Integrating op-amp not understood
Large signal slew rate of LT1001 is about 0,25V/us. All gain will vanish when the frequency (sinusoidal) rises to 1Mhz. You may have knocked your head to the non-ideality of the opamp. 

You can test it. Change R9 to 10 kOhm and C3 to 10 nF. Make the input pulses 100 times longer and see if the result is still the same. If it's ok after the changes, then I guessed right.

There's also another hitch. The time constant of the integrator is only 1us. 1V input causes 1V drift in 1us. You can meet the max output (=about 13,5V) too soon. But you cannot see it if the problem is too small slew rate, which prevents the linear operation.

ADD due the comment:
You changed R9 and C3. You made the integrator 100x slower. 1 V input causes now only 0,01 V change to output in 1 us.

Slew rate:

The opamp can change its output max. 0,25 V/us and you ask only 0,01 V/us, the slew rate 0,25 V/us is enough.

About 1 us time constant: I have no idea what is your final application. At least this opamp cannot handle it if you apply 0,5 or 1 volt input. Maybe 100 mV input would be better.