}{D f \rho_0 \left(\cos \left(\frac{2 h \pi }{D}\right)+\cosh \left(\frac{2 h \pi }{D}\right)\right) \left(\cos ^2\left(\frac{h \pi }{D}\right) \cosh ^2\left(\frac{h \pi }{D}\right)+\sin ^2\left(\frac{h \pi }{D}\right) \sinh ^2\left(\frac{h \pi }{D}\right)\right) \left(\frac{\cosh \left(\frac{h \pi }{D}\right) \sinh \left(\frac{h \pi }{D}\right)}{\cos ^2\left(\frac{h \pi }{D}\right) \cosh ^2\left(\frac{h \pi }{D}\right)+\sin ^2\left(\frac{h \pi }{D}\right) \sinh ^2\left(\frac{h \pi }{D}\right)}-\frac{\cos \left(\frac{h \pi }{D}\right) \sin \left(\frac{h \pi }{D}\right)}{\cos ^2\left(\frac{h \pi }{D}\right) \cosh ^2\left(\frac{h \pi }{D}\right)+\sin ^2\left(\frac{h \pi }{D}\right) \sinh ^2\left(\frac{h \pi }{D}\right)}\right)}+\frac{4 h \pi  u_g \cosh \left(\frac{h \pi }{D}\right) \sin \left(\frac{\pi  z}{D}\right) \sinh \left(\frac{\pi  z}{D}\right) \cos \left(\frac{h \pi }{D}\right)}{D \left(\cos \left(\frac{2 h \pi }{D}\right)+\cosh \left(\frac{2 h \pi }{D}\right)\right) \left(\frac{\cosh \left(\frac{h \pi }{D}\right) \sinh \left(\frac{h \pi }{D}\right)}{\cos ^2\left(\frac{h \pi }{D}\right) \cosh ^2\left(\frac{h \pi }{D}\right)+\sin ^2\left(\frac{h \pi }{D}\right) \sinh ^2\left(\frac{h \pi }{D}\right)}-\frac{\cos \left(\frac{h \pi }{D}\right) \sin \left(\frac{h \pi }{D}\right)}{\cos ^2\left(\frac{h \pi }{D}\right) \cosh ^2\left(\frac{h \pi }{D}\right)+\sin ^2\left(\frac{h \pi }{D}\right) \sinh ^2\left(\frac{h \pi }{D}\right)}\right)}+\frac{4 \pi  \tau_y \cosh \left(\frac{h \pi }{D}\right) \sin \left(\frac{\pi  z}{D}\right) \sinh \left(\frac{\pi  z}{D}\right) \cos \left(\frac{h \pi }{D}\right)}{D f \rho_0 \left(\cos \left(\frac{2 h \pi }{D}\right)+\cosh \left(\frac{2 h \pi }{D}\right)\right) \left(\frac{\cosh \left(\frac{h \pi }{D}\right) \sinh \left(\frac{h \pi }{D}\right)}{\cos ^2\left(\frac{h \pi }{D}\right) \cosh ^2\left(\frac{h \pi }{D}\right)+\sin ^2\left(\frac{h \pi }{D}\right) \sinh ^2\left(\frac{h \pi }{D}\right)}-\frac{\cos \left(\frac{h \pi }{D}\right) \sin \left(\frac{h \pi }{D}\right)}{\cos ^2\left(\frac{h \pi }{D}\right) \cosh ^2\left(\frac{h \pi }{D}\right)+\sin ^2\left(\frac{h \pi }{D}\right) \sinh ^2\left(\frac{h \pi }{D}\right)}\right)}+\frac{2 \pi  \tau_y \cos \left(\frac{\pi  (h+z)}{D}\right) \cosh \left(\frac{h \pi }{D}\right) \sinh \left(\frac{\pi  (h+z)}{D}\right) \cos \left(\frac{h \pi }{D}\right)}{D f \rho_0 \left(\cos \left(\frac{2 h \pi }{D}\right)+\cosh \left(\frac{2 h \pi }{D}\right)\right)}-\frac{2 u_g \cos \left(\frac{\pi  z}{D}\right) \cosh \left(\frac{h \pi }{D}\right) \cosh \left(\frac{\pi  z}{D}\right) \cos \left(\frac{h \pi }{D}\right)}{\cos \left(\frac{2 h \pi }{D}\right)+\cosh \left(\frac{2 h \pi }{D}\right)}-\frac{2 \pi  \tau_y \cosh \left(\frac{h \pi }{D}\right) \cosh \left(\frac{\pi  (h+z)}{D}\right) \sin \left(\frac{\pi  (h+z)}{D}\right) \cos \left(\frac{h \pi }{D}\right)}{D f \rho_0 \left(\cos \left(\frac{2 h \pi }{D}\right)+\cosh \left(\frac{2 h \pi }{D}\right)\right)}+\frac{2 u_g \cos \left(\frac{\pi  z}{D}\right) \cosh \left(\frac{\pi  z}{D}\right) \sin ^2\left(\frac{h \pi }{D}\right) \sinh \left(\frac{h \pi }{D}\right) \cos \left(\frac{h \pi }{D}\right)}{\left(\cos \left(\frac{2 h \pi }{D}\right)+\cosh \left(\frac{2 h \pi }{D}\right)\right) \left(\cos ^2\left(\frac{h \pi }{D}\right) \cosh ^2\left(\frac{h \pi }{D}\right)+\sin ^2\left(\frac{h \pi }{D}\right) \sinh ^2\left(\frac{h \pi }{D}\right)\right) \left(\frac{\cosh \left(\frac{h \pi }{D}\right) \sinh \left(\frac{h \pi }{D}\right)}{\cos ^2\left(\frac{h \pi }{D}\right) \cosh ^2\left(\frac{h \pi }{D}\right)+\sin ^2\left(\frac{h \pi }{D}\right) \sinh ^2\left(\frac{h \pi }{D}\right)}-\frac{\cos \left(\frac{h \pi }{D}\right) \sin \left(\frac{h \pi }{D}\right)}{\cos ^2\left(\frac{h \pi }{D}\right) \cosh ^2\left(\frac{h \pi }{D}\right)+\sin ^2\left(\frac{h \pi }{D}\right) \sinh ^2\left(\frac{h \pi }{D}\right)}\right)}-\frac{2 u_g \cosh ^2\left(\frac{h \pi }{D}\right) \sin \left(\frac{\pi  z}{D}\right) \sinh \left(\frac{h \pi }{D}\right) \sinh \left(\frac{\pi  z}{D}\right) \cos \left(\frac{h \pi }{D}\right)}{\left(\cos \left(\frac{2 h \pi }{D}\right)+\cosh \left(\frac{2 h \pi }{D}\right)\right) \left(\cos ^2\left(\frac{h \pi }{D}\right) \cosh ^2\left(\frac{h \pi }{D}\right)+\sin ^2\left(\frac{h \pi }{D}\right) \sinh ^2\left(\frac{h \pi }{D}\right)\right) \left(\frac{\cosh \left(\frac{h \pi }{D}\right) \sinh \left(\frac{h \pi }{D}\right)}{\cos ^2\left(\frac{h \pi }{D}\right) \cosh ^2\left(\frac{h \pi }{D}\right)+\sin ^2\left(\frac{h \pi }{D}\right) \sinh ^2\left(\frac{h \pi }{D}\right)}-\frac{\cos \left(\frac{h \pi }{D}\right) \sin \left(\frac{h \pi }{D}\right)}{\cos ^2\left(\frac{h \pi }{D}\right) \cosh ^2\left(\frac{h \pi }{D}\right)+\sin ^2\left(\frac{h \pi }{D}\right) \sinh ^2\left(\frac{h \pi }{D}\right)}\right)}+\frac{4 \pi  \tau_y \cos \left(\frac{\pi  z}{D}\right) \cosh \left(\frac{h \pi }{D}\right) \cosh \left(\frac{\pi  z}{D}\right) \sin \left(\frac{h \pi }{D}\right) \sinh \left(\frac{h \pi }{D}\right) \cos \left(\frac{h \pi }{D}\right)}{D f \rho_0 \left(\cos \left(\frac{2 h \pi }{D}\right)+\cosh \left(\frac{2 h \pi }{D}\right)\right) \left(\cos ^2\left(\frac{h \pi }{D}\right) \cosh ^2\left(\frac{h \pi }{D}\right)+\sin ^2\left(\frac{h \pi }{D}\right) \sinh ^2\left(\frac{h \pi }{D}\right)\right) \left(\frac{\cosh \left(\frac{h \pi }{D}\right) \sinh \left(\frac{h \pi }{D}\right)}{\cos ^2\left(\frac{h \pi }{D}\right) \cosh ^2\left(\frac{h \pi }{D}\right)+\sin ^2\left(\frac{h \pi }{D}\right) \sinh ^2\left(\frac{h \pi }{D}\right)}-\frac{\cos \left(\frac{h \pi }{D}\right) \sin \left(\frac{h \pi }{D}\right)}{\cos ^2\left(\frac{h \pi }{D}\right) \cosh ^2\left(\frac{h \pi }{D}\right)+\sin ^2\left(\frac{h \pi }{D}\right) \sinh ^2\left(\frac{h \pi }{D}\right)}\right)}+u_g

