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Tantalum caps are normally very low leakage devices as compared to aluminum electrolytics. For this reason, they are preferred over high-value aluminum electrolytics for RC timing applications. A tantalum cap normally fails to a dead short vs. an open or leakage.
If the current is too low to measure with the normal multimeter low current range, then just use the multimeter in the DC voltage measure mode.
First, connect the cap to the power supply voltage at which you want to test the leakage, to precharge the cap. Then connect the multimeter in series with the power supply and the capacitor. For a typical multimeter input impedance of 10megΩ, the leakage current will be Vm/10meg, where Vm is the voltmeter reading.