4.415   ODE No. 1415

$$\boxed{{\displaystyle \frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}y\left(x\right)=-2\frac{n\cosh \left(x\right)\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)}{\sinh \left(x\right)}-(-a^2+n^2)y\left(x\right)=0}}$$

1. Problem in Latex
2. Mathematica input
3. Maple input

Mathematica: cpu = 0.841107 (sec), leaf count = 273 $$\left\{\{y(x)\to \frac{c_2(-1{)}^{\frac{1}{2}(-2n-1)+1}{\tanh }^2(x{)}^{\frac{1}{4}(-2n-1)+1}{({\tanh }^2(x)-1)}^{\frac{1}{2}(\frac{a+n}{2}+\frac{1}{2}(a+n+1)+\frac{1}{2}(-2n-1)+1)}{}_2{F}_1(\frac{1}{2}(-2n-1)+\frac{a+n}{2}+1,\frac{1}{2}(-2n-1)+\frac{1}{2}(a+n+1)+1;\frac{1}{2}(-2n-1)+2;{\tanh }^2(x))\exp (\frac{1}{2}(n-1)\log (1-{\tanh }^2(x))-n\log (\tanh (x)))}{\sqrt{\tanh (x)}}+\frac{c_1{\tanh }^2(x{)}^{\frac{1}{4}(2n+1)}{({\tanh }^2(x)-1)}^{\frac{1}{2}(\frac{a+n}{2}+\frac{1}{2}(a+n+1)+\frac{1}{2}(-2n-1)+1)}{}_2{F}_1(\frac{a+n}{2},\frac{1}{2}(a+n+1);\frac{1}{2}(2n+1);{\tanh }^2(x))\exp (\frac{1}{2}(n-1)\log (1-{\tanh }^2(x))-n\log (\tanh (x)))}{\sqrt{\tanh (x)}}\}\right\}$$

Maple: cpu = 0.109 (sec), leaf count = 43 $$\{y\left(x\right)=\mathit{\_}\mathit{C}\mathit{1}{(\sinh \left(x\right))}^{-n+\frac{1}{2}}\mathit{L}\mathit{e}\mathit{g}\mathit{e}\mathit{n}\mathit{d}\mathit{r}\mathit{e}\mathit{P}(a-\frac{1}{2},n-\frac{1}{2},\cosh \left(x\right))+\mathit{\_}\mathit{C}\mathit{2}{(\sinh \left(x\right))}^{-n+\frac{1}{2}}\mathit{L}\mathit{e}\mathit{g}\mathit{e}\mathit{n}\mathit{d}\mathit{r}\mathit{e}\mathit{Q}(a-\frac{1}{2},n-\frac{1}{2},\cosh \left(x\right))\}$$