有什么建议么？

答案1

像这样吗？

在此处输入图片描述

\documentclass{article}
\usepackage[letterpaper,margin=1in]{geometry} % set page parameters appropriately
\usepackage{amsmath} % for 'align*' env.
\begin{document}
Put $\lambda=h\pi/D$, $\mu=\pi z/D$, and $\nu=\lambda+\mu$. Put 
$P=\cos(2\lambda)+\cosh(2\lambda)$, 
$Q=\cos^2\lambda \cosh^2\lambda + \sin^2\lambda \sinh^2\lambda$, and
$R=\cosh\lambda \sinh\lambda - \cos\lambda \sin\lambda$. Then
\begin{align*}
u_g
&-\frac{2 u_g \cosh\lambda \sin\lambda \sin\mu \sinh\mu \cos^2\lambda}{PR}
 -\frac{4\pi \tau_y \cosh^2\lambda \sin\mu \sinh\mu \cos^2\lambda}{f\rho_0 DPR} \\
&+\frac{4 h\pi u_g \cosh\lambda \sin\mu \sinh\mu \cos\lambda}{DPR/Q}
 +\frac{4\pi \tau_y \cosh\lambda \sin\mu \sinh\mu \cos\lambda}{f\rho_0 DPR/Q}\\
&+\frac{2\pi \tau_y \cos\nu \cosh\lambda \sinh\nu \cos\lambda}{f\rho_0 DP}
 -\frac{2 u_g \cos\mu \cosh\lambda \cosh\mu \cos\lambda}{P}\\
&-\frac{2\pi \tau_y \cosh\lambda \cosh\nu \sin\nu \cos\lambda}{f\rho_0 DP}
 +\frac{2 u_g \cos\mu \cosh\mu \sin^2\lambda \sinh\lambda \cos\lambda}{PR} \\
&-\frac{2 u_g \cosh^2\lambda \sin\mu \sinh\lambda \sinh\mu \cos\lambda}{PR}
 +\frac{4\pi \tau_y \cos\mu \cosh\lambda \cosh\mu \sin\lambda \sinh\lambda \cos\lambda}{f\rho_0 DPR}\,.
\end{align*}
\end{document}

附录，受到@Thev 的后续评论的启发：一旦证明了 Mathematica 的大喇叭公式可以显示为 10 个\frac表达式（加上一个单独u_g项）的总和，就可以（应该？？）寻找进一步的方法使公式更易于理解。例如，可以注意到 10 个\frac表达式中有 5 个是 的倍数2u_g，而其他 5 个是 的倍数\frac{2\pi\tau_y}{f\rho_0 D}。还可以对分子进行更多组织；例如，可以强制按\lambda-项在\mu-项之前在\nu-项之前进行排序，同时按\cos、\cos^2、\cosh、\sin、\sin^2、进行二次\sinh排序。收集这些想法，并根据@Thev 的建议增加行距，最终可能会得到以下结果（屏幕截图中的水平线表示文本块的宽度）：

在此处输入图片描述

%% (compile with the same preamble as above)
\begin{align*}
u_g+2u_g \smash{\biggl\{}
&{-}\frac{\cos^2\lambda \cosh\lambda \sin\lambda \sin\mu \sinh\mu}{PR} 
 +\frac{2\pi h \cos\lambda \cosh\lambda \sin\mu \sinh\mu}{DPR/Q}
 -\frac{\cos\lambda \cosh\lambda \cos\mu \cosh\mu}{P}\\[0.75ex]
&\quad+\frac{\cos\lambda \sin^2\lambda \sinh\lambda \cos\mu \cosh\mu}{PR} 
 -\frac{\cos\lambda \cosh^2\lambda \sinh\lambda \sin\mu \sinh\mu}{PR}
 \smash{\biggr\}} \\[1.5ex]
{}+\frac{2\pi\tau_y}{f\rho_0 D} \smash{\biggl\{}
&{-}\frac{2 \cos^2\lambda \cosh^2\lambda \sin\mu \sinh\mu}{PR} 
 +\frac{2\pi \cos\lambda \cosh\lambda \sin\mu \sinh\mu}{PR/Q} 
 +\frac{\cos\lambda \cosh\lambda \cos\nu \sinh\nu}{P}\\[0.75ex]
&\quad-\frac{\cos\lambda \cosh\lambda \cosh\nu \sin\nu}{P}
 +\frac{2 \cos\lambda \cosh\lambda \sin\lambda \sinh\lambda \cos\mu \cosh\mu}{PR}
 \smash{\biggr\}}\,.
\end{align*}

