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As pointed out above --- why is this information important?
If it is really important, then get an electronic wattmeter with a voltage output. Create a circuit that gives you binary 1 output during the startup time. Create an integrating circuit, op amp with feedback capacitor. You need a manual reset of the integrator (short the capacitor to bleed to 0 volts). A gating circuit at the integrator input controlled by the binary 1 signal to gate the wattmeter voltage output to the integrator input. A voltmeter on the integrator output.
When the input gate is off (binary gating signal is 0) and the integrator is reset, then the output voltmeter reads 0. At turn on of the motor the gate closes, and the integrator adds up the incoming watt signal. The voltmeter will gradually increase during the startup time. At the end of the startup time the gate is turned off and the integrator output remains constant at its last value. This is a measure of the watt-seconds of energy used during startup.
Obviously you will need to calibrate the circuit. If you are talking about motors in the 1 HP range, then you might calibrate with a 100 W to 1000 W resistor.
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