Check number of invocations within class in case of JUnit testing

I have a code which calculates something, caches is, and if already calculated, then reads from the cache; similar to this:

public class LengthWithCache {
    private java.util.Map<String, Integer> lengthPlusOneCache = new java.util.HashMap<String, Integer>();

    public int getLenghtPlusOne(String string) {
        Integer cachedStringLenghtPlusOne = lengthPlusOneCache.get(string);
        if (cachedStringLenghtPlusOne != null) {
            return cachedStringLenghtPlusOne;
        }
        int stringLenghtPlusOne = determineLengthPlusOne(string);
        lengthPlusOneCache.put(string, new Integer(stringLenghtPlusOne));
        return stringLenghtPlusOne;
    }

    protected int determineLengthPlusOne(String string) {
        return string.length() + 1;
    }
}

I want to test if function determineLengthPlusOne has been called adequate number of times, like this:

public class LengthWithCacheTest {
    @Test
    public void testGetLenghtPlusOne() {
        LengthWithCache lengthWithCache = new LengthWithCache();

        assertEquals(6, lengthWithCache.getLenghtPlusOne("apple"));
        // here check that determineLengthPlusOne has been called once

        assertEquals(6, lengthWithCache.getLenghtPlusOne("apple"));
        // here check that determineLengthPlusOne has not been called
    }
}

Mocking class LengthWithCache does not seem a good option, as I want to test their functions. (According to my understanding we mock the classes used by the tested class, and not the tested class itself.) Which is the most elegant solution for this?

My first idea was to create another class LengthPlusOneDeterminer containing function determineLengthPlusOne, add pass it to function getLenghtPlusOne as parameter, and mock LengthPlusOneDeterminer in case of unit testing, but that seems a bit strange, as it has unnecessary impact on the working code (the real clients of class LengthWithCache).

Basically I am using Mockito, but whatever mock framework (or other solution) is welcome! Thank you!