我毫不怀疑可以进行进一步的调整...

答案2

在此处输入图片描述

让 tex 做一些内联替换和内联分数。

\documentclass{article}

\begin{document}
\begin{flushleft}

$\displaystyle
\alpha=\frac{h \pi }{D},
\beta=\frac{\pi z}{D}
\gamma=\frac{2 h \pi }{D}
$

\def\za{h \pi}
\def\zb{D}
\def\zc{\pi z}
\def\zd{2 h \pi }

In

$\displaystyle
\let\left\relax
\let\right\relax
\def\frac#1#2{%
\def\zz{#1}\def\zzz{#2}%
\ifx\zzz\zb
  \ifx\zz\za
     \alpha
   \else
  \ifx\zz\zc
      \beta
     \else
  \ifx\zz\zd
      \gamma
     \else
    (#1)/D
   \fi
   \fi
    \fi
\else
\penalty-1000(#1)/(#2)%
\fi}
 -\frac{2 u_g \cosh \left(\frac{h \pi }{D}\right) \sin
  \left(\frac{h \pi }{D}\right) \sin \left(\frac{\pi z}{D}\right)
  \sinh \left(\frac{\pi z}{D}\right) \cos ^2\left(\frac{h \pi
    }{D}\right)}{\left(\cos \left(\frac{2 h \pi }{D}\right)+\cosh
    \left(\frac{2 h \pi }{D}\right)\right) \left(\cos ^2\left(\frac{h
        \pi }{D}\right) \cosh ^2\left(\frac{h \pi }{D}\right)+\sin
    ^2\left(\frac{h \pi }{D}\right) \sinh ^2\left(\frac{h \pi
      }{D}\right)\right) \left(\frac{\cosh \left(\frac{h \pi
        }{D}\right) \sinh \left(\frac{h \pi }{D}\right)}{\cos
      ^2\left(\frac{h \pi }{D}\right) \cosh ^2\left(\frac{h \pi
        }{D}\right)+\sin ^2\left(\frac{h \pi }{D}\right) \sinh
      ^2\left(\frac{h \pi }{D}\right)}-\frac{\cos \left(\frac{h \pi
        }{D}\right) \sin \left(\frac{h \pi }{D}\right)}{\cos
      ^2\left(\frac{h \pi }{D}\right) \cosh ^2\left(\frac{h \pi
        }{D}\right)+\sin ^2\left(\frac{h \pi }{D}\right) \sinh
      ^2\left(\frac{h \pi }{D}\right)}\right)}-\frac{4 \pi \tau_y
  \cosh ^2\left(\frac{h \pi }{D}\right) \sin \left(\frac{\pi
      z}{D}\right) \sinh \left(\frac{\pi z}{D}\right) \cos
  ^2\left(\frac{h \pi }{D}\right)}{D f \rho_0 \left(\cos \left(\frac{2
        h \pi }{D}\right)+\cosh \left(\frac{2 h \pi }{D}\right)\right)
  \left(\cos ^2\left(\frac{h \pi }{D}\right) \cosh ^2\left(\frac{h \pi
      }{D}\right)+\sin ^2\left(\frac{h \pi }{D}\right) \sinh
    ^2\left(\frac{h \pi }{D}\right)\right) \left(\frac{\cosh
      \left(\frac{h \pi }{D}\right) \sinh \left(\frac{h \pi
        }{D}\right)}{\cos ^2\left(\frac{h \pi }{D}\right) \cosh
      ^2\left(\frac{h \pi }{D}\right)+\sin ^2\left(\frac{h \pi
        }{D}\right) \sinh ^2\left(\frac{h \pi }{D}\right)}-\frac{\cos
      \left(\frac{h \pi }{D}\right) \sin \left(\frac{h \pi
        }{D}\right)}{\cos ^2\left(\frac{h \pi }{D}\right) \cosh
      ^2\left(\frac{h \pi }{D}\right)+\sin ^2\left(\frac{h \pi
        }{D}\right) \sinh ^2\left(\frac{h \pi
        }{D}\right)}\right)}+\frac{4 h \pi u_g \cosh \left(\frac{h \pi
    }{D}\right) \sin \left(\frac{\pi z}{D}\right) \sinh
  \left(\frac{\pi z}{D}\right) \cos \left(\frac{h \pi }{D}\right)}{D
