This paper investigates the performance of using various non-quadratic adaptive algorithms in the adaptation of a non-linear receiver, coupled with a second-order phase tracking subsystem, for asynchronous DS-CDMA communication system impaired by double-spread multipath channel and Gaussian mixture impulsive noise. These algorithms are the lower order (where the power of the cost function is lower than 2), the least-mean mixed norm (where a mixed-norm function is introduced, which combines the LMS and the LMF functions), and the least mean square-fourth switching (where this algorithm switches between LMS and LMF depending on the value of the error). The non-linear receiver comprises feed-forward filter (FFF), feedback filter (FBF), and an equalizer/second order phase locked loop (PLL). The investigations study the effect of using the proposed algorithms on the performance of the non-linear receiver in terms of the mean-square error (MSE) and bit-error-rate (BER). Computer simulation results indicate that the least-mean mixed proposed receiver's algorithm gives the fastest convergence rate and similar BER performance, in comparison with the NLMS adaptive receiver. Furthermore, extensive computer simulation tests have been carried out to determine the optimum values of the step-size, the power of the cost function, and the adaptation parameter of the proposed algorithms. Results show that the optimum values of the step-size for the lower-order, least-mean square fourth, least-mean mixed norm, and the NLMS algorithms are 5x10-4, 10-6, 5x10-4, and 0.01, respectively. The optimum value of the power of the lower-order algorithm is 1.9 and the optimum value of the adaptation parameter of the least-mean mixed algorithm is 0.9.