Is there a way to late-initialize a member variable (a class) in C++?

MY suggestion: Use a function:

private: static int calculate_first(int input) {return input*5;}
explicit Second(int input) : first(calculate_first(input)) {}

Base classes will be initialized in the order they're declared in the class inheritance list, and then members will be initialized in the order that they're listed in the class, so the calculation can depend on non-static member-variables and base classes if they have already been initialized.

---

Alternatively:

Default constructor, then reassign:

explicit Second(int input) { first = input*5; }

Dummy value, then reassign:

explicit Second(int input) : first(0) { first = input*5; }

Use boost::optional (or std::optional as of C++17):

boost::optional<First> first;
explicit Second(int input) { first = input*5; }

Use the heap:

std::unique_ptr<First> first;
explicit Second(int input) { first.reset(new First(input*5));}
Second(const Second& r) first(new First(*(r->first))) {}
Second& operator=(const Second& r) {first.reset(new First(*(r->first)));}

Placement new:

This is tricky and not suggested 
and worse in every way than boost::optional
So sample deliberately missing.
But it is an option.

Initialize first in the member initializer list.

It may help to perform your calculations in a helper function and use a forwarding constructor:

class Second {
public:
    Second() : Second(helper_function()) {}

private:
    Second(int calc): first(calc) {}
    static int helper_function() { return ...; }

    First first;
};

You can just do what you said in the comments, or, you can make first a pointer to First and give it memory whenever you like, although i don't recommend this way

This sentence is the core of the problem:

I can't initialize first in the initializer-list with the right value, since the value is obtained after some operation.

You should know that what you want to do here is not perfect programming style in Java, either. Leaving the field with some default value and then assigning it a bit later after some calculations have been done effectively prevents it from being final, and consequently the class from being immutable.

In any case, your goal must be to push those calculations directly into the initialization of the member, using private helper functions (which may be static):

class Second {
private:
    First first;

    static int getInitializationData()
    {
        // complicated calculations go here...
        return result_of_calculations;
    }
public:
    Second() : first(getInitializationData()) {}
};

In my opinion, everything else is just a workaround and will complicate your life in the long run.