  \left(\cos \left(\frac{2 h \pi }{D}\right)+\cosh \left(\frac{2 h \pi
      }{D}\right)\right) \left(\frac{\cosh \left(\frac{h \pi
        }{D}\right) \sinh \left(\frac{h \pi }{D}\right)}{\cos
      ^2\left(\frac{h \pi }{D}\right) \cosh ^2\left(\frac{h \pi
        }{D}\right)+\sin ^2\left(\frac{h \pi }{D}\right) \sinh
      ^2\left(\frac{h \pi }{D}\right)}-\frac{\cos \left(\frac{h \pi
        }{D}\right) \sin \left(\frac{h \pi }{D}\right)}{\cos
      ^2\left(\frac{h \pi }{D}\right) \cosh ^2\left(\frac{h \pi
        }{D}\right)+\sin ^2\left(\frac{h \pi }{D}\right) \sinh
      ^2\left(\frac{h \pi }{D}\right)}\right)}+\frac{4 \pi \tau_y
  \cosh \left(\frac{h \pi }{D}\right) \sin \left(\frac{\pi
      z}{D}\right) \sinh \left(\frac{\pi z}{D}\right) \cos
  \left(\frac{h \pi }{D}\right)}{D f \rho_0 \left(\cos \left(\frac{2 h
        \pi }{D}\right)+\cosh \left(\frac{2 h \pi }{D}\right)\right)
  \left(\frac{\cosh \left(\frac{h \pi }{D}\right) \sinh \left(\frac{h
          \pi }{D}\right)}{\cos ^2\left(\frac{h \pi }{D}\right) \cosh
      ^2\left(\frac{h \pi }{D}\right)+\sin ^2\left(\frac{h \pi
        }{D}\right) \sinh ^2\left(\frac{h \pi }{D}\right)}-\frac{\cos
      \left(\frac{h \pi }{D}\right) \sin \left(\frac{h \pi
        }{D}\right)}{\cos ^2\left(\frac{h \pi }{D}\right) \cosh
      ^2\left(\frac{h \pi }{D}\right)+\sin ^2\left(\frac{h \pi
        }{D}\right) \sinh ^2\left(\frac{h \pi
        }{D}\right)}\right)}+\frac{2 \pi \tau_y \cos \left(\frac{\pi
      (h+z)}{D}\right) \cosh \left(\frac{h \pi }{D}\right) \sinh
  \left(\frac{\pi (h+z)}{D}\right) \cos \left(\frac{h \pi
    }{D}\right)}{D f \rho_0 \left(\cos \left(\frac{2 h \pi
      }{D}\right)+\cosh \left(\frac{2 h \pi
      }{D}\right)\right)}-\frac{2 u_g \cos \left(\frac{\pi
      z}{D}\right) \cosh \left(\frac{h \pi }{D}\right) \cosh
  \left(\frac{\pi z}{D}\right) \cos \left(\frac{h \pi
    }{D}\right)}{\cos \left(\frac{2 h \pi }{D}\right)+\cosh
  \left(\frac{2 h \pi }{D}\right)}-\frac{2 \pi \tau_y \cosh
  \left(\frac{h \pi }{D}\right) \cosh \left(\frac{\pi (h+z)}{D}\right)
  \sin \left(\frac{\pi (h+z)}{D}\right) \cos \left(\frac{h \pi
    }{D}\right)}{D f \rho_0 \left(\cos \left(\frac{2 h \pi
      }{D}\right)+\cosh \left(\frac{2 h \pi
      }{D}\right)\right)}+\frac{2 u_g \cos \left(\frac{\pi
      z}{D}\right) \cosh \left(\frac{\pi z}{D}\right) \sin
  ^2\left(\frac{h \pi }{D}\right) \sinh \left(\frac{h \pi }{D}\right)
  \cos \left(\frac{h \pi }{D}\right)}{\left(\cos \left(\frac{2 h \pi
      }{D}\right)+\cosh \left(\frac{2 h \pi }{D}\right)\right)
  \left(\cos ^2\left(\frac{h \pi }{D}\right) \cosh ^2\left(\frac{h \pi
      }{D}\right)+\sin ^2\left(\frac{h \pi }{D}\right) \sinh
