\begin{align*} \frac {\partial }{\partial r} \left (r^2 \frac {\partial u}{\partial r} \right ) + \frac {1}{\sin \theta } \frac {\partial }{\partial \theta } \left (\sin \theta \frac {\partial u}{\partial \theta } \right ) + \frac {1}{\sin ^2\theta } \frac {\partial ^2 u}{\partial \phi ^2}=0 \end{align*}
Mathematica ✓
ClearAll["Global`*"];
lap = Laplacian[f[r, theta, phi], {r, theta, phi}, "Spherical"];
sol = AbsoluteTiming[TimeConstrained[DSolve[lap == 0, f[r, theta, phi], {r, theta, phi}], 60*10]];
\[\left \{\left \{f(r,\theta ,\phi )\to \begin {array}{cc} \{ & \begin {array}{cc} \sqrt {2} r^{-\frac {1}{2} \sqrt {4 c_7+1}-\frac {1}{2}} \left (c_1 r^{\sqrt {4 c_7+1}}+c_2\right ) \left (c_4 \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (-\sqrt {4 c_7+1}+2 \sqrt {c_8}+1\right ),\frac {1}{4} \left (\sqrt {4 c_7+1}+2 \sqrt {c_8}+1\right ),\frac {1}{2},\cos ^2(\theta )\right )+c_3 \cos (\theta ) \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (-\sqrt {4 c_7+1}+2 \sqrt {c_8}+3\right ),\frac {1}{4} \left (\sqrt {4 c_7+1}+2 \sqrt {c_8}+3\right ),\frac {3}{2},\cos ^2(\theta )\right ) \text {sgn}(\sin (\theta ))\right ) \left (-\sin ^2(\theta )\right )^{\frac {\sqrt {c_8}}{2}} \left (c_6 \cos \left (\phi \sqrt {c_8}\right )+c_5 \sin \left (\phi \sqrt {c_8}\right )\right ) & -\pi \leq \theta \leq \pi \land 0\leq \phi \leq \pi \\ \text {Indeterminate} & \text {True} \\\end {array} \\\end {array}\right \}\right \}\]
Maple ✓
restart;
PDE := diff(r^2*diff(F(r,theta,phi),r),r)+ 1/sin(theta)*diff(sin(theta)*diff(F(r,theta,phi),theta),theta)+ 1/sin(theta)^2*diff(F(r,theta,phi),phi$2) = 0;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(PDE,F(r,theta,phi),'build')),output='realtime'));
\[F \left (r , \theta , \phi \right ) = \frac {\left (-\sin \left (\theta \right )^{2}\right )^{\frac {\sqrt {\textit {\_c}_{2}}}{2}} \left (\operatorname {hypergeom}\left (\left [\frac {\sqrt {\textit {\_c}_{2}}}{2}+\frac {\sqrt {1+4 \textit {\_c}_{1}}}{4}+\frac {3}{4}, \frac {\sqrt {\textit {\_c}_{2}}}{2}-\frac {\sqrt {1+4 \textit {\_c}_{1}}}{4}+\frac {3}{4}\right ], \left [\frac {3}{2}\right ], \frac {\cos \left (2 \theta \right )}{2}+\frac {1}{2}\right ) \cos \left (\theta \right ) c_{4} +\operatorname {hypergeom}\left (\left [\frac {\sqrt {\textit {\_c}_{2}}}{2}+\frac {\sqrt {1+4 \textit {\_c}_{1}}}{4}+\frac {1}{4}, \frac {\sqrt {\textit {\_c}_{2}}}{2}-\frac {\sqrt {1+4 \textit {\_c}_{1}}}{4}+\frac {1}{4}\right ], \left [\frac {1}{2}\right ], \frac {\cos \left (2 \theta \right )}{2}+\frac {1}{2}\right ) c_{3} \right ) \left (r^{\frac {\sqrt {1+4 \textit {\_c}_{1}}}{2}} c_{1} +r^{-\frac {\sqrt {1+4 \textit {\_c}_{1}}}{2}} c_{2} \right ) \left (c_{5} \sin \left (\sqrt {\textit {\_c}_{2}}\, \phi \right )+c_{6} \cos \left (\sqrt {\textit {\_c}_{2}}\, \phi \right )\right )}{\sqrt {r}}\]
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________