    ^2\left(\frac{h \pi }{D}\right)\right) \left(\frac{\cosh
      \left(\frac{h \pi }{D}\right) \sinh \left(\frac{h \pi
        }{D}\right)}{\cos ^2\left(\frac{h \pi }{D}\right) \cosh
      ^2\left(\frac{h \pi }{D}\right)+\sin ^2\left(\frac{h \pi
        }{D}\right) \sinh ^2\left(\frac{h \pi }{D}\right)}-\frac{\cos
      \left(\frac{h \pi }{D}\right) \sin \left(\frac{h \pi
        }{D}\right)}{\cos ^2\left(\frac{h \pi }{D}\right) \cosh
      ^2\left(\frac{h \pi }{D}\right)+\sin ^2\left(\frac{h \pi
        }{D}\right) \sinh ^2\left(\frac{h \pi
        }{D}\right)}\right)}-\frac{2 u_g \cosh ^2\left(\frac{h \pi
    }{D}\right) \sin \left(\frac{\pi z}{D}\right) \sinh \left(\frac{h
      \pi }{D}\right) \sinh \left(\frac{\pi z}{D}\right) \cos
  \left(\frac{h \pi }{D}\right)}{\left(\cos \left(\frac{2 h \pi
      }{D}\right)+\cosh \left(\frac{2 h \pi }{D}\right)\right)
  \left(\cos ^2\left(\frac{h \pi }{D}\right) \cosh ^2\left(\frac{h \pi
      }{D}\right)+\sin ^2\left(\frac{h \pi }{D}\right) \sinh
    ^2\left(\frac{h \pi }{D}\right)\right) \left(\frac{\cosh
      \left(\frac{h \pi }{D}\right) \sinh \left(\frac{h \pi
        }{D}\right)}{\cos ^2\left(\frac{h \pi }{D}\right) \cosh
      ^2\left(\frac{h \pi }{D}\right)+\sin ^2\left(\frac{h \pi
        }{D}\right) \sinh ^2\left(\frac{h \pi }{D}\right)}-\frac{\cos
      \left(\frac{h \pi }{D}\right) \sin \left(\frac{h \pi
        }{D}\right)}{\cos ^2\left(\frac{h \pi }{D}\right) \cosh
      ^2\left(\frac{h \pi }{D}\right)+\sin ^2\left(\frac{h \pi
        }{D}\right) \sinh ^2\left(\frac{h \pi
        }{D}\right)}\right)}+\frac{4 \pi \tau_y \cos \left(\frac{\pi
      z}{D}\right) \cosh \left(\frac{h \pi }{D}\right) \cosh
  \left(\frac{\pi z}{D}\right) \sin \left(\frac{h \pi }{D}\right)
  \sinh \left(\frac{h \pi }{D}\right) \cos \left(\frac{h \pi
    }{D}\right)}{D f \rho_0 \left(\cos \left(\frac{2 h \pi
      }{D}\right)+\cosh \left(\frac{2 h \pi }{D}\right)\right)
  \left(\cos ^2\left(\frac{h \pi }{D}\right) \cosh ^2\left(\frac{h \pi
      }{D}\right)+\sin ^2\left(\frac{h \pi }{D}\right) \sinh
    ^2\left(\frac{h \pi }{D}\right)\right) \left(\frac{\cosh
      \left(\frac{h \pi }{D}\right) \sinh \left(\frac{h \pi
        }{D}\right)}{\cos ^2\left(\frac{h \pi }{D}\right) \cosh
      ^2\left(\frac{h \pi }{D}\right)+\sin ^2\left(\frac{h \pi
        }{D}\right) \sinh ^2\left(\frac{h \pi }{D}\right)}-\frac{\cos
      \left(\frac{h \pi }{D}\right) \sin \left(\frac{h \pi
        }{D}\right)}{\cos ^2\left(\frac{h \pi }{D}\right) \cosh
      ^2\left(\frac{h \pi }{D}\right)+\sin ^2\left(\frac{h \pi
        }{D}\right) \sinh ^2\left(\frac{h \pi }{D}\right)}\right)}+u_g
$

\end{flushleft}

\end{document}

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